Ideal Gas Law (Chain Rule)

Total Page：16

File Type：pdf, Size：1020Kb

Application of the Chain Rule in an Application of the Ideal Gas Law Background: In Chemistry, the Ideal Gas Law plays an important role. It states the relationship between the pressure, volume, temperature and number of moles of a gaseous material. The Ideal Gas Law can be stated as PV nRT where P= pressure, V= volume, n = number of moles of an ideal gas, T= temperature, and R= constant. Experiment: Suppose a sample of n moles of an ideal gas is held in an isothermal (constant temperature, T) chamber with initial volume V0. The ideal gas is compressed by a piston as shown in the diagram. As the gas is compressed, the volume decreases. According to the Ideal Gas Law, the pressure will increase. Let us assume that the volume of the gas changes at a constant rate as defined by the equation V(t) V0 kt , where t is the time and k is a constant. Problem Statement: Use the chain rule can to find the time rate of change of the pressure with respect to time. Solution: The ideal gas law can be solved for the pressure, P(t) to give: where P(t) and V(t) have been written as explicit functions of time and the other symbols are constant. Let us differentiate each side of the equation with respect to time. dP(t) d nRT dt dt V (t) where the constant terms, n, R, and T, have been moved to the left of the derivative operator. Applying the chain rule gives where the power rule has been used to differentiate the term (1/V). Now consider the volume and its derivative dV k dt Substituting in for dV/dt and V yields dP/dt. V(t) V0 kt dP 1 nRT k dt V 2 1 nkRT 2 (V0 kt) This result represents the rate of change of pressure with respect to time. .

Copy Link

Recommended publications

Is Colonic Propionate Delivery a Novel Solution to Improve Metabolism and Inflammation in Overweight Or Obese Subjects?
Commentary in IgG levels in IPE-treated subjects versus Is colonic propionate delivery a novel those receiving cellulose supplementa- tion. This interesting discovery is the Gut: first published as 10.1136/gutjnl-2019-318776 on 26 April 2019. Downloaded from solution to improve metabolism and first evidence in humans that promoting the delivery of propionate in the colon inflammation in overweight or may affect adaptive immunity. It is worth noting that previous preclinical and clin- obese subjects? ical data have shown that supplementation with inulin-type fructans was associated 1,2 with a lower inflammatory tone and a Patrice D Cani reinforcement of the gut barrier.7 8 Never- theless, it remains unknown if these effects Increased intake of dietary fibre has been was the lack of evidence that the observed are directly linked with the production of linked to beneficial impacts on health for effects were due to the presence of inulin propionate, changes in the proportion of decades. Strikingly, the exact mechanisms itself on IPE or the delivery of propionate the overall levels of SCFAs, or the pres- of action are not yet fully understood. into the colon. ence of any other bacterial metabolites. Among the different families of fibres, In GUT, Chambers and colleagues Alongside the changes in the levels prebiotics have gained attention mainly addressed this gap of knowledge and of SCFAs, plasma metabolome analysis because of their capacity to selectively expanded on their previous findings.6 For revealed that each of the supplementa- modulate the gut microbiota composition 42 days, they investigated the impact of tion periods was correlated with different 1 and promote health benefits.
[Show full text]
Entropy: Ideal Gas Processes
Chapter 19: The Kinec Theory of Gases Thermodynamics = macroscopic picture Gases micro -> macro picture One mole is the number of atoms in 12 g sample Avogadro’s Number of carbon-12 23 -1 C(12)—6 protrons, 6 neutrons and 6 electrons NA=6.02 x 10 mol 12 atomic units of mass assuming mP=mn Another way to do this is to know the mass of one molecule: then So the number of moles n is given by M n=N/N sample A N = N A mmole−mass € Ideal Gas Law Ideal Gases, Ideal Gas Law It was found experimentally that if 1 mole of any gas is placed in containers that have the same volume V and are kept at the same temperature T, approximately all have the same pressure p. The small differences in pressure disappear if lower gas densities are used. Further experiments showed that all low-density gases obey the equation pV = nRT. Here R = 8.31 K/mol ⋅ K and is known as the "gas constant." The equation itself is known as the "ideal gas law." The constant R can be expressed -23 as R = kNA . Here k is called the Boltzmann constant and is equal to 1.38 × 10 J/K. N If we substitute R as well as n = in the ideal gas law we get the equivalent form: NA pV = NkT. Here N is the number of molecules in the gas. The behavior of all real gases approaches that of an ideal gas at low enough densities. Low densitiens m= enumberans tha oft t hemoles gas molecul es are fa Nr e=nough number apa ofr tparticles that the y do not interact with one another, but only with the walls of the gas container.
