ILPAC Unit P1: the Gaseous State 1

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UNIT ~ 1] The Gaseous State

11I~~llltrl]nl Pl-A N27913 c Inner London Education Authority 1963

First published 1963 by John Murray (Publishers) Ltd 50 Albemarle Street, London W1X 4BD

Printed and bound in Great Britain by Martin's of Berwick

British Library Cataloguing in Publication Data ILPAC Unit P1: The gaseous state 1. Science 500 Q161.2

ISBN 0 7195 4039 9

ii CONTENTS

PREFACE v Acknowledgements vi Key to ILPAC activity symbols and hazard symbols vii

INTRODUCTION 1

Pre-knowledge 2 PRE-TEST 3

LEVEL ONE

STATES OF MATTER 5 Translation, rotation and vibration 6 THE GAS LAWS 7 Boyle's law and Charles' law 7 The combined gas law 10 Gay-Lussac's law of combining volumes 10 AVOGADRO'S THEORY AND ITS APPLICATIONS 11 Avogadro's theory 11 Using Avogadro's theory to calculate reacting volumes 11 Determining the formula of a gaseous hydrocarbon 13 Molar volume of gases 15 Another method for determining the Avogadro constant 1B THE IDEAL GAS EQUATION 19 Deriving the ideal gas equation 19 The gas constant, R 20 Calculations using the ideal gas equation 21 Determination of molar mass of a gas 23 Experiment 1 - determining the molar mass of a gas 23 Determination of molar mass of a volatile liquid 26 Experiment 2 - determining the molar mass of a volatile liquid 27 LEVEL ONE CHECKLIST 31 LEVEL ONE TEST 33

LEVEL TWO

TWO MORE GAS LAWS 37 Graham's law of effusion 37 Experiment 3 - determining the molar mass of a gas by effusion 3B Dalton's law of partial pressures 41 THE KINETIC THEORY OF GASES 44 A theoretical model of a gas 44 Deriving the gas laws from kinetic theory 46 DEVIATIONS FROM IDEAL BEHAVIOUR 49 Non-ideal behaviour 49 The van der Waals equation 51

iii THE VARIATION OF MOLECULAR SPEEDS 53 The Maxwell distribution 54 Use of the Maxwell distribution curve 55 Measuring molecular speeds - the Zartman experiment 58 The root-mean-square speed of gas molecules 59 LEVEL TWO CHECKLIST 60 END-OF-UNIT TEST 61

APPENDIX ONE THE LIQUEFACTION OF GASES 65 Andrews' isotherms and the critical point 65 Cooling of gases by expansion 66

APPENDIX TWO ADDITIONAL EXERCISES 69

ANSWERS TO EXERCISES 74

iv PREFACE

This volume is one of twenty Units produced by ILPAC, the Independent Learning Project for Advanced Chemistry, written for students preparing for the Advanced Level examinations of the G.C.E. The Project has been sponsored by the Inner London Education Authority and the materials have been extensively tested in London schools and colleges. In its present revised form, however, it is intended for a wider audience; the syllabuses of all the major Examination Boards have been taken into account and questions set by these boards have been included.

Although ILPAC was initially conceived as a way of overcoming some of the difficulties presented by uneconomically small sixth forms, it has frequently been adopted because its approach to learning has certain advantages over more traditional teaching methods. Students assume a greater responsibility for their own learning and can work, to some extent, at their own pace, while teachers can devote more time to guiding individual students and to managing resources.

By providing personal guidance, and detailed solutions to the many exercises, supported by the optional use of video-cassettes, the Project allows students to study A-level chemistry with less teacher-contact time than a conventional course demands. The extent to which this is possible must be determined locally; potentially hazardous practical work must, of course, be supervised. Nevertheless, flexibility in time-tabling makes ILPAC an attractive propo- sition in situations where classes are small or suitably-qualified teachers are scarce.

In addition, ILPAC can provide at least a partial solution to other problems. Students with only limited access to laboratories, for example, those studying at evening classes, can concentrate upon ILPAC practical work in the laboratory, in the confidence that related theory can be systematically studied elsewhere. Teachers of A-level chemistry who are inexperienced, or whose main discipline is another science, will find ILPAC very supportive. The materials can be used effectively where upper and lower sixth form classes are timetabled together. ILPAC can provide 'remedial' material for students in higher education. Schools operating sixth form consortia can benefit from the cohesion that ILPAC can provide in a fragmented situation. The project can be adapted for use in parts of the world where there is a severe shortage of qualified chemistry teachers. And so on.

A more detailed introduction to ILPAC, with specific advice both to students and to teachers, is included in the first volume only. Details of the Project Team and Trial Schools appear inside the back cover.

LONDON 1983

v ACKNOWLEDGEMENTS

Thanks are due to the following examination boards for permission to reproduce questions from past A-level papers:

Oxford Delegacy of Local Examinations End-of-Unit Test 10(1977), 11(1977)

Southern Universities Joint Board Level One Test 9(1976)

The Associated Examining Board Teacher-marked exercise, p6S(1977) Level One Test 6(1975)

University of London Entrance and Schools Examinations Council Exercise 24(L1976) Teacher-marked Exercise, p53(L19S1) Level One Test 1(N1976), 2(L19S0), 3(N197S), 4(N1974), 5(L19S0), S(N197S) End-of-Unit Test 1(N19S0), 4,5&6(N1974), 9(L19S1), 12(L1977)

Where answers to these questions are included, they are provided by ILPAC and not by the examination boards. Questions from papers of other examining boards appear in other Units.

Photographs are included by permission as follows:

Explo8ion, p 22 - Popperfoto Snooker table, p 53 - Colorsport Photographs of students - Tony Langham

vi SYMBOLS USED IN ILPAC UNITS 0 W Reading Revealing exercises

Exercise Discussion ~ ® ~\;lTest ~ Computer programme

~ Experiment 'A' Level question

I[ 11 Video programme 'A' Level part question 00

[Q8 Film loop 'A' Level question Special paper Model-making

Worked example

Teacher-marKed exercise

INTERNATIONAL HAZARD SYMBOLS

Harmful Toxic Radioactive ~ ~ [iJ Flammable II Explosive

~ ,:v J!t Corrosive Oxidising -- ~

vii

INTRO DUCTION

In this Unit. we look first at the properties which distinguish gases from liquids and solids. Then we examine some of the experimental laws which describe the behaviour of gases. The most important of these laws is expressed by what is known as the ideal gas equation:

pV == nRT Only hypothetical 'ideal' gases observe these laws precisely under all conditions, but we often apply them successfully to real gases. Level One ends with the application of the ideal gas equation to methods for determining molar mass.

In Level Two we first consider two more gas laws and then we develop the simple kinetic theory of gases to explain their behaviour. We show how the gas laws can be derived from the theory by making some assumptions about the nature of molecules in gases. We then explain how the approximate truth of these assumptions accounts for the observed deviations from the gas laws.

Finally we consider the measurement and calculation of molecular speeds and the way molecular speeds vary in a sample of gas.

In Appendix One we consider the liquefaction of gases, and in Appendix Two we give some additional numerical exercises.

There are three experiments in this Unit; two are in Level One and one is in Level Two.

There are two ILPAC video-programmes designed to accompany this Unit. They are not essential, but you should try to see them at the appropriate times if they are available.

Volumes of reacting gases The distribution of molecular speeds

1 PRE-KNOWLEDGE

Before you start work on this Unitl you should be able to:

(1) obtain a value of temperature in K from one in °C;

3 (2) obtain a value of volume in dm from one in cm3; (3) obtain a value of pressure in atm from one in mmHg; (4) explain the abbreviation s.t.p; (5) calculate the relative molecular mass and the molar mass of a substance from its chemical formula; (6) calculate the amount of a substanc? from its mass and molar massl using the correct symbols and units; (7) calculate the density of a substance from its mass and volumel using the correct symbols and units;

(8) define the Avogadro constant I L.

PRE-TEST

To find out whether you are ready to start Level Onel try the following testl which is based 011 the pre-knowledge items. You should not spend more than 30 minutes on this test. Hand your answers to your teacher for marking.

2 PRE-TEST

1. Express the following quantities in different units as indicated: (a) 23 DC (in K),

(b) 59.0 cm3 (in dm3), (c) 758 mmHg (in atrn l. (3 )

2. Explain the terms: (a) s.t.p., (b) the Avogadro constant. (4 )

3. (a) Calculate the relative molecular mass of chloroethane, C2H5Cl. (1) (b) 25.0 cm3 of liquid chloroethane weighs 22.5 g. What amount of chloroethane is this? Use correct symbols and units. (2) (c) What is the density, p, of chloroethane? Use correct symbols and units.

(C = 12.0, H = 1.0, Cl = 35.5) (2) (Total 12 marks)

LEVEL ONE STATES OF MATTER

Gases have properties which enable us to distinguish them easily from solids and liquids, and these distinctive properties can be simply explained in terms of the spacing and movement of molecules.

Objectives. When you have finished this section, you should be able to:

(1) state the properties which distinguish gases from liquids and solids; (2) explain these distinctive properties in terms of the spacing and movement of molecules.

You have already studied the differences between solids, liquids and gases in your pre-A-level course, but in order to make the distinctions clear*, you should read the appropriate section in your text- book(sJ which describes the differences. You will probably find this at the beginning of a chaper on gases, perhaps under the heading "States of Matter"'.

The headings in Table 1 will help to guide your reading - look particularly for explanations of behaviour in terms of simple kinetic theory. However, do not dwell on any mathematical treatment of the theory - this will come later. (This is what we mean by taking a 'qualitative' approach - we usually leave the 'quantitative' aspects of a topic, involving some mathematics and numerical problems, till later in the Unit.J

*As with other classification systems, the distinctions may become blurred, e.g. liquid crystals have some of the features of both liquids and solids. Also, other states of matter, e.g. plasma, have been defined, but the simple classification of matter into solids, liquids and gases is adequate for A-level work.

5 Exercise 1 Illustrate the essential difference between solids, liquids and gases by completing Table 1, using as few words as possible. (Answer on page 74 )

Table 1

Solids Liqu ids Gases

Volume (fixed or variable)

Shape (fixed or variable)

Compressibility (relative)

Density (relative)

Spacing of molecules

Strength of attractive forces

Brief description of molecular motion

We now look more closely at the motion of molecules in gases and distinguish three different types of motion.

Translation~ rotation and vibration

When we consider the motion of a gas molecule, we think mainly about movement from place to place, which is known as translation.

Rotation occurs when a molecule spins about an axis through its centre, like a top. Rotation is independent of translation, i.e. one can occur with or without the other.

Vibration only occurs in a molecule if it has more than one atom. Vibration in a diatomic molecule is shown in Fig. 1. The distance between two atoms changes rapidly about an average distance, which is the one we can measure. Molecules consisting of three or more atoms can have several different modes of vibration, as shown in Fig. 2. Vibration is independent of translation and rotation.

6 --•------0

rotation vibration

Fig.1. Rotation and vibration of a diatomic molecule

0 • 0 t -- 0 • 0 0 • 0 ~ ~ stretching vibrations bending vibration

Fig.2. Different modes of vibration of a triatomic molecule

In general, molecules in a gas have all three types of motion and, therefore, three types of kinetic energy. However, nearly all the kinetic energy in a gas is translational, and this is the only type we consider in this Unit. The others are important in infra-red and microwave spectroscopy, which you may study later.

Having compared solids, liquids and gases, we now look at the behaviour of gases. Because the molecules of a gas are so far apart, the laws which describe their behaviour are remarkably simple.

THE GAS LAWS

We look in turn at several laws which describe quantitatively the behaviour of gases. These laws were discovered empirically (i.e. by studying the results of experiments) but were later shown to have the same theoretical basis.

Objectives. When you have finished this section, you should be able to:

(3) state Boyle's law and Charles' law; (4) state the combined gas law and use it to calculate the effect of changing pressure, temperature or volume of a sample of gas.

Boyle's law and Charles' law

These laws were formulated by measuring how the volume of a sample of <=) gas varies with temperature and pressure. You have probably studied ~ these laws in your pre-A-Ievel course and will therefore be able to do ~ Exercises 2 and 3. If not, you should read about them in your text- book.

7 Exercise 2 (a) Write down Boyle's law and Charles' law both in words and mathematically. (bJ Which of the graphs below illustrate the laws?

M M E E M "0 o <, <, <,E Q.) Q.) Q.) E E E :J :J :J (5 (5 (5 > > > /~ // / // /' L/ / ./ pressure/atm temperaturaz 'C pressu rei atm

M M E M E o E "0 <, <, <, Q.) Q.) Q.) E E E :J :J :J "0 / / (iv) "0 (5 / (vi) -: / > / > > -: -: / / .,/ /

temperature/K temperaturezC temperature/ K

Fig.3

(Answers on page 74 J

Exercise 3 (a) A meteorological balloon is to be filled with helium at ~ atmospheric pressure. What will the volume of the balloon be if it is to hold all the gas from a 25.0 dm3 ~~ gas cylinder at 150 atm? Assume constant temperature.

(bJ The balloon will burst if its volume exceeds 4000 dm3• If the filling temperature is 15 °C, what is the maximum temperature the balloon can stand at atmospheric pressure? (Answers on page 74 J

(More problems on Boyle's law and Charles· law for practice and/or revision are in Appendix 2A on page 69 • )

If you really were launching a meteorological balloon, you would want to know hqw its volume would change when both temperature and pressure vary, as they do when the balloon rises. You can do this by combining Boyle·s law and Chrles. law, as we show in the following Worked Example.

Worked Example A meteorological balloon has a volume of 6.15 m3 when filled with helium at 14 °c and 762 mmHg. What will its volume be if the temperature rises to 19 °c and the pressure falls to 749 mmHg.

