Chapter 5 Measurements and Calculations

1 5.1 Scientific Notation

Objective: To show how very large or very small numbers can be expressed as the product of a number between 1 and 10 and a power of 10.

2 Scientific Measurements

In science we have two types of observations:

Qualitative: observations of color, odor, appearance, ect.

Quantitative: observations of measurement.

3 Scientific Measurements

Scientific measurements must always be represented as a number and a unit. If units are not included mistakes can be easily made. During a lab, if units are not included, or improperly included, it could result in drastically different results or even danger to yourself and your classmates.

4 Scientific Notation

In science we often make measurements that are very large or small. To make things easier to write we use scientific notation. Numbers such as 93,000,000 or 0.000054 can become tedious to rewrite when making long calculations. Also you run the risk of losing track of zeros causing miscalculations. Therefore, we have devised a better, easier system.

5 Scientific Notation

When using scientific notation for very large numbers: 1. Move the decimal to the left until there is only one digit between 1 and 10. 2. Count the number of places you move the decimal and make this the power of 10. 3. Rewrite your number as a decimal times 10 to the power of x.

6 Scientific Notation

EXAMPLE: 930 1. Move the decimal to make the number between 1 and 10. 930 => 9.30 2. Multiply by 10 raised to a power of 2, because the decimal was moved 2 places. 9.3x 102

7 Scientific Notation

When a very small number is involved, the decimal must be moved to the right instead of the left, in this case we make the power of 10 a negative number.

0.0000093 => move the decimal 6 places to the right. 9.3 x 10-6

8 Scientific Notation

EXAMPLES: Convert to scientific notation: 1. 8,000 2. 75,600,000,000 3. 0.000000546 4. 0.0004876

9 Scientific Notation

Multiplication and Division of Scientific Notation: When multiplying 2 numbers with exponents, you must add the exponents. EXAMPLE: (3.2 x 104)(2.8 x 103)= (3.2 x 2.8) x 104+3= 9.0 x 107 When dividing you subtract the exponents. EXAMPLE: 6.4 x 103 = 6.4/8.3 x 103-5 8.3 x 105 =0.77 x 10-2 = 7.7 x 10-3

10 Scientific Notation

Addition and subtraction of scientific notation requires that the 2 numbers are raised to the same power of 10 EXAMPLE: 1.31 x 105 + 4.2 x 104 = 13.1 x 104 + 4.2 x 104 =(13.1 + 4.2) x 104 = 17.3 x 104 = 1.73 x 105

11 5.2 Units

Objective: To learn the English, metric, and SI systems of measurements. The units part of a measurement tells us what scale or standard is being used to represent the results of the measurement. The need for common units is necessary so that scientists may have a universal language.

12 Units

The English system of units is the system used in the United States and includes inches, feet, miles. The metric system is used internationally and is known as the International System or SI Units. Below are some basic SI Units. Mass kilogram kg Length meter m Time second s Temperature Kelvin K

13 Units Below are the prefixes used for the metric system.

14 5.3 Measurements of Length, Volume, and Mass Objective: To use the metric system to measure length, volume, and mass.

To measure the length or distance, we commonly use a ruler. The unit used for length is the meter.

15 Measurements

16 Measurements Volume is the amount of 3- dimensional space occupied by a substance. If you have a cube that is 1m x 1m x 1m, the volume of this cube would be 1m3 or 1 cubic meter. If the cube was divided into 1000 cubes, the volume of 1 cube would be 1 dm3 or 1 Liter (L). Each liter contains 1000 cm3. 1 cm3 = 1 mL. 1000 mL = 1 L

17 Measurements

What this instrument is called? Graduated cylinder What would you measure with this? Liquid volume

18 Measurements

When measuring mass we use the unit, kilogram. The base unit for this would be the gram, however, this is a small unit, so the kilogram is the standard SI unit. In this class we will be measuring in grams. What type of instrument is used to measure mass?

19 5.4 Uncertainty in Measurement

Objectives: To learn how uncertainty in a measurement arises. To learn to indicate a measurement’s uncertainty by using significant figures.

20 Uncertainty Whenever a measurement is made with instruments such as a ruler, graduated cylinder or thermometer some estimation must be made. If you and your group must measure the length of a pin, you may get results such as the following: 2.84 cm, 2.85 cm, 2.86 cm, 2.85 cm, 2.86 cm Why aren’t these results all exactly the same?

21 Uncertainty If we look at fig. a we see that the pin is between 2.8 and 2.9 cm. So if we enlarge that distance and break the space between 2.8 and 2.9 into 10 even pieces, we can see that it lands on precisely 5. This means that the distance is 2.85. You must always estimate a digit beyond the marked reading.