[Show full text]
THE SOLUBILITY of GASES in LIQUIDS Introductory Information C
THE SOLUBILITY OF GASES IN LIQUIDS Introductory Information C. L. Young, R. Battino, and H. L. Clever INTRODUCTION The Solubility Data Project aims to make a comprehensive search of the literature for data on the solubility of gases, liquids and solids in liquids. Data of suitable accuracy are compiled into data sheets set out in a uniform format. The data for each system are evaluated and where data of sufficient accuracy are available values are recommended and in some cases a smoothing equation is given to represent the variation of solubility with pressure and/or temperature. A text giving an evaluation and recommended values and the compiled data sheets are published on consecutive pages. The following paper by E. Wilhelm gives a rigorous thermodynamic treatment on the solubility of gases in liquids. DEFINITION OF GAS SOLUBILITY The distinction between vapor-liquid equilibria and the solubility of gases in liquids is arbitrary. It is generally accepted that the equilibrium set up at 300K between a typical gas such as argon and a liquid such as water is gas-liquid solubility whereas the equilibrium set up between hexane and cyclohexane at 350K is an example of vapor-liquid equilibrium. However, the distinction between gas-liquid solubility and vapor-liquid equilibrium is often not so clear. The equilibria set up between methane and propane above the critical temperature of methane and below the criti cal temperature of propane may be classed as vapor-liquid equilibrium or as gas-liquid solubility depending on the particular range of pressure considered and the particular worker concerned.
[Show full text]
The Lower Critical Solution Temperature (LCST) Transition
Copyright by David Samuel Simmons 2009 The Dissertation Committee for David Samuel Simmons certifies that this is the approved version of the following dissertation: Phase and Conformational Behavior of LCST-Driven Stimuli Responsive Polymers Committee: ______________________________ Isaac Sanchez, Supervisor ______________________________ Nicholas Peppas ______________________________ Krishnendu Roy ______________________________ Venkat Ganesan ______________________________ Thomas Truskett Phase and Conformational Behavior of LCST-Driven Stimuli Responsive Polymers by David Samuel Simmons, B.S. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin December, 2009 To my grandfather, who made me an engineer before I knew the word and to my wife, Carey, for being my partner on my good days and bad. Acknowledgements I am extraordinarily fortunate in the support I have received on the path to this accomplishment. My adviser, Dr. Isaac Sanchez, has made this publication possible with his advice, support, and willingness to field my ideas at random times in the afternoon; he has my deep appreciation for his outstanding guidance. My thanks also go to the members of my Ph.D. committee for their valuable feedback in improving my research and exploring new directions. I am likewise grateful to the other members of Dr. Sanchez’ research group – Xiaoyan Wang, Yingying Jiang, Xiaochu Wang, and Frank Willmore – who have shared their ideas and provided valuable sounding boards for my mine. I would particularly like to express appreciation for Frank’s donation of his own post-graduation time in assisting my research.