8 Solution 1. Calculate the effect on the volume of gas dUB to prBssurB changB alonB. using BoylB's law:

3 762 mmHg 6.15 m x 749 mmHg

NoticB that bBcausB a prBssurB dBcrBasB givBs a volumB increasB. we multiply thB volumB by a factor grBatBr than one. i.B. 762/749. 2. Calculate the BffBct on this nBW volumB of gas dUB to a tBmpBraturB changB alone. using Charles' law. V V constant or 2 ~ T T2 T3 !3.. V3 V2 X T wherB V2 is thB volumB calculatBd in stBP 1 . 1.

3 762 (273 + 19)K (6.15 749) x V3 cm x (273 + 14)K Notice that bBcausB a tBmpBraturB incrBasB givBs a volumB incrBasB WB again multiply thB volume by a factor grBatBr than onB. i.B. 292/287.

The final volumB. V3• is therBforB given by

V3 = 6.15 m3 x ;~~ x ~~~ =16.37 m31

Now try some exercisBs. You will soon find that. with a little practicB. you can do both StBps in onB. New volume = original volume x pressure factor x tBmperaturB factor

Hint: always check whether the factors should bB greatBr or IBss than onB.

EXBrcise 4 What is thB" volume at s.t.p. of a samplB of gas which occupies 3B.2 cm3 at 18 °c and 765 mmHg? (Answer on page 74 )

Exercise 5 79.0 cm3 of hydrogen were. collected from a reaction between ~ zinc and sulphuric acid. at 21 °c and 756 mmHg. What volume would the gas occupy at s.t.p.? ~\\\ (Answer on page 74 )

The combination of Boyle's law and Charles' law which you have been using is often quoted sBparately as the combined gas law.

9 The combined gas law

In the previous exercises you applied Boyle's law and Charles' law to problems where p~ V and T all vary. The final relationship in the worked example:

V = V X Pl X ~ 2 1 P2 Tl

is a mathematical statement of the combined gas law, more commonly stated in the form:

You will use the combined gas law most often for calculating volume changes, but it can of course be used to calculate pressure or temperature changes in a very similar way.

More problems on the combined gas law are in Appendix 2B on page 69 .

The last of the gas laws in this section refers to volume relationships in reactions between gases.

Gay-Lussac's law of combining volumes

Gay-Lussac studied a great many reactions involving gases, and discovered that the volumes of all reactants and products were simply related.

Objective. When you have finished this section, you should be able to:

(5) state Gay-Lussac's law of combining volumes, and give some examples.

Read about Gay-Lussac's law in your text-book, and look at some specific examples so that you can do the next exercise.

If it is available, view the ILPAC videotape 'Volumes of reacting gases'. Make sure you have the worksheet which goes with the videotape.

Exercise 6 ( a) State Gay-Lussac's law.

( b) 100 cm3 of ammonia decomposes to give 50 cm3 of nitrogen and 150 cm3 of hydrogen. Show how these experimental facts illustrate Gay-Lussac's law, stating any assumptions you make. Can the equation for the reaction be deduced from this information alone? (Answers on page 74 )

Having looKed at some empirical laws, we now consider a very important theory which was developed to help explain them, particularly Gay-Lussac's law.

10 AVOGADRO'S THEORY AND ITS APPLICATIONS

This theory not only explains Gay-Lussac's law, but has many other useful applications, as we will show.

Avogadro's theory

Objectives. When you have finished this section, you should be able to:

(6) state Avogadro's theory; (7) describe how Avogadro's theory was used to establish simple chemical formulae.

Read AaboutdAV~gadlro)'s.theOry t(somtetbimeksclalkl~dAVOg~~ro~s ~yp~the~~s or GJc=2, even voga ro s aw ln your ex - 00, 00 lng par lCU ar y or e way it explained Gay-Lussac's law and helped to establish simple chemical formulae. Then you should be able to do the next exercise.

Exercise 7 One example of a gas reaction which observes Gay-Lussac's law is described by the following word-equation.

hydrogen + chlorine -+ hydrogen chloride (1 volume) (1 volume) (2 volumes) (a) Use Avogadro's theory to show that molecules of hydrogen and chlorine must contain even numbers of atoms. (b) Assume that both hydrogen and chlorine are diatomic. What then must be the formula of hydrogen chloride? (c) Suppose that hydrogen was shown to have a formula H4, and chlorine C16. What then would be the formula of hydrogen chloride? Cd) What evidence is there that hydrogen is H2 and not H4 or H6 etc? (Hint: consider the variety of compounds that could be formed from hydrogen molecules and from hydrogen chloride.) (Answers on page 75 )

You have seen how Avogadro's theory was used initially to help establish chemical formulae from measurements of reacting volumes of gases. However, we now often use the theory the other way round, i.e. to calculate reacting volumes of gases of known formulae.

Using Avogadro's theory to calculate reacting volumes

Objective. When you have finished this section you shOUld be able to:

(8) use Avogadro's theory to calculate reacting volumes of gases.

We show the procedure for this type of calculation in a worked example.

11 Worked example What volume of oxygen would be required to burn 3 completely 20 cm of ethane, C2H6, and what volume of carbon dioxide would be produced? (Volumes measured at the same room temperature and pressure.)

Solut ion 1.. Write the equation for the reaction:

C2H6(g) + 3!02(g) ~ 2C02(g) + 3H20(1) 2. Write the amounts of each gas under the equation:

C2H6(gJ + 3!02(gJ ~ 2C02(gJ + 3H20(lJ 1 mol 3! mol 2 mol 3. Write under the amounts the relative volumes of each gas, by applying Avogadro's theory. Since equal volumes contain equal amounts, it follows that equal amounts occupy equal volumes. ~ C2H6(gJ + 3!02(gJ 2C02(g) + 3H2OCI) 1 mol 3! mol 2 mol 1 volume 3! volumes 2 volumes V 3! x V 2 x V 4. Use the volume given in the question to calculate the other volumes. Volume of ethane V = 20 cm3 Volume of oxygen 3! x V = 3! x 20 cm3 =170 cm31 3 3 Volume of carbon dioxide = 2 x V = 2 x 20 cm =140 cm 1

Now try the following exercise in the same way.

Exercise 8 In each of the following reactions the volume of some of ~ the gases involved is given. Calculate the volumes of the ~ gases which react and are produced. All gas volumes are measured at the same temperature and pressure. (a) 2N20(g) ~ 2N2(g) + 02(g) 15 ern"

(Answers on page 75 J

More problems on calculating volumes, for practice and for revision, are in Appendix 2C on page 70 .

Another application of Avogadro's theory is in a method for finding the formula of a gaseous hydrocarbon by combustion.

12 Determining the formula of a gaseous hydrocarbon

We can easily measure the volumes of gases involved in the combustion of a hydrocarbon. From these volumes, and the equation for the reaction, we can obtain the formula of the hydrocarbon.

Objectives. When you have finished this section, you should be able to:

(9) outline a method for measuring reacting volumes of gases in the combustion of a hydrocarbon; (10) calculate the formula of a gaseous hydrocarbon from reacting volumes of gases involved in its combustion.

Read about measuring reacting volumes in the combustion of hydrocarbons in your text-book. You do not need a detailed experimental method - an outline of the steps involved in sufficient. You are asked about these steps in the next exercise. We deal with the calculation in a worked example after the exercise.

Exercise 9 (a) Why is the hydrocarbon mixed with an excess of oxygen rather than just the amount required for combustion Ctwo reasons) ? (b) Why is the gaseous mixture after combustion treated with a concentrated alkali? (c) All the volume measurements must be made at the same temperature. What difference does it make to the use of the measurements if the temperature is much higher, say 150 oCr Cd) What advantage might there be in working at a high temperature? (Answers on page 75 )

A worked example shows you how to work out a formula from the results of a combustion experiment. Then you can try some problems for yourself.

Worked Example 10 cm3 of a gaseous hydrocarbon at room temperature was mixed with 100 cm3 of oxygen (an excess). After sparking the mixture and allowing it to cool to its original temperature the total volume was found to be 95 cm3• This contracted to 75 cm3 when in contact with a concentrated solution of sodium hydroxide. What is the formula of the hydrocarbon?

Solution: 1. Write an equation for the reaction in words: hydrocarbon(g) + oxygen(g) ~ carbon dioxide(g) + water(l) 2. Calculate the volume of carbon dioxide produced. This is equal to the contraction in volume as the sodium hydroxide absorbs carbon dioxide .

3 3 3 ... volume of CO2 formed = 95 cm - 75 cm = 20 cm

13 3. Calculate the volume of water vapour (if anyJ produced. In this example, the volumes are measured at room temperature. Thus the water is liquid and has negligible volume. (In other problems, you may have to consider the volume of water vapour.J 4. Calculate the volume of oxygen used up in the reaction. The final volume, 75 cm3, must be unused oxygen . ... volume of oxygen used

5. Write down the volume of each gas under the word-equation, and divide each by the smallest volume to obtain the relative volumes. hydrocarbon(gJ + oxygen(gJ ~ carbon dioxide(gJ + water(lJ 10 cm3 25 cm3 20 cm3 1 volume 2~ volumes 2 volumes

6. Apply Avogadro's theory to write down the relative amount of each gas. You may find it easier, though not strictly necessary, to convert these to whole numbers. hydrocarbon(gJ + oxygen(gJ carbon dioxide(gJ + water(lJ 10 cm3 25 cm3 20 cm3 1 volume 2~ volumes 2 volumes 1 mol 2~ mol 2 mol ?mol or 2 mol 5 mol 4 mol 7. Call the hydrocar~on CxHy, and the amount of water z mol. Write a full equation in the form: 2C H (g) + 50 (g) ~ 4C0 (g) + zH 0(1) x y 2 2 2 B. Calculate x, y and z by noting that the equation must balance, i.e. the same number of atoms of each sort must appear on each side of the equation. It is convenient to do this, one line at a time, in tabular form.

Table 2

Left-hand side Right-hand side of equation of equation Calculation

C atoms 2x 4 2x = 4 x = 2 . atoms 2 x 5 = 10 (2 x 4) + z = 8 + z 10 = 8 + z z = 2 ° . H atoms 2y 22 2y = 22 . Y = 2

(Always consider the atoms in this order - C, 0, H, - it makes the calculation simpler. )

The formula of the hydrocarbon is therefore C2H2, and the equation is: 2C2H2(g) + 502(g) ~ 4C02(g) + 2H20(I)

Note that we could equally well have taken the ratio of volumes as 1 :2~:2, and arrived at the equivalent equation:

C2H2(g) + 2~02(g) ~ 2C02(g) + H20(I)

Now you should try some similar problems yourself.

14 Exercise 10 When 15 cm3 of a gaseous hydrocarbon was exploded with 60 cm3 of oxygen (an excess), the final volume was 45 cm3• This decreased to 15 cm3 on treatment with sodium hydroxide solution. What was the formula of the hydrocarbon? All measurements were made at the same room temperature and pressure. (Answer on page 75 )

The next exercise is more difficult because the information given is different. Also you cannot this time ignore the volume of water! However, the same general method should be used. If you cannot do it without help, look for clues in the answer page and then try again.

Exercise 11 When 10 cms of a gaseous hydrocarbon CsH were exploded ~ with an excess of oxygen at 105 °c the v~lume measured ~ under the same physical conditions expanded by 5.00 cm3• What is the value of x, and what volume of oxygen was used under these conditions? (Answer on page 76 )

More problems on the formulae of gaseous hydrocarbons, for practice and for revision, are in Appendix 20 on page 70.

In the next section we show how Avogadro's theory leads to an important and useful statement about the volume of one mole of gas.

Molar volume of gases

The converse of Avogadro's theory (i.e. putting it the other way round) is that if we have samples of different gases at the same temperature and pressure and containing the same number of molecules, then they must occupy equal volumes.

Objectives. When you have finished this section, you should be able to:

(11) explain the term molar volume of a gas; (12) calculate the molar volume of a gas from measurements of mass and volume; (13) use the molar volume of a gas at s.t.p. in calculations of amount.

Read about molar volume in your text-book, looking particularly for its o connection with Avogadro's theory. OJ

If the fixed number of molecules referred to above is the number in one mole, then the corresponding fixed volume is known as the molar volume, Vm. Molar volume, like other volumes, varies with temperature and pressure but its value at s.t.p. is most often used.

15 An expression which is very useful in calculations involving molar volume is:

amount of gas volume of gas or E1Jn = - molar volume Vm

where both volumes refer to the same conditions of temperature and pressure. This expression is very similar to the one you have already used often: mass amount of substance or molar mass and is easily derived from it by dividing each mass by the appropriate density,

n - m/p v - M/p V m

We use this expression in the following worked example which shows you how to calculate molar volume from experimental data for a gas of known formula.

Worked Example Calculate the molar volume at s.t.p. for carbon dioxide I I 3 w given that 2.50 g occupies 0.450 dm at 3.00 atm and 16 °C.

Solution

1. Calculate the volume at s.t.p. as in previous examples using the combined gas law.

p~V~ P2V2 ~ T2 Pl T2 or V2 V1 x X P2 Tl 3.00 atm 273 K 0.450 dm3 x x 1 .28 dm3 1 .00 atm (273 + 16)K

2. Calculate the amount, n, of CO2 using the expression: m n M 2.50 g n 0.0568 mol 44.0 g moll 3. Calculate the molar volume, Vm, using the expression n V Vm V 1 .28 dm3 or 22 5 dm3 mol-1 Vm n 0.0568 mol 1 . I

Now you should be able to calculate some molar volumes for yourself.

16 Exercise 12 Give the following experimental results, calculate the molar volume of each gas at s.t.p. (a) 0.122 g of hydrogen, H2(g), occupies 0.211 dm3 at 7.00 atm and 20.0 °c. (b) 1.10 g of butane, C4H~0(g), occupies 34.4 dm3 at 600°C and 30.0 mmHg. (1 atm = 760 mmHg.) (c) 2.00 g of oxygen, 02(g), at 5.00 kPa* and 500 K occupies 3 51.9 dm • (1.00 atm = 101 kPa.) (Answers on page 76 )

*The pascal, symbol Pa, is the official SI unit for pressure and is equivalent to a force of one newton acting over an area of one square metre. i.e. 1 Pa = 1 N m-2 The pascal is a very small unit of pressure, so the kilopascal, symbol kPa, is more often used. However chemists still usually use the units atmospheres and millimetres of mercury.