22 Uncertainty

So if we look back at your group’s readings we see that the first 2 digits are all the same, it is the estimated third digit is where things vary. 2.84 cm, 2.85 cm, 2.86 cm, 2.85 cm, 2.86 cm The third digit is called an uncertain number. When making a measurement you must always record the certain digits (those marked) plus the first uncertain number.

23 Uncertainty

Any measurement always has some degree of uncertainty. The uncertainty of a measurement depends of the measuring device. The ruler has uncertainty that occurs in the hundredths place. A beaker would have uncertainty that occurs in the 10s or ones place.

24 Uncertainty

The numbers recorded in a measurement are called significant figures. The number of significant figures is determined by the uncertainty of the measuring device. With the ruler for measuring the pin, the uncertainty fell in the hundredths place. The uncertainty is usually assumed to be + 1 unless otherwise indicated.

25 5.5 Significant Figures

Objective: To learn to determine the number of significant figures in a calculated result.

26 Significant Figures

Chemistry requires doing many types of calculations. With each calculation we must consider what happens when arithmetic is done wit numbers that contain uncertainties. Mathematicians have kindly studied the statistical background for how uncertainty accumulates and have come up with some rules for uncertainty.

27 Significant Figures

When we are using our calculators, we get an answer with as many as 10 numbers after the decimal, however, we only multiplied 10.345 x 8.98. So are all of those numbers in our answer really significant? The following rules explain how to determine what numbers are significant and necessary to include in our answer.

28 Significant Figures

Rules for Counting Significant Figures: 1. Nonzero integers. Nonzero integers always count as significant figures. For example, the number 1457 has four nonzero integers, all of which count as significant figures. 2. Zeros. There are three classes of zeros. a. Leading zeros are zeros that precede all of the nonzero digits. They never count as significant figures. For example, in the number 0.0025, the 3 zeros simply indicate the position of the decimal point. The number only has 2 significant figures, the 2 and the 5.

29 Significant Figures

Rules for Counting Significant Figures cont.: 2. b. Captive zeros are zeros that fall between the nonzero digits. They always count as significant figures. For example, the number 1.008 has 4 significant figures. c. Trailing zeros are zeros at the right end of the number. They are significant only if the number is written with a decimal point. The number one hundred written as 100 has only one significant figure, but written as 100. has 3 significant figures. 30 Significant Figures Rules for Counting Significant Figures cont.: 3. Exact numbers. Often calculations involve numbers that were not obtained using measuring devices but were determined by counting: 10 experiments, 3 apples, 8 molecules. Such numbers are called exact numbers. They can be assumed to have an unlimited number of significant figures. Exact numbers can also arise from definitions. For example, 1 inch is defined as exactly 2.54 cm. Thus, in the statement 1in. = 2.54 cm, neither 2.54 nor 1 limits the number of significant figures when it is used in a calculation.

31 Significant Figures

Use your rules to determine the number of significant figures: 1. 0.0108 2. 0.0050060 3. 5.030 x 103 4. 480

32 Significant Figures

On your calculator you will find you have several extra numbers and you must round off. Rules for Rounding Off: 1. If the digit to be removed a. is less than 5, the preceding digit does not change. Example: 1.33 = 1.3 b. Is equal to or greater than 5, the preceding digit is increased by 1. Example: 1.36 = 1.4 2. In a series of calculations, carry the extra digits through to the final result and then round off. 33 Significant Figures

Rules for Using Significant Figures in Calculations: 1. For multiplication or division, the number of significant figures in the result is the same as that in the measurement with the smallest number of significant figures. We say this measurement is limiting, because it limits the number of significant figures in the result. For example: 4.56 x 1.4 = 6.384 => 6.4 2 sig figs You try: 8.315 = ?

298 34 Significant Figures

2. For addition or subtraction, the limiting term is the one with the smallest number of decimal places. For example: 12.11+18.0+1.013 = 31.123 The correct answer would be 31.1 You try: 0.6875 – 0.1 = ?

NOTE: For multiplication/division the significant figures are counted. For addition/subtraction the decimal places are counted.

35 5.6 Problem Solving and Dimensional Analysis Objective: To learn how dimensional analysis can be used to solve various types of problems.

The problem: Your mother asks you to buy 3 dozen cookies. The cookies come in packages of 6 cookies. How many packages of cookies must you buy?