[Show full text]
Pressure Diffusion Waves in Porous Media
Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Pressure diffusion waves in porous media Permalink https://escholarship.org/uc/item/5bh9f6c4 Authors Silin, Dmitry Korneev, Valeri Goloshubin, Gennady Publication Date 2003-04-08 eScholarship.org Powered by the California Digital Library University of California Pressure diffusion waves in porous media Dmitry Silin* and Valeri Korneev, Lawrence Berkeley National Laboratory, Gennady Goloshubin, University of Houston Summary elastic porous medium. Such a model results in a parabolic pressure diffusion equation. Its validity has been Pressure diffusion wave in porous rocks are under confirmed and “canonized”, for instance, in transient consideration. The pressure diffusion mechanism can pressure well test analysis, where it is used as the main tool provide an explanation of the high attenuation of low- since 1930th, see e.g. Earlougher (1977) and Barenblatt et. frequency signals in fluid-saturated rocks. Both single and al., (1990). The basic assumptions of this model make it dual porosity models are considered. In either case, the applicable specifically in the low-frequency range of attenuation coefficient is a function of the frequency. pressure fluctuations. Introduction Theories describing wave propagation in fluid-bearing porous media are usually derived from Biot’s theory of poroelasticity (Biot 1956ab, 1962). However, the observed high attenuation of low-frequency waves (Goloshubin and Korneev, 2000) is not well predicted by this theory. One of possible reasons for difficulties in detecting Biot waves in real rocks is in the limitations imposed by the assumptions underlying Biot’s equations. Biot (1956ab, 1962) derived his main equations characterizing the mechanical motion of elastic porous fluid-saturated rock from the Hamiltonian Principle of Least Action.
[Show full text]
IB Questionbank
Topic 3 Past Paper [94 marks] This question is about thermal energy transfer. A hot piece of iron is placed into a container of cold water. After a time the iron and water reach thermal equilibrium. The heat capacity of the container is negligible. specific heat capacity. [2 marks] 1a. Define Markscheme the energy required to change the temperature (of a substance) by 1K/°C/unit degree; of mass 1 kg / per unit mass; [5 marks] 1b. The following data are available. Mass of water = 0.35 kg Mass of iron = 0.58 kg Specific heat capacity of water = 4200 J kg–1K–1 Initial temperature of water = 20°C Final temperature of water = 44°C Initial temperature of iron = 180°C (i) Determine the specific heat capacity of iron. (ii) Explain why the value calculated in (b)(i) is likely to be different from the accepted value. Markscheme (i) use of mcΔT; 0.58×c×[180-44]=0.35×4200×[44-20]; c=447Jkg-1K-1≈450Jkg-1K-1; (ii) energy would be given off to surroundings/environment / energy would be absorbed by container / energy would be given off through vaporization of water; hence final temperature would be less; hence measured value of (specific) heat capacity (of iron) would be higher; This question is in two parts. Part 1 is about ideal gases and specific heat capacity. Part 2 is about simple harmonic motion and waves. Part 1 Ideal gases and specific heat capacity State assumptions of the kinetic model of an ideal gas. [2 marks] 2a. two Markscheme point molecules / negligible volume; no forces between molecules except during contact; motion/distribution is random; elastic collisions / no energy lost; obey Newton’s laws of motion; collision in zero time; gravity is ignored; [4 marks] 2b.
[Show full text]
Exercise and Cellular Respiration Lab
California State University of Bakersfield, Department of Chemistry Exercise and Cellular Respiration Lab Standards: MS-LS1-7 Develop a model to describe how food is rearranged through chemical reactions forming new molecules that support growth and/or release energy as this matter moves through an organism. Introduction: I. Background Information. Cellular respiration (see chemical reaction below) is a chemical reaction that occurs in your cells to create energy; when you are exercising your muscle cells are creating ATP to contract. Cellular respiration requires oxygen (which is breathed in) and creates carbon dioxide (which is breathed out). This lab will address how exercise (increased muscle activity) affects the rate of cellular respiration. You will measure 3 different indicators of cellular respiration: breathing rate, heart rate, and carbon dioxide production. You will measure these indicators at rest (with no exercise) and after 1 and 2 minutes of exercise. Breathing rate is measured in breaths per minute, heart rate in beats per minute, and carbon dioxide in the time it takes bromothymol blue to change color. Carbon dioxide production can be measured by breathing through a straw into a solution of bromothymol blue (BTB). BTB is an acid indicator; when it reacts with acid it turns from blue to yellow. When carbon dioxide reacts with water, a weak acid (carbonic acid) is formed (see chemical reaction below). The more carbon dioxide you breathe into the BTB solution, the faster it will change color to yellow. The purpose of this lab activity is to analyze the effect of exercise on cellular respiration. Background: I.