You will nave noticed that the answers to the exercise are not equal. Even with the most accurate experimental data, slight differences in molar volumes occur. This is because the gases do not observe the gas laws precisely - only a hypothetical ideal gas does this.

However, we very often assume that gases are ideal, and therefore assume that the molar volume of any gas is a constant. The value we use at s.t.p. is 22.4 dm3 mol-1, a figure well worth remembering even though it is usually given in examinations when required.

! Vm (s.t .p.) = 22.4 mol dm-3!

The molar volume is often used to calculate amounts of gas from volume measurements. In the following exercises, you can use almost the same method as in the last worked example.

Exercise 13 Calculate the amount of gas contained in a 10.0 dm3 globe at 27 °c and 350 mmHg. (Answer on page 76

Exercise 14 What is the volume of 8.00 g of oxygen at 23 °c and 0.200 atm? (Answer on page 77

We do not always take s.t.p. as our reference conditions. You may well have met the statement that the molar volume of any gas at 'room temperature and pressure' is 24.0 dm3 mol-1• The next exercise refers to this value.

17 Exercise 15 If the pressure is 1.00 atm, at what temperature is the molar volume of a gas 24.0 dm~ mol-1? (Answer on page 77 )

More problems on molar volumes, for practice and for revision, are in Appendix 2E on page 70 .

We now turn to an application of molar volume in calculating the Avogadro constant.

Another method for determining the Avogadro constant

In Unit S1, you used the monomolecular layer experiment to obtain an approximate value for L, the Avogadro constant. There are many independent methods for determining L, and we now outline one of the more accurate methods. The molar volume of a gas is used in conjunction with measurements of radioactivity.

Objective. When you have finished this section, you should be able to:

(14) use the molar volume of a gas to obtain Avogadro's constant from the results of radioactivity experiments.

In the following exercise, you need only apply what you have already learned about the Avogadro constant (Unit S1), radioactivity (Unit S2) and molar volume.

Exercise 16 A sample of radium chloride was kept in a sealed tube and~ was found to produce helium gas at the rate of 4.13 x 10-4 cm3 in one hour. Another sample, of the same mass ~~ was found to emit a-particles at a rate of 3.07 x 1012 S-l.

(a) How was the helium formed? (b) Calculate a value for the Avogadro constant. (Answers on page 77 )

You should now be able to see the great value of Avogadro's theory in a variety of useful calculations.

Next we consider a very important relationship, known as the ideal gas equation, which embodies both Avogadro's theory and the combined gas law.

18 THE IDEAL GAS EQUATION

The ideal gas equation is usually in the form:

pV == nRT where R, the gas constant, has the same value for all gases. In the following section we show how the ideal gas equation is related to the combined gas law and Avogadro's theory. Then we show you how to use it in a number of calculations, particularly in determining the molar mass of a gas.

Objectives. When you have finished this section, you should be able to:

(15) state the ideal gas equation; (16) recognise that the gas constant, R, has different numerical values according to the units chosen for pressure and volume; (17) use the ideal gas equation in simple calculations.

You should read about the ideal gas equation in your text-book, looking for its relationship to the combined gas law and Avogadro's theory, and also the use of the equation in calculations. You may find an explanation of why real gases do not observe the ideal gas equation precisely, but we will deal with this in Level Two.

For the moment, we assume ideal behaviour in real gases.

Deriving the ideal gas equation

The ideal gas equation follows from the combined gas law and Avogadro's theory, as we now show. The combined gas law is written

or constant

For one mole of gas, the volume is written as Vm, while the constant is written as R, and is known as the gas constant. pV m i.e. R T We have already used Avogadro's theory to show that the molar volume, Vm, is the same for all gases under the same conditions. It follows, therefore, that R is the same for all gases. Rearranging the expression, we have

pVm RT This applies for one mole of gas, but for n mol it becom8s:

pV nRT

19 The gas constantJ R

The gas constant~ R, is a proportionality constant relating p, V, nand T. Its numerical value depends on the units chosen for the variables. T is always expressed in kelvin (K) and n in mol, but a variety of units are in use for p and V. Chemists usually express p in atmospheres (atm) and V in dm3 - use these units in the next exercise.

3 Exercise 17 Use the fact that one mole of ideal gas occupies 22.4 dm ~ at s.t.p. to dalculate a value for R, the gas constant. (Answer on page 77 )

In the next exercise, you calculate some other numerical values for R, using a variety of units. This involves using a conversion factor, as we show in a Worked Example.

3 1 1 Worked Example R is given as 0.0821 atm dm K- mol- • What value I ~ I should be used if the volume of gas were given in m3?

Solution 1. Rearrange the expression for R to isolate the unit to be changed.

3 R = 0.0821 atm (dm ) K-~ mol-~

2. Make the appropriate substitution, given that 1.00 m3 1.00 x 103 dm3

Dividing by 103, 1.00 X 10-3 m3 1.00dm3

3 3 R = 0.0821 atm (1.00 x 10- m ) K-~ mol-~

=J 8.21 x 10-5 atm m3 K-1 mol-1 I

Alternatively, the substitution can be made during the calculation using the ideal gas equation. pV nET

1.00 atm x 22.4 x 1 .00 X 10-3 m3 R E! nT 1 .00 mol x 273 K

Note that when a quantity is expressed in a combination of units, those with positive indices are usually written first, in alphabetical order, followed by those with negative indices, also in alphabetical order. However, this is not a hard-and-fast rule.

20 Exercise 18 Obtain two more values of R, in different units. (a) Substitute 1 .00 atm 760 mmHg in the expression R 0.0821 atm dm3 K-1 mol-1 (b) Repeat the calculation of R as in Exercise 17, using the following conversion factors. 1.000 atm = 1.013 kPa = 101.3 kN m-2 1 .00 N m = 1 .00 J (Answers on page 77 J

The most widely used values for Rare 0.0821 atm dm3 K-1 mol-1 and 8.314 J mol-1 K-1• It is most important in calculations that the value for R is consistent with the units of the data you are using. You must choose a value for R in the exercises which follow, but first we give a worked example of a typical calculation involving the ideal gas equation.

Calculations using the ideal gas equation

Worked example Assuming ideal behaviour, calculate the volume occupied by 2.00 g of carbon monoxide at 20 °c 2 under a pressure of 6250 N m- •

Solution 1. Calculate the molar mass of carbon monoxide: M = (12.0 + 16.0) g mol-1 = 28.0 g mol-1 2. Identify the values to be substituted in the ideal gas equation, pV nRT.

2 P 6250 N m- n 2.00 g / 28.0 g mol-l. 0.0714 mol T (273 + 20)K 293 K R 8.314 J K-1 mol-l. (This value for R is appropriate because the pressure is in N m-2 , and 1 J = 1 N m) 3. Substitute the values in the equation V = nRT pV nRT in the form p 0.0714 mol x 8.314 J K-l. mol-l. x 293 K v 6250 N m-2 0.0714 x 8.314 J x 293 6250 N m 2 but 1.00 J 1 .00 N m

V = 0.0714 x 8.314 N m x 293 =\0.0278 m3 or 27.8 dm3 I 6250 N m-2 Note that the units used in the calculation determine the unit of the 3 answer. It would be possible to do the calculation using R = 0.0821 atm dm K-1 mol-1 but the answer would be expressed in a very peculiar unit of volume! (atm dm5 N-1)

21 The next two exercises are straightforward applications of the ideal gas equation.

Exercise 19 (a) What is the volume of 0.500 mol of sulphur dioxide, S02' at s.t.p?

(b) What is the volume of 1.50 g of hydrogen, H2, at 15 °c and a pressure of 750 mmHg? (c) At what temperature will 4.71 g of nitrogen occupy 12.0 dm3 at 760 mmHg? (Answers on page 77)

In the next exercise, you have to find reacting amounts from a chemical equation before applying the ideal gas equation.

Exercise 20 What mass of zinc is required to produce 2.00 dm3 of hydrogen at 15 °c and 755 mmHg by reacting with acid? (Answer on page 77 )

In the next exercise, you use the ideal gas equation to get an idea of the pressure and volume changes which occur in an explosion.

An explosion occurs when a chemical reaction produces large volumes of gas very rapidly. The gas cannot expand rapidly enough to avoid a very high pressure immediately around the explosive, and it is the spread of shock waves from this small region of high pressure which causes the noise and damage associated with explosions.

22 Exercise 21 A widely-used explosive is TNT* which has a formula ~

C7H5N306• This is mixed with a solid oxidant so that oxygen required for combustion can be supplied rapidly. ~~, (a) Write an equation for the combustion of TNT. Assume that C and H atoms are completely oxidized and that

N atoms emerge as nitrogen gas. N2• ' (b) What amount of gas is produced from 1.00 mol of TNT. and what volume would it occupy at 1.00 atm and 400 °C. (c) Assuming that 1.00 mol of TNT mixed with oxidant occupies 3 0.500 dm • what is the increase in volume expressed as a percentage? (d) Assume that the reaction occurs so fast that the gaseous products occupy only 2.00 dm3 at 6000C. What would be the resulting pressure? (Answers on page 78 )

*TNT is an abbreviation of the old name trinitrotoluene. The modern systematic name is methyl-2.4.6-trinitrobenzene.

These calculations could also have been done using a known value for molar volume. but the use of the ideal gas equation is more general.

More problems on the ideal gas equation. for more practice and for revision. are in Appendix 2F on page 71 .

One of the most important applications of the ideal gas equation is in the determination of molar mass. This is the subject of the next section.

Determination of molar mass of a gas

The ideal gas equation can be transformed into an equation involving the mass of the gas. You will use this form of the equation to determine the molar masses of a gas and of a volatile liquid.

Objectives. When you have finished this section. you should be able to:

(18) derive an expression for the molar mass of a gas from the ideal gas equation; (19) use this expression to calculate molar mass; (20) describe and carry out experiments to determine the molar mass of a gas and of a volatile liquid.

You1 tShOluldread a~out the detterbmi2atll'on,/~fmOlart.maslses Off gatsehsand Wc=2, vo a i e liqui ds In your tex - oo~. oO~lng par lCU ar yl or 8 way . in which the ideal gas equation is used. Focus your attention on the general principles of the experimental methods rather than the practical details.

23 You should do the following exercises to ensure that you have grasped the principles before doing the experiments which follow them.

Exercise 22 (a ) Write down an expression for the amou nt, n, of a gas ~ in terms of the mass, m, and the molar mass, M. (b) Substitute your expression into the· ideal gas equation to obtain an expression for the molar mass. (Answers on page 78)

In the next exercise, use the expression you have just derived.

Exercise 23 Calculate the molar mass of a substance, given that 3.72 g of its vapour occupies 2.00 dm3 at 740 mmHg and 100 °C. (Answer on page 78 )

More problems on molar mass determination, for practice and for revision, are in Appendix 2G on page 71.

You should now be ready for the next two experiments. The first should be done at the earliest opportunity, but the second can be left till later if you want to spread out your practical work. Ask your teacher about this.

EXPERIMENT 1 Determining the molar mass of a gas

Aim The purpose of this experiment is to measure the volume and mass of a sample of carbon dioxide, and to use these values to determine the molar mass.

Introduction You weigh a clean dry flask full of air and then full of carbon dioxide. By filling the flask with water and reweighing, you can find its volume. Knowing the density of air, you calculate the mass of air filling the flask and use it to find the mass of the empty flask, and hence the mass of carbon dioxide. You then use the ideal gas equation to determine the molar mass of carbon dioxide.

24 Requirements

3 volumetric flask, 100 cm , dry, with stopper balance(s) capable of taking volumetric flask and of weighing - (a) up to 100 g with accuracy of 0.001 g (b) up to 200 g with accuracy of 0.1 g carbon dioxide cylinder or generator delivery tube, glass, 20-30 cm long rubber tubing, 30-90 cm, to connect gas cylinder to delivery tube thermometer, 0-100 °c access to barometer (or telephone number of local meteorological office)

Procedure 1. Get instructions from your teacher on how to operate the gas cylinder - there will be some valves you must not touch. Alternatively, you can use a simple carbon dioxide generator, provided you purify the gas from acid spray and dry it before use. Again, ask your teacher. 2. Weigh the dry volumetric flask together with its stopper to the nearest 0.001 g. Enter the mass in Results Table 1. 3. Remove the stopper, insert the glass delivery tube from the carbon dioxide cylinder or generator so that it reaches the bottom of the flask, and open the valve so that gas passes through for at least one minute. Keep the flask upright throughout. 4. Slowly remove the delivery tube, quickly close the flask with the stopper and close the valve on the cylinder or generator. 5. Weigh the flask with stopper again to the nearest 0.001 g. 6. Repeat steps 3, 4 and 5, and check that there is no further change in mass (i.e. that the carbon dioxide has indeed displaced all the air from the flask). If this is not the case, repeat these steps again - and then yet again, if necessary, until the mass is constant. 7. Fill the flask with water and insert the stopper, so that excess water is pushed out. Dry the outside of the flask and weigh it, full of water, on a robust balance to the nearest 0.1 g. 8. Note room temperature and atmospheric pressure. 9. Complete the results table below.

Results Table 1

Mass of flask filled with air g

Mass of flask filled with CO2 a b

Mass of flask filled with water g Room temperature °c

Atmospheric pressure mmHg

Density of air under conditions of experiment g cm-3

The following table gives values for the density of the air under various conditions of temperature and pressure. If your conditions do not corres- pond to any of those quoted, you should estimate the appropriate value.

P1-B 25 Table 3 Density of air (g cm-3) at different temperatures and pressures.