36 Dimensional Analysis

In order to solve this problem and many others in chemistry we are going to use a process called dimensional analysis. We are going to analyze the dimensions or units involved here. First we must figure out the conversion factors. These are the numbers that help us convert from one unit to another. 1 dozen = ? cookies 1 package = ? cookies

37 Dimensional Analysis So the conversion factor for the first is 1 dozen = 12 cookies, also if we read the problem we can learn that 1 package = 6 cookies We can now use this information to calculate how many packages we need. 3 dozen x 12 cookies x 1 package = 6 pkgs. 1 1 dozen 6 cookies We can cancel units that are the top and bottom of the equation because this equals 1.

38 Dimensional Analysis If you divide 2 this equals 1. The same rule 2 applies when dividing units. Cookies = 1 cookies Or elephants = 1 elephants It does not matter what the unit is, if it is on the top and bottom of the equation, they equal 1. And anything times one is anything. 2x1= 2 or cookies x 1 = cookies

39 Dimensional Analysis

So let’s look at our problem again. 3 dozen x 12 cookies x 1 package = 6 pkgs. 1 1 dozen 6 cookies The dozen cancels, the cookies cancel and we are left with only packages as a unit. IT IS VERY IMPORTANT TO ALWAYS CLEARLY WRITE DOWN YOUR UNITS!!! If you do not, you WILL lose track and your answer will become incorrect.

40 Dimensional Analysis

Earlier we measured the length of a pin. We found this pin to be 2.85 cm, how many inches is this? There are 2.54 cm in 1 inch. 1. Before we start the problem, read carefully and find what we piece of information we are trying to calculate here. How many inches is the pin? 2. Then write down the information you are given. The pin is 2.85 cm and 2.54 cm = 1 in.

41 Dimensional Analysis

Now we can set up our problem. 3. Always start with the given value, in this case it would be 2.85 cm. 2.85 cm x 1 inch = ? 1 2.54 cm 4. Next write in your conversion factor(s). 5. Now check your units and see if they cancel. If they don’t….then you need to find the missing units.

42 Dimensional Analysis

6. Draw a light line through your canceled units. 2.85 cm x 1 inch = 1.12 inches 1 2.54 cm 7. Multiply and divide you answer and add your final units to your answer. 8. Check your significant figures. 9. Ask whether your answer makes sense.

43 Dimensional Analysis

1. The length of a marathon race is approximately 26.2 miles. What is this distance in kilometers? HINT: 1 mi = 1760 yd and 1 m = 1.094 yd

44 Dimensional Analysis

Your set up: 26.2 mi x 1760 yd x 1 m = 1 1 mi 1.094 yd Do the units cancel? What units is the answer supposed to be in?

45 Dimensional Analysis

26.2 mi x 1760 yd x 1 m x 1km = 1 1 mi 1.094 yd 1000 m Do your units all cancel? How many significant figures? What is the correct answer? Does this seem reasonable?

46 5.7 Temperature Conversions: An Approach to Problem Solving Objective: To learn the three temperature scale. To learn to convert from one scale to another. To continue to develop problem-solving skills.

47 Temperature Conversions

There are three scales used for measuring temperature:

Fahrenheit – Part of the English System Celsius – Used in the metric system Kelvin – The base unit in the SI units and also known as the absolute scale.

48 Temperature Conversions

The thermometers indicate the freezing pts and boiling pts on each scale. Notice there is an equal distance between the 2 pts on the Celsius and Kelvin scales, but not the Fahrenheit.

49 Temperature Conversions There are simple equations you can use to convert temperatures from one scale to another. Celsius -> Kelvin Because these units are equal, all we have to do is add 273 to the Celsius temperature.

T C + 273 = TK Kelvin -> Celsius

TK - 273 = T C

50 Temperature Conversions When converting with the Fahrenheit scale we must do more math because the scales are not equal units. Celsius ->Fahrenheit

T F=1.80(T C)+32 Fahrenheit -> Celsius

T C= T F – 32 1.80

51 Temperature Conversions

Convert the following: 1. 38 C = F 2. 145 C = K 3. 373 K = C 4. 25 C = K 5. 85 F = K

52 5.8 Density

Objective: To define density and its units.

Density: the amount of matter present in a given volume of substance or is mass per unit volume. In mathematical terms this means: Density = mass or is the ratio of mass volume to volume

53 Density

What type of units would you use for density? Mass => ? Units Volume => ? Units Mass = grams (g) Volume = milliliters (ml) = cubic cm = cm3/cc So density units = g/ml or g/cm3

54 Density

So what does the term density really mean? If something is very dense...what does that mean? What if something is not very dense? You know how to calculate density, but how would you measure density? What type of instruments would you need?

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