[Show full text]
Thermodynamics of Ideal Gases
D Thermodynamics of ideal gases An ideal gas is a nice “laboratory” for understanding the thermodynamics of a ﬂuid with a non-trivial equation of state. In this section we shall recapitulate the conventional thermodynamics of an ideal gas with constant heat capacity. For more extensive treatments, see for example [67, 66]. D.1 Internal energy In section 4.1 we analyzed Bernoulli’s model of a gas consisting of essentially 1 2 non-interacting point-like molecules, and found the pressure p = 3 ½ v where v is the root-mean-square average molecular speed. Using the ideal gas law (4-26) the total molecular kinetic energy contained in an amount M = ½V of the gas becomes, 1 3 3 Mv2 = pV = nRT ; (D-1) 2 2 2 where n = M=Mmol is the number of moles in the gas. The derivation in section 4.1 shows that the factor 3 stems from the three independent translational degrees of freedom available to point-like molecules. The above formula thus expresses 1 that in a mole of a gas there is an internal kinetic energy 2 RT associated with each translational degree of freedom of the point-like molecules. Whereas monatomic gases like argon have spherical molecules and thus only the three translational degrees of freedom, diatomic gases like nitrogen and oxy- Copyright °c 1998{2004, Benny Lautrup Revision 7.7, January 22, 2004 662 D. THERMODYNAMICS OF IDEAL GASES gen have stick-like molecules with two extra rotational degrees of freedom or- thogonally to the bridge connecting the atoms, and multiatomic gases like carbon dioxide and methane have the three extra rotational degrees of freedom.
[Show full text]
AP Chemistry Chapter 5 Gases Ch
AP Chemistry Chapter 5 Gases Ch. 5 Gases Agenda 23 September 2017 Per 3 & 5 ○ 5.1 Pressure ○ 5.2 The Gas Laws ○ 5.3 The Ideal Gas Law ○ 5.4 Gas Stoichiometry ○ 5.5 Dalton’s Law of Partial Pressures ○ 5.6 The Kinetic Molecular Theory of Gases ○ 5.7 Effusion and Diffusion ○ 5.8 Real Gases ○ 5.9 Characteristics of Several Real Gases ○ 5.10 Chemistry in the Atmosphere 5.1 Units of Pressure 760 mm Hg = 760 Torr = 1 atm Pressure = Force = Newton = pascal (Pa) Area m2 5.2 Gas Laws of Boyle, Charles and Avogadro PV = k Slope = k V = k P y = k(1/P) + 0 5.2 Gas Laws of Boyle - use the actual data from Boyle’s Experiment Table 5.1 And Desmos to plot Volume vs. Pressure And then Pressure vs 1/V. And PV vs. V And PV vs. P Boyle’s Law 3 centuries later? Boyle’s law only holds precisely at very low pressures.Measurements at higher pressures reveal PV is not constant, but varies as pressure varies. Deviations slight at pressures close to 1 atm we assume gases obey Boyle’s law in our calcs. A gas that strictly obeys Boyle’s Law is called an Ideal gas. Extrapolate (extend the line beyond the experimental points) back to zero pressure to find “ideal” value for k, 22.41 L atm Look Extrapolatefamiliar? (extend the line beyond the experimental points) back to zero pressure to find “ideal” value for k, 22.41 L atm The volume of a gas at constant pressure increases linearly with the Charles’ Law temperature of the gas.
[Show full text]
What Is High Blood Pressure?
ANSWERS Lifestyle + Risk Reduction by heart High Blood Pressure BLOOD PRESSURE SYSTOLIC mm Hg DIASTOLIC mm Hg What is CATEGORY (upper number) (lower number) High Blood NORMAL LESS THAN 120 and LESS THAN 80 ELEVATED 120-129 and LESS THAN 80 Pressure? HIGH BLOOD PRESSURE 130-139 or 80-89 (HYPERTENSION) Blood pressure is the force of blood STAGE 1 pushing against blood vessel walls. It’s measured in millimeters of HIGH BLOOD PRESSURE 140 OR HIGHER or 90 OR HIGHER mercury (mm Hg). (HYPERTENSION) STAGE 2 High blood pressure (HBP) means HYPERTENSIVE the pressure in your arteries is higher CRISIS HIGHER THAN 180 and/ HIGHER THAN 120 than it should be. Another name for (consult your doctor or immediately) high blood pressure is hypertension. Blood pressure is written as two numbers, such as 112/78 mm Hg. The top, or larger, number (called Am I at higher risk of developing HBP? systolic pressure) is the pressure when the heart There are risk factors that increase your chances of developing HBP. Some you can control, and some you can’t. beats. The bottom, or smaller, number (called diastolic pressure) is the pressure when the heart Those that can be controlled are: rests between beats. • Cigarette smoking and exposure to secondhand smoke • Diabetes Normal blood pressure is below 120/80 mm Hg. • Being obese or overweight If you’re an adult and your systolic pressure is 120 to • High cholesterol 129, and your diastolic pressure is less than 80, you have elevated blood pressure. High blood pressure • Unhealthy diet (high in sodium, low in potassium, and drinking too much alcohol) is a systolic pressure of 130 or higher,or a diastolic pressure of 80 or higher, that stays high over time.