15 °C 17 DC 19 0C 21 °c 23 °c 25 °c

740 mmHg 0.00119 0.00119 0.00118 0.00117 0.00116 0.00115 750 mmHg 0.00121 0.00120 0.00119 0.00119 0.00118 0.00117 760 mmHg 0.00123 0.00122 0.00121 0.00120 0.00119 0.00119 770 mmHg 0.00124 0.00123 0.00123 0.00122 0.00121 0.00120 780 mmHg 0.00126 0.00125 0.00124 0.00123 0.00122 0.00122

Calculation You need to calculate the mass of carbon dioxide from your experimental results, before using the ideal gas equation as you have done in the preceding exercises. The steps in the calculation are as follows. Calculate 1 • the volume of the flask (from the mass and qensity of water) ; 2. the mass of air in the flask; 3. the mass of the empty stoppered flask (i.e. with no air in it) ; 4. the mass of carbon dioxide in the flask; 5. the molar mass of carbon dioxide.

lSpecimen results on page 78 )

Questions 1. What value does the experiment give for the relative molecular mass of

CO2?

2. Calculate the density of CO2 at s.t.p. from your results 3. In step (4) why were you told to remove the delivery tube slowly? 4. Why is a less accurate balance adequate for weighing the flask full of water? (Answers on page 78 )

Determination of the molar mass of a volatile liquid

There are several different experimental methods for finding the molar ~I mass of a volatile liquid, using the ideal gas equation. You may ~I have details of some of these in your text-books, e.g. Victor Meyer's method, or Dumas' method.

In all of these methods, the sample is weighed as a liquid (much easier than weighing gases) and then heated to vaporise it completely. The volume of vapour at a known temperature is then measured, either directly or by displacing an equal volume of air, and the ideal gas equation applied.

INe now give details ot one of the simpler methods, although your teacher may prefer that you use one of the others, if you have the special apparatus required.

26 EXPERIMENT 2 Determining the molar mass of a volatile liquid

Aim The aim of the experiment is to determine the molar mass of 1,1,1-trichloroethane, CH3CC13, at the temperature of boiling water and at atmospheric pressure. The same method can be used for other liquids which boil at a temperature below 80 °C, but we have chosen this one because it is non-toxic and non-flammable.

Introduction You obtain the mass of a sample of the liquid by weighing a small hypodermic syringe before and after injection into a large gas syringe. The large syringe is heated in a steam jacket (see Fig. 4J and you measure the volume of the vapour at the temperature of condensing steam. Finally you apply the ideal gas equation as before to calculate the molar mass.

Requirements safety spectacles 100 cm3 gas syringe, glass self sealing rubber cap for gas syringe steam jacket for gas syringe thermometer, 0 °C to 105 DC, to fit steam jacket Bunsen burner, tripod, gauze and bench mat steam generator, with safety tube 2 cm3 hypodermic syringe, glass, and needle

1,1,1-trichloroethane, CH3CC13 filter paper self sealing silicone rubber to make temporary seal for needle balance (accuracy 0.001 gJ access to barometer (or telephone number of local meteorological office)

steam

rubber cap

hypodermic syringe

Fig. 4. steam and water

27 Hazard warning steam can cause serious scalding. Make sure your generator has a safety tube so that pressure cannot build up if the outlet tube becomes blocked. Also, make sure the steam inlet to the jacket is secure and the outlet from the jacket is directed downwards. Wear safety spectacles.

Procedure 3 1. Place the gas syringe into the steam jacket and draw in about 5 cm air before sealing it with the rubber cap. 2. Pass steam through the steam jacket until the temperature reading and the volume of air in the syringe reach steady values. You can begin the next step while you are waiting for this steady state to be established. 3. Draw about 1 cm3 of 1,1,1-trichloroethane into the hypodermic syringe through the needle, rinse the syringe with the liquid and expel it into the sink. Draw in another 1 cm3 of liquid and holding the syringe vertically with the needle uppermost, slowly push in the piston till every bubble of air is expelled and a few drops of liquid emerge. 4. Dry the outside of the needle with filter paper and seal it with a small piece of silicone rubber.

5. Weigh the hypodermic syringe with its cap and liquid contents and record the mass in Results Table 2. Keep the syringe horizontal and avoid touching the piston or warming the barrel with your hand, either of which could result in loss of liquid. 6. When the temperature and volume of the air in the gas syringe are constant, record the volume of air and, with steam still passing through the jacket, push the hypodermic needle through its own seal and through the rubber seal of the gas syringe so that its tip projects well into the air space. Inject about 0.2 cm3 of the liquid into the gas syringe. See Fig. 5.

end of gas syringe

self-sealing • / rubber

(3;====.;:::::=

hypodermic needle

rubber cap

Fig. 5.

7. Withdraw the needle into its own self sealing cap, and re-weigh the syringe, cap and contents immediately so that no more liquid escapes. Record the mass. 8. Make sure, by twirling it, that the piston in the gas syringe can move freely so that the pressure inside is the same as atmospheric pressure, which should also be recorded. 9. Watch the temperature, and the volume of air and vapour in the syringe, until both reach steady values. Record these steady values.

28 10. Remove the cap from the gas syringe and push the piston in and out several times to expel the vapour. 11. If you, or another student, wish to use the apparatus again immediately, leave the steam generator going. Otherwise turn off the Bunsen burner.

Results Table 2

Mass of hypodermic syringe and liqu id before inj ect ion g

Mass of hypodermic syringe and liquid after injection g

Temperature of vapour °c

Atmospheric pressure mmHg

Volume of air in syringe cm3

Volume of air and vapour in syringe cm3

Calculation From your results, calculate the molar mass of 1,1,1-trichloroethane by the method you have already used in the preceding exercises. (Specimen results on page 79

Questions 1. What value does the experiment give for the relative molecular mass of 1,1,1-trichloroethane? 2. What might happen if the hypodermic needle were shorter than the nozzle of the gas syringe, and what effect would this have on your final results? 3. Calculate the molar mass from your experimental results in a different way, using the known value for molar volume at s.t.p. 4. The results obtained using the alternative method is very slightly different. Can you suggest why this is so? (Answers on page 79

The method in Experiment 2 gives good results for liquids which boil below about 80 °C. For liquids which boil above 80 °C, the steam jacket can be replaced by a small electric oven, thermostatically controlled.

However, there are several other methods. One of these, Victor Meyer's method, is the subject of the next exercise, taken from a past A-level paper. Candidates were expected to understand the principles behind the method, -wh.Lch are just the same as those we have already discussed, but not necessarily to have used it.

29 Exercise 24 A weighed sample of a volatile measuring liquid, which is contained in cylinder, F the sample tube A, is introduced ~ into the apparatus as shown. The main part of the apparatus is sample surrounded by a heating jacket, tube, A B, the temperature of which is maintained by the gently boiling liquid C. The side tube 0 is pulled back and the sample tube drops on to some sand, E, where- upon the sample vaporises. The water vapour thus formed displaces an equivalent volume of air from the apparatus which is collected over the water in the measuring cylinder F. Choose the most appropriate answer from those listed for each question.

sand,E

heati ng jacket, B

liquid, C

HEAT

Fig.6. Victor Meyer's apparatus.

(a) The most likely reason for the sand inside the apparatus is to A prevent the decomposition of the sample liquid B absorb the liquid and prevent its vaporisation C protect the apparatus from damage when the sample tube is dropped o allow the inner tube of the apparatus to attain rapidly the temperature of the boiling liquid, C E absorb any water that may be sucked back from the gas collection apparatus into the main apparatus. (b) In carrying out the experiment, which one of the following conditions is least necessary for obtain- ing an accurate result? A The boiling point of liquid C in the heating jacket must be accurately known. B The boiling point of the volatile liquid must be lower than that of liquid C. C The apparatus must be well stoppered before the sample tube is allowed to drop on to the sand. o The vapour formed by the volatile liquid must be unreactive with air. E The apparatus must be heated well before the liquid sample is dropped on to the sand. 30 (c) Assuming the measuring cylinder F to have a volume capacity of 150 cm3, what is the maximum quantity of liquid sample that may be used in A? A 0.1 mol, B 0.05 mol, C 0.01 mol, 0 0.005 mol, E 0.001 mol. (Answers on page 79

LEVEL ONE CHECKLIST

You have now reached the end of Level One of this Unit. The following is a summary of the objectives in Level One. Read carefully ~hrough it and check that you have adequate notes.

At this stage, you should be able to:

(1) & (2) state the properties which distinguish gases from liquids and solids and explain them in terms of the spacing and movement of molecules; (3) state Boyle's law and Charles' law;

(4) state the combined gas law and use it to calculate the effect of changing pressure, temperature or volume of a sample of gas; (5) state Gay-Lussac's law of combining volumes, and give some examples; (6) & (7) state Avogadro's theory and describe how it was used to establish simple chemical formulae; (8) use Avogadro's theory to calculate reacting volumes of gases; (9) outline a method for measuring the vDlumes of gases produced in the combustion of hydrocarbons; (10) calculate the formula of a gaseous hydrocarbon from the volumes of gases involved in its combustion; (11) s (12) calculate the molar volume of a gas from measurements of mass and volume; (13) use the molar volume of a gas at s.t.p. in calculations of amount; (14) use the molar volume of a gas to obtain Avogadro's constant from the results of radioactivity experiments; (15) s (17) state the ideal gas equation, and use it in simple calculations; (16) recognise that the gas constant, R, has different numerical values according to the units chosen for pressure and volume; (18) & (19) derive an expression for the molar mass of a gas from the ideal gas equation and use it to calculate the molar mass; (20) describe and carry out experiments to determine the molar mass of a gas and of a volatile liquid.

31 LEVEL ONE TEST

To find out how well you have learned the material in Level One, try the test which follows. Read the notes below before starting.

1. You should spend about 1 hour on this test. 2. You will need graph paper. 3. Hand your answers to your teacher for marking.

32 LEVEL ONE TEST

1. A sample of gas occupied 400 cms at s.t.p. Its volume, in cms, at 45 °C and 700 mmHg pressure would be 700 273 760 318 A 400 x -- x 0 400 x -- x 760 318 700 298

700 298 760 318 B 400 x x E 400 x x 760 318 700 273

760 298 C 400 x 700 x 318 (1 )

2. Which sample of gas contains the SMALLEST number of molecules? A 25 Iitres of N2 at 1 atm and 600 K B 15 litres of CO2 at 1 atm and 300 K C 10 Iitres of H2 at 2 atm and 300 K

0 5 i : tres of CH4 at 2 atm and 150 K E 5 litres of HCl at 4 atm and 450 K. (1 )

3. 20 cms of a gaseous element X reacts with excess of an element Y to form 40 cms of a gaseous compound of X and Y, all volumes being measured under the same conditions of temperature and pressure. From this information, it can be deduced that A the molecule of X contains at least two atoms of X B the formula of the compound formed is XY C equal volumes of gases contain equal numbers of molecules o molecules of X cannot consist of more than two atoms E X is less dense than the compound of X and Y. (1 )

In Question 4, one, more than one, or none of the suggested responses may be correct. You should answer as follows: A if only 1 ,2 and 3 are correct B if only 1 and 3 are correct C if only 2 and 4 are correct 0 if only 4 is correct E if some other response, or combination, is correct.

4. Which of the following masses of gas would occupy about 3 dm3 at 25 °c and 1 atmosphere? (1 mol of gas occupies 24 dm3 at 25 °C and 1 atm; C = 12, 0 = 16, S = 32, Ar = 40) 1. 8.0 g of sulphur dioxide 3. 4 g of oxygen 2. 5.5 g of carbon dioxide 4. 10 g of argon (1 )

33 For Question 5 choose an answer from A to E as follows: A Both statements true: second explains first B Both statements true: second does not explain first C First true: second false D First false: second true E Both false

5. First statement Second stat2ment 1 mol of carbon dioxide occupiBs Molecules of carbon dioxide a greater volume than 1 mol c~ are larger than those of carbon monoxide at room temper- carbon monoxide. ature and atmosphere pressure. ( 1J

6. The following information refers to 5.60 g of a gas at a constant temperature of DoC. Table 4

Pressure /atm 2.25 1 .90 1 .60 1 .12 0.75

Volume /dm3 2.00 2.35 2.S5 4.00 6.00 1 (a) Plot a graph of pressure against volume ( 2 )

(b) Is the gas behaving ideally? Give your reasons. (1 ) (c) Name and state the law which the information illustrates. (1) (d) Determine the gradient of the graph and write an expression relating the gradient, temperature and molar gas constant R. (2) (e) Hence, given that the molar volume is 22.4 dm3 mol-1 at s.t.p., calculate the relative molecular mass of the gas. (2) (f) Calculate the density of the gas at 13 oC and O.SOO atm. (2)

7. When 30 em" of a gaseous hydrocarbon were exploded with 350 em" I I of oxygen, the volume of gas remaining, after cooling to room A temperature, was 290 cm3• This was reduced to SO cm3 by the addition of aqueous potassium hydroxide. Assume that all volumes are recorded at the same temperature and pressure.

(a ) What volume of carbon dioxide is produced in the explosion? ( 1 ) (b ) What volume of oxygen reacts with the hydrocarbon? ( 1 ) (c ) What is the molecular formula of the hydrocarbon? ( 4 ) (d ) State one chemical principle used in your calculation of the molecular formula of the hydrocarbon. ( 1 )

34 8. In an experiment to determine the composition of a mixture of the gases propene~ C3H6~ and butene, C4HS' a student attempted to find the average relative molecular mass of the hydrocarbon mixture by finding the mass of a certain volume of it. The apparatus used was a standard volumetric flask, inverted and securely clamped in a well ventilated fume cupboard. See Fig. 7.

The results of the experiment were: standard ..- volumetric g Mass of hydrocarbon mixture 0.546 flask Volume of hydrocarbon mixture 300 cm3 calibration Pressure 1 .00 atmosphere - mark Temperature 27 DC (C = 12.0, H = 1 .00, molar volume at 0 DC and 1 atm = 22.4 dm3 mol-1)

Fig.7.

(a) Would you expect the mass of the stoppered flask when filled with the hydrocarbon mixture to be greater or less than its initial mass when filled with air? (1)

(b) What fault is there in the arrangement shown in the diagram for filling the flask with the hydrocarbon mixture? (1) (c) How would you determine the TOTAL volume of the volumetric flask? (1) (d) (i) Calculate the volume of the hydrocarbon mixture at 0 °c and 1 .00 atm pressure. (ii) Calculate the average mass of one mole of the particles in the hydrocarbon mixture. (3) (e) From your calculation state which hydrocarbon is more abundant in the mixture. (1)

9. Using the ideal gas equation, calculate the number of molecules of neon in a bulb of capacity 2.48 dm3, evacuated to a pressure of 0.001 mmHg at 27 °C.

1 mmHg 13.60 x 980.7 x 10-2 N m-2 R 8.31 J K-1 mol-1 J L 6.02 x 1023 mol-1 (4) (Total 33 marks)

LEVEL TWO

In Level Two we first consider two more empirical gas laws and then show how all the gas laws can be interpreted in terms of a theoretical model of a gas. This model~ known as the kinetic theory of gases~ also helps to explain why gases are not ideal~ i.e. why they do not observe the gas laws precisely. Finally we look at the range of different molecular speeds in a sample of gas.

Since the requirements of the different examlnlng boards vary widely in this area we suggest that you ask your teacher whether you should study the whole of Level Two. You may be advised to cover only those topics mentioned in your particular syllabus.

TWO MORE GAS LAWS

In the two sections which follow~ you extend your knowledge of the way gases behave.

Graham's law of effusion

This law arose from experiments which compare the rates at which different gases pass through a very small hole (or series of holesJ in a container. This process is known as effusion~ and is very similar to diffusion, which you have probably studied in your pre-A-level course. Graham's law provides us with an alternative method for determining the molar mass of a gas.

Objectives. When you have finished this section you should be able to:

(21J state Graham's law of effusion; (22J use Graham's law in calculations; (23J describe a simple experiment which uses Graham's law to determine the molar mass of a gas.

Look up Graham's law of effusion in your text-book and read about its application to determining the density and molar mass of a gas. You o may find the law referred to as Graham's law of diffusion. Qr

The next exercises are based on Graham's law.

37 Exercise 25 A volumetric flask Was filled with hydrogen, H2, and another with sulphur dioxide, S02' for an experiment. The flasks were stoppered, but not perfectly sealed, and put aside for use later. (a) Use molar masses to calculate the ratio of the densities of hydrogen and sulphur dioxide under the same conditions. (b) State Graham's law of effusion. (c) Assuming the flasks and the gaps between stoppers and flasks were identical, how much more rapidly would the hydrogen escape (to be replaced by air) than the sulphur dioxide? (Answers on page 79

Exercise 26 A gas syringe, set up as in Fig. 8 (page 39), was filled ~ first with hydrogen and then with carbon monoxide, and ~ the gases were allowed to effuse through a tiny pin-hole under the weight of the piston. The time taken for 75 cm3 of hydrogen to escape was 25 s, compared with 93 s for carbon monoxide. Given the molar mass of hydrogen, calculate the molar mass of carbon monoxide. (Answer on page 79 )

More problems on Graham's law, for practice and/or revision, are in Appendix 2H on page 72

In the next experiment you determine the molar mass of a gas using the method described in the last exercise.

EXPERIMENT 3 Determining the molar mass of a gas by effusion

Aim The purpose of this experiment is to apply Graham's law to determine the molar mass of domestic gas.

Introduction You fill a gas syringe with hydrogen, and allow it to escape through a small pin-hole under the weight of the piston. You measure the rate of escape and repeat the experiment using domestic gas. Assuming the molar mass of hydrogen, you use Graham's law to calculate the molar mass of domestic gas.

38 Requirements small piece of aluminium foil quick-setting glue three-way tap and connector to fit syringe gas syringe, 100 cm3 retort stand and clamp needle or pin stopclock or stopwatch (preferably to 0.1 sec) hydrogen cylinder or generator with rubber delivery tube tubing to gas tap

Hazard warning Hydrogen and domestic gas are not only flammable, but they form explosive mixtures with air. THERE MUST BE NO FLAMES IN THE LABORATORY DURING THIS EXPERIMENT

Procedure (steps 1 to 3 may already be done Tor you) 1. Cut a piece of smooth aluminium foil about 1 cm square and lay it on a flat surface, ready to be glued to the three-way tap. Do not pierce the foil yet.

o 2. Apply a little glue to the flat end of one of =-0 the tubes of the three-way tap. There must be enough to form a complete seal between the foil t-g -- gas syringe = 0 and the glass, but not so much as to close the =--" aperture. 3. Lower the three-way tap vertically on to the foil, press gently together, and leave until the glue has dried. gas under test 4. Check that the syringe piston moves freely (it must not be greased), attach the three-way tap and clamp the syringe carefully as shown in Fig. 8. Turn the tap so that there is a channel between side tube and syringe (position A - Fig. 9) and fill the syringe with air. 5. Turn the tap so that the side tube is closed but gas can pass between syringe and foil, (position B - Fig. 10) '- pin-hole in foil Fig.8. 6. Watch the volume reading of the syringe for half

a minute - it should not change. If the volume Q) OJ does change, air must be escaping from the three C .~ way tap, or from the connector, or from the III joint between foil and glass. Check these in B turn, regreasing the tap if necessary, until you are satisfied that the apparatus does not leak. (There is always some leakage between the piston and the syringe, but in a good syringe this will be so slow as to be negligible over half a A minute. If it is not negligible, you can dispense with the three-way tap and the foil, seal the nozzle with a bung, and use the gap between piston and syringe for the effusion!) '- pin-hole in foil Fig.9.

39 7. With the tap still in position 8, carefully Q) OJ pierce the foil with a needle making a very C .~ small hole at first. Watch the rate of fall (f) of the piston and increase the size of the E hole as necessary until the piston falls at a rate equivalent to about 1 cm3 air expelled each second. This sets the size of hole for channel the complete experiment - do not touch it again! B B. Turn the tap to position A and expel the air from the syringe through the side tube. 9. Check with your teacher on any instructions

for using the hydrogen cylinder or generator '- pin-hole in foil (NO FLAMES NEAR!) and obtain a slow flow of

gas through the delivery tube before Fig. 10. connecting it to the side tube of the three way tap.

10. Fill the syringe to about 50 cm3, remove the delivery tube and expel the gas from the syringe to ensure that no air remains. Refill to about 75 cm3, turn off the cylinder or generator and turn the tap to position B. 11. Allow the piston to fall to the 60 cm3 mark and start the stop clock at the moment it passes the mark. 12. Stop the clock when the piston passes the 10 cm3 mark and record the time in Results Table 3. 13. Repeat steps 9 to 12 at least twice. The times should not vary by more than 10% - if they d~, check again for leaks, particularly from the tap. and for a sticking piston. Repeat if necessary to obtain reproducible results. 14. Expel the hydrogen from the syringe, obtain a slow flow of domestic gas through a rubber tube from a gas tap and fill the syringe as before, repeating steps 10 to 13 as necessary to complete the results table.

Results Table 3

1 2 3 4 Mean

Time for effusion of 50 cm3 of hydrogen/s

Time for effusion of 50 cm3 of domestic gas/s

Calculation Use your results to calculate the molar mass of domestic gas by the method used in the preceding exercise. (Specimen resul ts on page 80)

40 Questions 1. What is the main constituent of domestic gas? Is your result for the molar mass consistent with your answer? 2. Does your result suggest that impurities in domestic gas have molar masses greater or less than the molar mass of the main constituent? What might these impurities be? 3. Why were you told to allow the piston to fall from the 75 cm3 mark to the 60 cm3 mark before starting the stopclock? 4. What difference would you expect if you repeated the experiment with the same apparatus and the same pinhole at a higher temperature? Explain. S. What difference would you expect if you repeated the experiment with the same apparatus at the same temperature but with a larger hole? (Answers on page 80 )

The gas laws you have studied so far relate mainly to pure gases. Now we consider a law which enables us to consider mixtures of gases more fully.

Dalton's law of partial pressures

You have learned that the pressure exerted by a gas is due to the combined effect of many collisions between individual molecules and the containing wall. Dalton invented the term 'partial pressure' to refer to the contribu- tion to the total pressure made by the molecules of one particular sort in a mixture of gases.

Objectives. When you have finished this section, you should be able to:

(24) state Dalton's law of partial pressures; (25) use the concept of mole fraction in calculating partial pressures; (26) use Dalton's law in simple calculations.

Read about Dalton's law of partial pressures in your text book. Find both a statement of the law in words and a mathematical expression. The following exercise will test your understanding.

Exercise 27 (a) state Dalton's law of partial pressures in your own words. (b) A globe contains oxygen at a pressure of 0.30 atm. Hydrogen is admitted until the total pressure is 0.80 atm, and then nitrogen is admitted until the total pressure is 0.90 atm. What is the partial pressure of each gas? (Answers on page 80 )

41 The last exercise was very simple. You are more likely to meet problems where you have to calculate partial pressures from the amounts of gas present.

To do this, it is convenient to introduce a quantity known as 'mole fraction'. In any mixture of substances (whether gaseous or not) XA,the mole fraction of A, is given by the expressions:

amount of A ~ ,,--_m_o_l_e_f_r_a_c_t_i_o_n_o_f_At_o_t_a_l_o_m_o_u_n_t~or ~

We now derive a simple and useful expression relating partial pressure to mole fraction. Dalton's law tells us that the partial pressure of a gas in a mixture is the same as if that gas alone occupied the entire volume. The ideal gas equation can then be applied to each gas in turn.

For example, in a mixture of three gases, A, Band C, with partial pressures PA, PB and PC' we can write for the gas A: RT PAV = nART or PA = nA x II IN We can express the quantity in terms of the total pressure, P, and the V total amount of gas, n. IN nJ{f E pV = or V- n Substituting into the expression for PA: ta nA = x - = x E = p x - PA nA V nA n n

Le·1 PA = P x XA X p in the same way, P = P x B and Pc x Xc or partial pressure = total pressure x mole fraction

By calculating mole fractions from the equation for the reaction you should be able to do the next exercise.

Exercise 28 Some ammonia in a syringe is completely decomposed to ~ nitrogen and hydrogen by passing it over heated iron wool. If the total pressure is then 760 mmHg, calculate ~~ the partial pressure of each gas. e Answer on page 80)

If you can calculate partial pressures from amounts, then you should also be able to calculate amounts from partial pressures in the next exercise.

42 Exercise 29 If the amount of oxygen in the globe referred to in /) Exer-c i se 27 W8re 0.25 mol, how much of the other gases ~ would b8 pr8s8nt? (Answer on pag8 80

An application of Avogadro's theory 8nab18s us to calculat8 mole fractions and partial pressures from the volumes of gases which make up a mixture.

Another way of stating Avogadro's theory is to say that the volume of a gas is proportional to its amount. Therefore, in a mixture of gases, A, B and C, we can express the mole fraction of A by: x _ amount of A volume of A A - total amount total volume We have already shown that amount is proportional to partial pressure. You will find it helpful therefor8 to r8member th8se symmetrical relationships.

partial pressure of A amount of A volume of A total pressure total amount total volume

Now try the following exercises.

Exercise 30 85.0 cm3 of moist air, at 1.00 atm, was dried carefully ~ and its volume, measured at the same temperature and pressure, fell to 82.0 cm3• What was the partial ~~ pressure of water vapour in the moist air? (Answer on page 81 )

Exercise 31 A diver should always breathe oxygen at a partial pressure of 2.0 x 104 N m-2• What percentage of oxygen by volume should his breathing mixture contain when he dives to a depth of 40 m and the pressure on him rises to 5.0 ,x 105 N m-2? (Answer on page 81 )

There are more exercises on Dalton's law of partial pressure$ in Appendix 21 on page 72.

HaVing looked at a number of empirical laws which describe how gases behave, we now turn to a theory which helps to explain this behaviour.

43 THE KINETIC THEORY OF GASES

You have already come across some aspects of the kinetic th8ory, for instance the idea that a gas consists of widely separated molecules in random motion. Collisions between the molecules and the container cause Molecule-to-wall collisions causing pressure. the pressure exerted by a gas, as Fig.11. shown in Fig. 11.

We now show that if we make certain assumptions about the nature of gas molecules and their motion, not only can we give a simple descriptive explanation of the gas laws, but we can also derive them mathematically.

You also know that gas laws are only approximately true. We show later that this too is explained by the kinetic theory if we examine some of the assumptions more carefully.

We begin with the assumptions which are made in order to construct a theoretical model of a gas.

A theoretical model of a gas

Objectives. When you have finished this section, you should be able to:

(27) state the assumptions made in the kinetic theory of gases; (28) explain why the pressure of a gas changes with volume and with temperature; (29) discuss briefly the validity of the assumptions of the kinetic theory of gases under different conditions of, temperature and pressure.

Before you read about the kinetic theory of gases, look at the following list of the assumptions which underlie it. Your reading should clarify each point for you.

1. Gas molecules move in straight lines, colliding occasionally with each other and with the wall of the container. 2. Gas molecules undergo perfectly elastic collisions, i.e. no translational kinetic energy is converted to other forms. 3. The average kinetic energy of the molecule in a sample of gas is proportional to the absolute temperature, i.e. Err T. 4. Gas molecules have negligible volume compared with the volume in which they move. (This is another way of saying that they are widely separated.) 5. There are no attractive forces between molecules.

44 You should read about the kinetic theory of gases in your text-books. c=) Look for an account of the assumptions, which may, of course, be ~ worded slightly differently in different books. Try to get a clear ~I mental picture of molecular motion in a gas and of how this motion is related to what we measure as temperature and pressure. Your text may include a mathematical derivation of the g~s laws but we leave this topic till later.