[Show full text]
THE SOLUBILITY of GASES in LIQUIDS INTRODUCTION the Solubility Data Project Aims to Make a Comprehensive Search of the Lit- Erat
THE SOLUBILITY OF GASES IN LIQUIDS R. Battino, H. L. Clever and C. L. Young INTRODUCTION The Solubility Data Project aims to make a comprehensive search of the lit erature for data on the solubility of gases, liquids and solids in liquids. Data of suitable accuracy are compiled into data sheets set out in a uni form format. The data for each system are evaluated and where data of suf ficient accuracy are available values recommended and in some cases a smoothing equation suggested to represent the variation of solubility with pressure and/or temperature. A text giving an evaluation and recommended values and the compiled data sheets are pUblished on consecutive pages. DEFINITION OF GAS SOLUBILITY The distinction between vapor-liquid equilibria and the solUbility of gases in liquids is arbitrary. It is generally accepted that the equilibrium set up at 300K between a typical gas such as argon and a liquid such as water is gas liquid solubility whereas the equilibrium set up between hexane and cyclohexane at 350K is an example of vapor-liquid equilibrium. However, the distinction between gas-liquid solUbility and vapor-liquid equilibrium is often not so clear. The equilibria set up between methane and propane above the critical temperature of methane and below the critical temperature of propane may be classed as vapor-liquid equilibrium or as gas-liquid solu bility depending on the particular range of pressure considered and the par ticular worker concerned. The difficulty partly stems from our inability to rigorously distinguish between a gas, a vapor, and a liquid, which has been discussed in numerous textbooks.
[Show full text]
5.5 ENTROPY CHANGES of an IDEAL GAS for One Mole Or a Unit Mass of Fluid Undergoing a Mechanically Reversible Process in a Closed System, the First Law, Eq
Thermodynamic Third class Dr. Arkan J. Hadi 5.5 ENTROPY CHANGES OF AN IDEAL GAS For one mole or a unit mass of fluid undergoing a mechanically reversible process in a closed system, the first law, Eq. (2.8), becomes: Differentiation of the defining equation for enthalpy, H = U + PV, yields: Eliminating dU gives: For an ideal gas, dH = and V = RT/ P. With these substitutions and then division by T, As a result of Eq. (5.11), this becomes: where S is the molar entropy of an ideal gas. Integration from an initial state at conditions To and Po to a final state at conditions T and P gives: Although derived for a mechanically reversible process, this equation relates properties only, and is independent of the process causing the change of state. It is therefore a general equation for the calculation of entropy changes of an ideal gas. 1 Thermodynamic Third class Dr. Arkan J. Hadi 5.6 MATHEMATICAL STATEMENT OF THE SECOND LAW Consider two heat reservoirs, one at temperature TH and a second at the lower temperature Tc. Let a quantity of heat | | be transferred from the hotter to the cooler reservoir. The entropy changes of the reservoirs at TH and at Tc are: These two entropy changes are added to give: Since TH > Tc, the total entropy change as a result of this irreversible process is positive. Also, ΔStotal becomes smaller as the difference TH - TC gets smaller. When TH is only infinitesimally higher than Tc, the heat transfer is reversible, and ΔStotal approaches zero. Thus for the process of irreversible heat transfer, ΔStotal is always positive, approaching zero as the process becomes reversible.
[Show full text]