To test your understanding so far, try the following exercise.

Exercise 32 (a) Explain, in terms of molecular motion, why a decrease in volume of a gas is accompanied by an increase in pressure (at constant temperature). (b) Explain similarly why an increase in temperature of a gas increases pressure (at constant volume). (c) What would happen to the pressure of a gas if molecular collisions were not perfectly elastic? (Answers on page 81 )

Now we look more closely at the validity of some of the five assumptions we listed above. The first three seem to be readily acceptable. You can test the validity of assumption 4 in the next exercise.

Exercise 33 (aJ Calculate the volume of 1 .00 g of water vapour at ~ 120 °C and 1.00 atm. ~\\\ (bJ What percentage of this volume is taken up by water molecules? (Hint: what is the density of liquid water?)

(Answer on page 81 J

Now test assumption 4 under more extreme conditions.

Exercise 34 The density of liquid nitrogen is approximately ~ 0.B10 g cm-3• One mole of gaseous nitrogen at -123 °c ~ and 50.0 atm occupies 156 cm3• What percentage of the total volume is taken up by the molecules under these conditions? (Answer on page 81 J

Similar results for other gases suggest that it is reasonable to ignore the volume of the molecules compared with the volume in which they move provided the pressure is not too high, and the temperature not too low.

Now consider assumption 5 in the next exercise.

45 Exercise 35 All gases can be liquefied by reducing the temperature ~ and/or increasing the pressure - though this happens much more readily for some than for others. What does ~~ this tell us about the assumption that there are no attractive forces between molecules?

(Answer on page 81 )

Some examination syllabuses require you to know more about the liquefaction of gases. We discuss this topic in Appendix 1.

You should now be prepared to accept the five assumptions which are the basis of the kinetic theory of gases, provided the pressure is not too high nor the temperature too low.

In the next section, we show by a revealing exercise that the gas laws can be derived mathematically from the five assumptions of the kinetic theory.

Deriving the gas laws from kinetic theory

You may not be examined on this derivation Cask your teacher). However, it is useful for you to know that the gas laws, which have been presented as summaries of experimental facts, also have a theoretical basis. Therefore you should work through the revealing exercise even if you do not need to be able to reproduce it.

Objective When you have finished this section, you should be able to:

(30) recognise that the gas laws can be derived mathematically from the five assumptions of the kinetic theory of gases.

We consider a cube of side a containing N gas molecules each of mass m. We show that the pressure-volume product, pV, depends only on m, N and the speeds of the molecules, and is therefore constant at constant temperature.

First we focus attention on one molecule travelling with speed C at right angles to a wall, as in Fig. 12.

I I I mel -+-----.----a ---+-- I I I ------l, ...... ••...... •.<,

Fig. 12. A single molecule moving in a cube-shaped box.

46 0.1. What will be the speed and direction of the molecule after its next collision with the wall?

A.1. The molecule has the same speed c but in the opposite direction. (Strictly speaking the v~locity changes from +c to -c.)

0.2. Which assumption of the kinetic theory have you used to answer 0.1?

A.2. The assumption that collisions are perfectly elastic.

Collisions with the wall occur so frequently that they appear to exert a steady force, which contributes to the total pressure on the wall. A force is equal to the rate of change of momentum, so to calculate the force exerted by the molecule we shall look at momentum.

0.3. If the momentum of the molecule is defined as me, what is the change in momentum at each collision?

A.3. Change momentum before collision - momentum after collision

mc - (-mc) = 2mc

0.4. What is the time interval between collisions at the same face of the cube?

A.4. The distance between collisions at the same face 2a Time = distance/speed = 2a/c.

0.5. How many collisions at the same face occur in unit time?

time A.5. Number of collisions interval between collisions

1 2a/c = c/2a

Q.6 What is the change in momentum in unit time (i.e. the rate of change of momentum)?

A.6. Rate change of one collision x no. of collisions in unit time.

2mc x c/2a = mc2/a This is the force exerted by one molecule on one face of the cube. It is not too difficult to show that if the molecule was moving in a different direction so that it struck each of the six faces in turn rather than just two, then the (average) force of each face would be mc2/3a

Now we consider all the other molecules, total N, in the cube. These have different velocities cl, c2' c3'········ .cN but the same mass m, (In this simple treatment we will ignore the fact that the molecules collide with each other as well as the wall, but assuming elastic collisions, it can be shown that this makes no difference to the calculation.)

47 0.7. What is the total force, due to all the molecules, on one face of the cube?

A.7. Total force

We now define a quantity called the 'root-mean-square speed', crms

Z C rms = Ie C 1 Z + C Z Z + C 3 Z + + CN ) = ~ Note that c, the mean value of c, is not the same as ~, the square root of the mean value of cZ, but the difference is not great.

0.8. Rewrite the expression for the force on one face of the cube, using cZ•

Z + c z + c z + + Z) m(c1 z 3 cN A.8. Force 3a

0.9. If pressure is the force per unit area, what is the pressure, p, on one face of the cube?

force mNcZ A.9. p = area ~

0.10. Rewrite the expression above in the form pV ...... , where V is the volume of the cube.

mNcZ A.10. pV = pa3 = or pV = 3

0.11. Show that the expression above is equivalent to Boyle's law. (Hint: refer to assumption 3 in the list quoted earlier.)

A.11. The average kinetic energy is proportional to temperature. But E = ~mcz :. at constant temperature, mcz is constant. For a fixed mass of gas, N is also constant. :. pV == ;mNcz constant (Boyle's law)

So we have this remarkable result - that by making some simple assumptions we can derive a law found to be true (within limits) by experiment. This suggests strongly that the assumptions are reasonable ones. Further support for the assumptions is provided by the fact that we can also derive Avogadro's theory, the ideal gas equation, and all the other gas laws, from the fundamental equation, pV = ~mNcz.

Ask your teacher whether you need these further derivations; if so, you could try them for yourself before checking in a text-book. However, if you find the derivations difficult, we suggest that you merely remember that the ideal gas equation (and all the gas laws) can be derived from the five simple assumptions of the kinetic theory.

48 However, you have also learned that the ideal gas equation is only approximately true for real gases. We now look more closely at deviations from ideal behaviour and See how the kinetic theory of gases also predicts such deviations.

DEVIATIONS FROM IDEAL BEHAVIOUR

You very often use the ideal gas equation in calculations, and when you do you assume that the gas concerned is 'ideal', or shows 'ideal behaviour'. This is a very reasonable assumption over a limited range of temperature and pressure, but you also need to consider the deviations from ideal hRhaviour which occur under nnore extreme conditions.

Non-ideal behaviour

Objectives. When you have finished this section. you should be able to:

(31) explain why gases generally show deviations from ideal behaviour; (32) suggest how the ideal gas equation could be modified to describe gas behaviour more accurately.

Read about deviations from ideal behaviour in your text-book. If you do not already know where to look from your previous reading, look at the end of the sections on the ideal gas equation or the kinetic theory of gases. A useful reference in some books is the van der Waals' equation - a modification of the ideal gas equation to describe real gases. You probably do not need to memorise this equation, but account of it often includes a useful discussion of non-ideal behaviour.

To test your understanding of deviations from ideal behaviour, do the following exercises.

Exercise 36 (a) Under what conditions of temperature and pressure ~ are deviations from ideal behaviour most marked? ~\\ (b) Which assumptions of the kinetic theory of gases are most likely to be false under those conditions? (e) For which of the following gases would you expect those assumptions to be (i) most nearly true, (ii) most clearly false? helium, carbon dioxide, methane, nitrogen (Hint: consider the approximate relative sizes of the molecules and look up boiling points.) (Answer on page 82 )

49 Exercise 37 Consider two gases which are at equal temperatures and which have molecules of equal mass. In one gas the attractive forces between molecules are appreciable (non-ideal) and in the other they are virtually nil (ideal). Which gas would exert the lower pressure? Explain. (Answer on page 82 )

To take account of the attractive forces in a non-ideal gas, it has been suggested that the ideal gas equation would describe gas behaviour better if the measured pressure, p, were increased by a quantity x. The equation wo~ld then be:

(p + x) V = nRT x would be very small for gases with small attractive forces, but would increase with pressure for all gases.

Exercise 38 Why would the value of x in the above equation increase as the pressure increases? (Answer on page 82 )

It has also been suggested that the measured volume, V, should be decreased by a quantity, y, so that

(p + x)( V - y) nRT

Exercise 39 (aJ Why should the measured volume be decreased? (b) Suggest how the quantity, y, might be calculated. (Answers on page 82 )

One way of describing graphically the deviations from ideal behaviour, is to V Measure the volume of one mole of gas, m, at constant temperature over a large range of pressures, and then to plot pVm against p, as shown in Fig. 13.

o 100 200 300 pressure, p/atm.

Fig. 13. Variation of P Vm with p at O°C.

50 When comparing deviations at different temperatures it is convenient to plot pVmlRT I'\J 2 against p, as shown in Fig. 14 ~ t) - this does not alter the ~ general shape of the curves .? :0 but makes it possible to ·Vi c./l superimpose curves at a.Q) different temperatures. The E o quantity pVmlRT is known as o the compressibility factor, Z, and is frequently used by engineers and chemists who o 300 660 pressure, p/atm. handle gases at high pressures. Fig. 14. Variation of compressibility factor, Z with pressure, p.

The next exercise asks questions about Figs. 13 and 14.

Exercise 40 (a) What is the significance of the horizontal line in 6.f Fig. 13? ~ (b) What is the value of the compressibility factor for an ideal gas?

(d) Suggest reasons why pVm, as measured for carbon dioxide, is less than the ideal value at moderate pressures, but then increases again at very high pressures. (Answers on page 82 )

In the next section we consider a way of modifying the ideal gas equation so that it describes the behaviour of real gases more precisely.

The van der Waals equation

We have shown that the ideal gas equation holds better for real gases if measured pressure is increased and the measured volume decreased:

(p + x)(V - y) = nRT

There have been several attempts to evaluate x and y in such a way as to write an equation which holds under all conditions. The best known of these is van der Waals' equation, which you should read about in your text-book(s) if your syllabus includes this topic.

Objectives. When you hav8 finished this section you should be able to:

(33) state van der Waals' equation; (34) explain why the van der Waals constants vary for different gases.

51 In the next exercise you examine the values for van der Waals constants given in Table 5. Table 5 van der Waals constants

Gas a/atm drn" mol-2 b/dm3 mol-l.

ammonia 4.17 0.0372 argon 1 .35 0.0323 carbon dioxide 3.60 0.0428

carbon monoxide 1 .49 0.0400

chlorine 6.50 0.0564 helium 0.034 0.0238 hydrogen 0.245 0.0267 methane 2.26 0.0430

nitrogen 1 .39 0.0392

oxygen 1 .36 0.0319 water 5.46 0.0330

Exercise 41 (a] State van der Waals' equation.

(b) What physical property do the four gases with the largest values for a have in common? (c] Why do hydrogen and helium have the smallest values for b? (Answers on page 82

The next exercise enables you to test van der Waals' equation.

Exercise 42 One mole of methane, CH4, at 300 K has values of pressure~ and volume shown in the table below. Calculate values for pV, nRT and (p + an2/V2)(V - nb). Is the van der ~~ Waals' equation better than the ideal gas equation?

Table 6

p V nET pV (p + an2/V2)(V - nb) /atm /dm3 /atm dm3 /atm dm3 /atm dm3

1.00 24.6 24.61 10.0 2.42

100 0.210 200 0.0960

500 0.0587 1000 0.0469

(Answer on page 82 )

52 At this stage, you should do the following Teacher-marked Exercise which is part of a recent A-level examination question. In part (b) a mathematical description is not required.

Teacher-marKed The Kinetic Theory of gases expounded in the last Exercise century sought to formulate our understanding of the behaviour of gases. However, real gases seldom behave as the ideal gas which conforms to the equation. pV = nRT

(aJ What are the assumptions made in deriving the ideal gas- equation? To what extent are these assumptions true for a real gas? State the conditions in which a gas is most nearly ideal. (bJ Explain how Dalton's law of partial pressures may be derived from the kinetic theory of gases. If nitrogen and hydrogen are introduced into a Haber converter in the molecular ratio 1 :3, and 15% of the nitrogen is converted into ammonia at 200 atmospheres, what will be the partial pressure of the ammonia gas in the gaseous mixture?

You have seen how the kinetic theory of gases successfully explained the gas laws and deviations from ideal gas behaviour, which were already known by experiment. Before a theory becomes respectable, however, it must not only explain observed results - it must also predict results which have not so far been observed, but which can be tested. A good example of this last step in what is known as 'the scientific method' is the subject of the next section.

THE VARIATION OF MOLECULAR SPEEDS

When two molecules collide in an elastic collision, their speeds generally change. One probably travels faster than before and one slower, just as happens with the balls on a snooker table.

53 At any given moment in a sample of gas there will therefore be a large range of molecular speeds. Consequently some molecules possess more than the average kinetic energy and some have less. The way in which the speeds are distributed was first calculated by Maxwell in 1860 from the kinetic theory of gases but was not confirmed experimentally~ by Zartman and others~ until the 1930's. In this section we look at the pattern of distribution of speeds known as the Maxwell distribution~ and also at the experimental confirmation.

The Maxwell distribution

Objectiv~s. When you have finished this section you should be able to:

(35) describe graphically the Maxwell distribution of molecular speeds in gas at different temperatures;

Read about the Maxwell distribution (sometimes known as the Maxwell- Boltzmann distribution) in your text-books. You do not need the mathematical formulation of the distribution~ but you must know the shape of the curve which describes how the number of molecules* with a given speed changes with speed. You must also be able to sketch a series of curves showing how the shape changes with temperature.

*This is often shown as the fraction of the total number~ N~ but the shape of the curve is the same. Strictly speaking~ it is the Maxwell function~ the fraction of molecules within a small range of speeds~ which is plotted.

To test your understanding of what you have read~ do the next two exercises.

Exercise 43 Which one of the following curves is a Maxwell distribution. and why are the others not Maxwell distributions? ~ In each case~ the number~ n~ of molecules with speed C (y-axis)~ is plotted against c (x-axis). ~~

(a) (b) (e)

(d) (e) (f) Fig. 15.

(Answers on page 83 )

54 Exercise 44 Fig. 16 shows a set of three Maxwell distribution curves /) for a sample of gas at different temperatures. ~

~en ::J o Q) "0 E '0 o c

speed, elm S-1 Fig. 16.

(a J What can you say about the temperatures T 1.' T 2 and T:3? (bJ What can you say about the areas under the three curves? (c) Describe in words the difference in distribution of speeds in a gas at two different temperatures. (Answers on page 83

Use of the Maxwell distribution curve

The Maxwell distribution curve is valuable in its application to the study of rates of reaction. For molecules to react together they must collide with a certain minimum energy - if the energy is less than this minimum the molecules bounce off each other unchanged.

Since kinetic energy depends on speed, the distribution of speeds will clearly help to determine the number of molecules which react, i.e. the rate of reaction.

You will study this in more detail in Unit P5, but to help your understanding of the Maxwell distribution, we now present a worked example. This shows you how to obtain from a Maxw8ll distribution curV8 th8 fraction of mo18cu18s with speeds (and therefore energies) in a given range.

55 Worked Example Calculate from the Maxwell distribution in Fig. 17 the fraction of molecules with soeeds between 500 m S-1 and 1000 m 5-1•

01··· ! ....•.••.•...... •...... !

~ I· '! •• ". " .•...... • ! •• ' i.! : m ~ ....! •.• ! •••.••• ~~ .••••. ; !•..•.•.. !~ ... '.'. )'" !I ~ I·· . C/) I '!!';" £ 1·.····· .~ ~C/) :J U m (5 E '0

1000 2000 3000 speed, elm s-l

Fig. 17.

Solution The total area under the curve represents the total number of molecules. The area under the curve between the two vertical lines represents the number of molecules with speeds between 500 m S-1 and 1000 m S-1. If the equation for the curve is known, the areas can be obtained by integration. If you study mathematics, you may know how to do this. However, a simple method is to count the number of small squares in each area. Fraction of molecules with number in range Ar8a 1 velocities in given range total number Area 2 87 small squares 0.40 217 small squares Alternatively, if your eyes cannot take the strain of counting squares, you can cut out the area with scissors and weigh the pieces of paper on an accurate balance. (Do this with copies - not with the Unit itself!) Area Mass 0.032 g The fraction is then: 0.40 Area 2 Mass 2 0.081 g Use either of the methods in the Worked Example to do the next exercise.

56 Exercise 45 Use the graph below to determine, at each of the temperatures, the fraction of molecules with speeds between the values indicated in Table 7, and complete this table.

Table 7

Fraction of molecules in speed range

Temperature < 500 m S-l 1000-1500 m S-l > 2000 m S-l

273 K

1273 K

2273 K

U "'0 Q) Q) 0. en -£ .~

•...... 0a> E~ c

1000 2000 3000 speed, clms-1

Pl-C 57 Measuring molecular speeds - the Zartman experiment

Objective. When you have finished this section you should be able to:

(36) describe how the Maxwell distribution curve can be obtained by means of the Zartman experiment.

Zartman (and others) devised an experiment to test directly whether molecular speeds in a gas really did follow the Maxwell distribution. Essentially, they fired repeated pulses of gas molecules at a moving target in a vacuum - see Fig. 18.

slit -- moving target area narrow beam ~-=-:!"_~ __ of molecules molecules travelling at different speeds

Fig. 18. The Zartman experiment.

If all the molecules travelled at the same speed, they would hit the target at the same point; if the speeds varied, there would be a 'scatter' of hits corresponding to the distribution of speeds.

For details of this experiment you should view the ILPAC videotape I[ 11 'The distribution of molecular speeds'. If this is not available,

read about it in your text-book(s) or ask your teacher about seeing 00 a film loop.

The following exercise tests your understanding of the Zartman experiment.

Exercise 46 (a) What gas was used in the experiment? Why do you think this type of substance was chosen? (b) Why was it necessary to perform the experiment in a vacuum? (c) How were 'pulses' of gas molecules fired into the' cylinder? (d) How was the position of arrival of the molecules at the target detected? (e) How did the results confirm the Maxwell distribution? (f) The point where the largest number of molecules reached the target is shown as T in Fig. 19. When these molecules left the aperture A, Twas 1.50 cm from their line of travel. If the cylinder was 10.0 cm in diameter and revolving at 125 r.p.s., calculate a value for the speed of these molecules. Answer in m s-~. (g) As practice in converting to different units, calculate the speed in m.p.h. (1.00 mile = 1.61 x 103 metre). (Answers on page 83 )

58 \ T line of travel +~ A :-.------1.5 em

250r.p.s.

Fig. 19.

In the next section, we compare the molecular speed calculated in the last exercise from experimental results with a speed calculated from the kinetic theory of gases.

The 'root-mean-square' speed of 90S molecules

Objective. When you have finished this section, you should be able to:

(37) calculate the root-mean-square speed for molecules of a gas at a given temperature.

Look back at the revealing exercise on page 4B for an explanation of root-mean-square speed, crms(~)' and the way we used it to derive an expression for the pressure volume product, pV. We showed that:

pV = ~mNc2 where m is the mass of each molecules and N the number of molecules. But we also showed that:

pV = nRT You can use these two expressions in the next exercise to calculate c rms

Exercise 47 Use the two expressions for pV to derive an expression for crms in terms of the molar mass, M, the gas constant, R, and the absolute temperature, T. (Answer on page 84 )

Exercise 4B ( a ) Calculate the root-mean-square speed of hydrogen molecules at 25°C. ( b) Calculate the root-mean-square speed of bismuth atoms in vapour at B50 °c. (Use R = B.31 J K-1 mol-1 and 1.00 J = 1.00 kg m2 S-2.) (Answers on page 84 )

59 LEVEL TWO CHECKLIST

You have now reached the end of this Unit. Look again at the checklist at the end of Level One. In addition, you should now be able to:

(21) & (22) state Graham's law of effusion and use it in calculations~ (23) describe a simple experiment which uses Graham's law to determine the molar mass of a gas; (24) & (26) state Dalton's law of partial pressures and use it in simple calculations; (25) use the concept of mole fraction in calculating partial pressures; (27) state the assumptions made in the kinetic theory of gases; (28) explain why the pressure of a gas changes with volume and with temperature; (29) discuss briefly the validity of the assumptions of the kinetic theor~ of gases under different conditions of temperature and pressure; (30) recognise that the gas laws can be derived mathematically from the five assumptions of the kinetic theory of gases; (31) explain why gases generally show deviations from ideal behaviour; (32) suggest how the ideal gas equation could be modified to describe gas behaviour more accurately; (33) state van der Waals' equation; (34) explain why the van der Waals constants vary for different gases; (35) describe graphically the Maxwell distribution of molecular speeds in a gas at different temperatures; (36) describe how the Maxwell distribution curve can be obtained by means of the Zartman experiment; , (37), calculate the root-mean-square speed for molecules of a gas at a given temperature.

END OF UNIT TEST

To find out how well you have learned the material in this Unit, try the test which follows. Read the notes below before starting. 1. You should spend about 1~ hours on this test. 2. Hand your answers to your teacher for marking.

60 END OF UNIT TEST

3 1. A flask contains 2 dm of NH3(g) initially at atmospheric pressure. Y is the total pressure in the flask as 0.7 dm3 of HCI(g) is slowly added to it. X is the volume of HCI(g) added. All volumes are measured at the same pressure and the temperature remains constant at room temperature throughout. Choose, from A to E, the graph which best shows the relationship between Y and X. y

E r

x -- -e- X ---+ X

-- ...•x -- ...•x ( 1)

In Questions 2 and 3, one, or more than one, of the suggested answers may be correct. You should answer as follows:

A if only 1, 2 and 3 are correct B if only 1 and 3 are correct C if only 2 and 4 are correct 0 if only 4 is correct E if some other response, or combination, is correct.

2. Which of the following statements about molecules of gaseous hydrogen and oxygen is/are correct? 1. The mean molecular velocity increases with temperature in each case. 2. The mean speed is directly proportional to the temperature in each case. 3. At the same temperature the hydrogen molecules have the greater speed. 4. At the same temperature the mean kinetic energies are equal. (1)

3. Which of the following statements about molecules in a gas is/are correct? 1 • As the temperature rises, a greater proportion acquire high energies. 2. The average kinetic energy is independent of the temperature. 3. The average kinetic energy is constant at a given temperature. 4. At a given temperature, all the molecules move at the same speed. ( 11 61 Questions 4, 5 and 6 refer to the experiment described below. The apparatus shown in the diagram can be used to determine the relative molecular mass of a volatile liquid. The liquid, L, is drawn into the hypodermic syringe and then introduced into the gas syringe through a self-sealing rubber cap. The liquid turns into vapour, and the volume of vapour produced is recorded. The thermometer reading is also noted.

steam

rubber cap

hypodermic syringe

steam and water

4. The mass of liquid introduced into the gas syringe would be found by A weighing the hypodermic syringe immediately before and after introducing the liquid into the gas syringe B weighing the hypodermic syringe before filling with liquid and again after introducing the liquid into the gas syringe C measuring the volume of liquid used and calculating the mass from its density o measuring the volume of vapour of L produced in the gas syringe and calculating the mass using its density E weighing the gas syringe before and after the experiment. ( 1)

5. The best way of making sure that the vapour of L was at the temperature recorded by the thermometer would be to A stop the steam supply and allow the gas syringe to cool down to room temperature before reading the volume B heat the liquid L to a temperature just below its boiling point before drawing it into the hypodermic syringe C ensure that the whole apparatus is well lagged with insulating material o wait until there was no further change in volume of the gas syringe before recording the volume E make sure that the bulb of the thermometer was touching the side of the gas syringe. ( 1 )

62 6. Which of the following liquids could NOT be successfully used in the same way as L in this experiment?

Relative Boiling Density Liquid molecular mass point/K /g cm-3 A 1,1,1-trichloroethane 133 347 1 .33 B trichloromethane 120 334 1.48 C toluene (methyl benzene) 92 383 0.86 D tetrachloromethane 154 349 1 .59 E ethyl methyl Ketone 72 352 0.80 (butanone) ( 1)

7. Ideal gas behaviour is most liKely when there is a large value of A temperature D molecular volume B pressure E atomicity C relative molecular mass (1 )

For Questions 8 & 9, choose an answer from A to E as follows: A Both statements true: second explains first B Both statements true: second does not explain first C First true: second false D First false: second true E Both false

First statement Second statement 8. 1 mol of helium occupies a Helium molecules are monatomic smaller volume than 1 mol of and hydrogen molecules are hydrogen at room temperature diatomic. and atmospheric pressure. ( 1)

9. At low pressure, the behaviour As the pressure of a gas is of a real gas approaches that decreased, the variation in the of an ideal gas. velocities of the molecules decreases. ( 1)

10. When 10 cm3 of a gaseous ether, (CxH )20, were exploded with an ~ 3 excess of oxygen there was a contrac¥ion of 20 cm , and a further ~ contraction of 20 cm3 on the addition of an aqueous solution of sodium hydroxide (all the measurements being made under the same conditions of room temperature and pressure). Deduce the molecular formula of the ether, being careful to explain your reasoning fully. (5)

11. On combustion of 20 cm3 of ammonia in excess oxygen at 100 °C, 10 cm3 of nitrogen and 30 cm3 of steam are formed, all volumes being recorded at the same pressure. Assuming that the nitrogen and steam molecules are N2 and H20 respectively, deduce the formula of ammonia. State any law you use. (3)

63 12. (a) (i) When 0.32 g of liquid bromine is volatilized, the volume 3 of vapour formed, measured at s.t.p., is 45 cm • Calculate the relative molecular mass of bromine.

S.t.p. is 273 K and 1 atm (i.e. 101 kN m-Z), 1 R = 0.082 atm dm3 K-1 mol-1 (i.e. 8.31 J K-1 mol- ). (ii) Does the relative molecular mass you have calculated refer to the substance in the liquid or the gaseous state? Explain.

(b) The diagram shows the distribution of velocities of molecules in a gas at a certain en ~ :J temperature. U Q) (5 E '0 c c

speed, C

(i) In what TWO ways would the graph differ at a higher temperature? (ii) In what way is the variation in rate of a gaseous reaction with temperature related to the change in velocity distribu-. tion? (3) (c) Describe an experiment, for example the 'Zartman experiment', which shows the distribution of speeds of the molecules visually. ( 5)

13. (a) Write the equation which describes the behaviour of an ideal gas in varying conditions. (1) (b) List five assumptions on which the kinetic theory of ideal (5) gases is based. Which of these assumptions are not valid for real gases? (2) (c) Why does a mixture of (ideal) gases exert a pressure equal to the sum of the pressures which each gas would exert if it alone occupied the container at the same temperature? (3) (d) What is the name given to the principle outlined in (c) above? (1) (e) Why does the pressure of a fixed mass of gas increase when its temperature is raised (at constant volume)? (2) (f) Why do different gases diffuse at different rates (under the same conditions). (2)

14. One of the equations which describe the behaviour of real gases rather better than does the ideal gas equation is known as the van der Waals equation: a (p + VZ)(l1 - b) = Hr a (a) Suggest reasons for the inclusion of the terms vY & b. ( 3) (b) Which of the two gases, sulphur dioxide and helium, has the greater value for (i) a, (ii) b? (2) (Total 50 marks)

64 APPENDIX ONE

THE LIQUEFACTION OF GASES

You already know from your study of gases that both increasing pressure and decreasing temperature tend to cause liquefaction, and that some gases liquefy more easily than others. We now consider in more detail the conditions under which liquefaction occurs. We also consider the principles of methods used to liquefy gases, because this is very important industrially to ease problems of large scale transport and storage.

Andrews' isotherms and the critical point

The conditions for liquefaction are best studied by looking at a series of plots of pressure, p, against volume, V, for a sample of gas at different temperatures. For each plot the temperature is constant, and for this reason they are known as isotherms (iso = same, therm = heat). Isotherms were first studied by Andrews and usually still bear his name.

Objectives. When you have finished this section you should be able to:

(38) sketch a typical series of isotherms covering a range of temperatures; (39) define critical point and critical temperature, and identify them on sketches of isotherms; (40) state the difference between a permanent gas and a vapour.

Read about Andrews's isotherms (or isothermals) and critical temper- 0 ature in your text-book. They will probably appear in a section on liquefaction of gases which includes methods of liquefaction - we ~ consider these later. Look for diagrams of typical isotherms and try ~ to understand the significance of the variation of shape at different temperatures.

To test your understanding, try the following exercise.

65 Exerclse 49 The following questions refer to the typical set of isotherms illustrated below.

80 S <,ro

::Je 70 CJ) CJ) c.Q) 60

50 A

volume

(al Write the temperatures Tl to T5 in increasing order. (b) Describe what happens to a sample of gas kept at temperature T2 as the pressure is increased from one atmosphere. (cl What names are given to the temperature T3 and the point X?

ld) At temperatures above T3, the gas is sometimes known as a permanent gas, and below T3 as a vapour. Why is this? [e) What is the critical pressure for the gas? (f) Which isotherm corresponds most closel~ to ideal behaviour? (g) Why is it that a gas cannot be liquefied by pressure alone at temperatures above the critical temperature? (Answers on page 84 )

Since many gases have critical temperatures well below 0 °C, it is necessary to cool them to cause liquefaction. This is true whether or not pressure is applied as well. The two most widely-used methods of cooling are associated with the expansion of gases, which we now consider.

Cooling of gases by expansion

You are probably familiar with the increase in temperature which accompanies an increase in pressure - this is why a bicycle pump gets hot. Cooling by expansion is less familiar, but is simply the reverse of this process. We distinguish two causes of this cooling, the Joule-Thomson effect and adiabatic expansion (isentropic expansion), and show how you can understand both of them in terms of the kinetic theory of gases.

Objectives. When you have finished this section, you should be able to:

(41) explain why a gas cools when it is allowed to expand, distinguishing carefully between the Joule-Thomson effect and adiabatic (or isentropic) cooling; (42) outline a process for the liquefaction of air.

66 Read about the Joule-Thomson effect and adiabatic (or isentropic) o cooling in your text book. We suggest you do not concern yourself with any mathematical account, but look for explanations in terms of ill the interconversion of work and energy. Look also for the way these effects are applied to the liquefaction of a gas (e.g. air) by one method such as the Claude process or the Linde process.

In the two exercises which follow, you relate what you have learned about these cooling processes to the kinetic theory of gases.

Exercise 50 Consider an ideal gas under pressure in a cylinder containing a piston which is first fixed and then allowed to move. molecules of ideal gas piston • I

cylinder

(a) By considering the kinetic energies of the molecules of gas and of the piston before and after collision with the piston surface, explain why expansion causes cooling. (b) Which of the two cooling effects is this? (c) Would a non-ideal gas show the same effect? (d) Complete this sentence: The gas performs on the .••••. and therefore loses ••••••

(Answers on page 84 )

Exercise 51 Consider a non-ideal gas being forced under pressure out of a nozzle into a large volume maintained at low pressure. /)

low pressure ~

high •...~/ pressure ~~, -.

(a) Which feature of non-ideal gases is important here? (Look at the assumptions of the kinetic theory of gases.) (b) Work is done when the point of application of a force moves. How do the gas molecules do work, and why does this lower the temperature? (c) Which of the two cooling effects is this? (d) Would an ideal gas show the same effect? (Answers on page 85 )

67 In order to consolidate the more descriptive aspects of the lique- [2] faction of gases, you should do the following Teacher-marked Exercise, ~ which is part of a recent A-level question. Since it was only part of a question, not much detail was required, but we suggest here that you write about 250 words. It is not difficult to trim down an answer for examination purposes

Teacher-marked State the principles underlying the liquefaction of Exercise gases and outline a process for the liquefaction of air.

68 APPENDIX TWO

ADDITIONAL EXERCISES

Answers to these exercises, together with abbreviated methods, are on pages 85 to 88.

A - Boyle's law and Charles' law

Exercise 52 100 cm3 of hydrogen, at 1.00 atm, is compressed to 37.0 cm3 at constant temperature. What is the new pressure?

Exercise 53 73.0 cm3 of nitrogen at 1456 mmHg is allowed to expand so that its pressure is 760 mmHg. What is its new volume at the same temperature?

Exercise 54 A 25.0 dm3 cylinder of compressed helium is sufficient to fill a balloon of volume 6.00 x 103 dm3 at 1.00 atm at the same temperature. What is the pressure in the cylinder?

Exercise 55 32.0 cm3 of carbon dioxide from a chemical reaction is collected in a syringe at 27 °C and 1.00 atm. What would be its volume at s.t.p?

B - The combined gas law

Exercise 56 The table below shows a number of gas volumes and the conditions under which they were collected and measured. Calculate the volumes the gases would occupy if the conditions were changed to those indicated.

volume of gas conditions of measurement new conditions

(a) 29.2 cm3 25 °C, 762 mmHg s.t.p. (b) 5.13 dm3 62 °c, 1.02 atm s.t.p. (c) 132 cm3 s.t.p. 25 °C, 758 mmHg (d) 42.1 cm3 31 °C, 752 mmHg 20 °C, 200 mmHg

Exercise 57 A sample of gas at 27 °c was heated so that both its pressure and its volume were doubled. What was the new temperature?

3 Exercise 58 A set of car tyres, each of volume 21.0 dm , were inflated till the pressure gauge read 28.5 lb in -2. During a journey, the temperature of the tyres increased from 15 °c to 42 °c and their volume increased to 21.6 dm3• What would the pressure gauge have read then? (The pressure of air in the tyre was the sum of the gauge pressure and atmospheric pressure, which may be taken as 14.7 lb in -2.)

69 C - Reacting volumes of gases

Exercise 59 What is the maximum volume of hydrogen chloride that can be obtained from 75 cm3 of hydrogen at the same temperature and pressure?

Exercise 60 From the equation given, calculate the maximum value of ammonia that can be oxidised by 15 cm3 of oxygen at the same temperature and pressure. Also calculate the maximum volume of the products under the same conditions. What is the percentage change in volume during the reaction?

4NH3(g) + 502(g) ~ 4NO(g) + 6H20(g)

3 Exercise 61 78.5 cm of ethanol vapour, C2H50H, at 110 °c was burnt in oxygen to carbon dioxide and water only. Calculate the volume of oxygen used and the volume of the products, all at the same temperature and pressure.

D - Formulae of gaseous hydrocarbons

Exercise 62 20 cm3 of a gaseous hydrocarbon was mixed with 200 cm3 of oxygen (an excess) and exploded. The final volume, at the same room temperature and 3 pressure, was 150 cm , but this was reduced to 70 cm3 after treat~ent with a concentrated alkali. Calculate the formula of the hydrocarbon.

Exercise 63 15 cm3 of a gaseous hydrocarbon at 100 °c was mixed with an excess of oxygen and exploded by sparking. Cooling the products caused a reduction in volume equivalent to 30 cm3 at the original temperature and pressure. A further reducti8n of 30 cm3 occurred on exposure to concentrated alkali. What was the formula of the hydrocarbon?

3 Exercise 64 30 cm of a gas C3Hx was mixed with twice the minimum volume of oxygen needed for complete combustion and the mixture was exploded. After cooling to the same room temperature, and at the same pressure, the volume was 90 cm3 less than before. Calculate the volume of oxygen used in the reaction, and the value of x.

Exercise 65 To 100 cm3 of a mixture of hydrogen, methane and nitrogen was added 196 cm3 of oxygen and the mixture exploded. The volume after explosion 3 was 165 cm and, after treatment with potassium hydroxide solution, 131 cm3• Calculate the percentage by volume of each gas in the initial mixture. (All gas volumes were measured at the same room temperature and pressure.) E - Molar volume

Exercise 6fi Calculate the molar volume of (a) ammonia at s.t.p. given that 0.0780 g occupies 86.1 cm3 at 1.25 atm and 17 °C.

0 3 (b) methane, CH4, at 25 C and 1.00 atm, given that 0.0591 g occupies 90.2 cm at s.t.p.

70 Exercise 67 1.75 g of chlorine occupies 0.545 dm3 at s.t.p. At what temperature is its molar volume 25.0 dm3 at 1 .00 atm?

Exercise 68 Calculate the amount of nitrogen in a 500 cm3 flask at 15 oC and 1.01 atm.

Exercise 69 What mass of oxygen occupies 152 cm3 at s.t.p.?

Exercise 70 What is the volume of 5.81 g of hydrogen at 19 DC and 0.928 atm?

F - Ideal gas equation

Exercise 71 What is the volume of (a) 0.250 mol of methane at 17 °c and 750 mmHg, (b) 0.521 g of ammonia at 50 °C and 762 mmHg?

Exercise 72 What volume of carbon dioxide, at 25 DC and 764 mmHg, could be obtained from 7.31 g of calcium carbonate?

Exercise 73 1 .52 g of pure zinc was dissolved in dilute acid and the resulting gas, after drying, occupied 557 cm3 at 1 .01 atm. What was the temperature?

Exercise 74 Excess magnesium was added to 50.0 cm3 of 0.102 M hydrochloric acid. What volume of dry gas would be produced at 14 DC and 751 mmHg?

Exercise 75 What is the pressure in a 500 cm3 vessel containing 0.325 g of nitrogen at 20 DC?

Exercise 76 What is the amount of gas in a 1.00 dm3 globe at 19 DC and 1765 mmHg?

Exercise 77 What volume of vapour would be formed by heating 2.10 g of ethanol, C2HsOH, to 120 DC at 757 mmHg?

Exercise 78 5.00 cm3 of water is introduced into an evacuated tube of volume 150 cm3• The tube is sealed and heated to 300 DC. What is the resulting pressure?

G - Molar mass determination

Exercise 79 A flask of 109 cm3 capacity weighed 78.521 g when evacuated and 78.719 g when filled with a gas at 21 °c and 759 mmHg. What was the molar mass of the gas?

71 Exercise 80 Calculate the molar masses of gases A, Band C from the data given below: Volume of gas Mass of gas Temperature Pressure A 257 cm3 0.672 g 25 °c 1 .00 atm

() B 111 cm3 0.128 g 17 °c 763 mmHg r' 3 LJ 526 cm 1 .60 g 21 °c 103 kPa (1.00 atm 101 kPa)

E,\t::lrcise81 0.25 cm3 of a liquid of density 0.91 g cm-3 was injected into a sYI'inge maintained at 16D DC. The liquid vaporised completely and occupied 81 cm3 at 764 mmHg. What was the molar mass of the liquid?

Exercise 82 Calculate the mol~r masses of the volatile liquids 0, E and F from the data given below: Mass of liquid Volume of vapour Temperature Pressure 0 0.184 g 0.0792 dm3 100 °c 0.984 atm E 0.295 g 121 cm3 140 °c 747 mmHg F 0.163 g 65.0 cm3 101 °c 105 kPa (1.00 atm = 101 kPa)

H - Graham's law

Exercise 83 The isotopes of uranium 235U and 238U can be separated by using the different rates of diffusion of the gaseous fluorides UF6• What is the ratio of these rates of diffusion?

Exercise 84 Oxygen was allowed to escape from a syringe through a small hole under the weight of the piston. It took 47 s for 50 cm3 of oxygen to escape. When the experiment was repeated for another gas, 50 cm3 took 17 s to escape. (a) What was the molar mass of the gas? (bJ How long would the same volume of ammonia take to escape?

Exercise 85 The times taken for the effusion of equal volumes of carbon dioxide and a mixture of carbon dioxide with carbon monoxide were 28 sand 24 s respectively. Calculate the apparent molar mass of the mixture and the mole fraction of each gas.

I - Partial pressures

Exercise 86 75 cm3 of hydrogen~ 15 cm3 of nitrogen~ and 90 cm3 of carbon dioxide~ all measured at the same temperature and pressure, were mixed in a vessel. The total pressure was 2.4 atm. Calculate the partial pressure of each gas in the mixture.

72 Exercise 87 A 500 cm3 globe contain~ oxygen at 1 .00 atm. 300 cm3 of nitrogen, measured at the same temperature and pressure, is added under pressure, and then carbon dioxide is added till the total pressure is 3.10 atm. Calculate the partial pressure of each gas in the mixture, and the volume of carbon dioxide used.

Exercise 88 A flask contains 200 cm3 of oxygen at 1 .00 atm, and another flask contains 500 cm3 of helium at 2.00 atm. The two flasks are connected to allow the gases to mix. What is the partial pressure of each gas in the mixture, and the total pressure?

Exercise 89 A 600 cm3 vessel contains 0.0232 g of nitrogen and 0.0417 g of carbon monoxide at 23°C. Calculate the total pressure and the partial pressure of each gas.

Exercise 90 A mixture of 82.2 g of helium and 27.4 g of oxygen was contained in a cylinder at 25.0 atm and 11°C. What was the volume of the cylinder, and the partial pressure of each gas?

Exercise 91 15 cm3 of a mixture of carbon monoxide and methane was mixed with excess oxygen and exploded. There was a contraction in volume of 21 cm3 at the same room temperature and 1.0 atm pressure. Calculate the mole fraction of each gas in the mixture and their partial pressures.

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