# THE IDEAL GAS LAW AND KINETIC THEORY

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1 Chapter 14 he Ideal Gas Law and Kinetic heory Chapter 14 HE IDEAL GAS LAW AND KINEIC HEORY REIEW Kinetic molecular theory involves the study of matter, particularly gases, as very small particles in constant motion. Because of the motion of the particles, an ideal gas has internal energy that can be transferred. We study gases by relating their pressure, volume, number of moles, and temperature in the ideal gas law. he content contained in sections 1,,, and 5 of chapter 14 of the textbook is included on the A hysics B exam. QUICK REFERENCE Important erms atomic mass unit one-twelfth the mass of a carbon-1 atom ideal gas law the law which relates the pressure, volume, number of moles, and temperature of an ideal gas internal energy the sum of the potential and kinetic energy of the molecules of a substance kinetic theory of gases the description of matter as being made up of extremely small particles which are in constant motion mole one mole of a substance contains Avogadro s number (6.0 x 10 ) of molecules or atoms 157

2 Chapter 14 he Ideal Gas Law and Kinetic heory Equations and Symbols nr where v i i i KE rms i f f f 1 f ( v) (constant pressure) mv (constant temperature) rms k B R M k B µ en Homework roblems Chapter 14 roblems 9, 10, 11, 14, 4, 6, 8, 9, 4, 5 pressure volume n number of moles R universal gas constant 8.1 J / (mol K) KE average kinetic energy of molecules v rms root-mean-square speed k B Boltzmann constant 1.8 x 10 - J/K Kelvin temperature M molecular mass µ mass of molecule DISCUSSION OF SELECED SECIONS 14.1 he Mole, Avogadro s Number, and Molecular Mass When we are dealing with small particles like atoms and molecules, it is convenient to express their masses in atomic mass units (u) rather than kilograms. Atomic mass units and kilograms are related by the conversion 1 u x 10-7 kg, which is approximately the size of a proton. When we buy 1 eggs we say we have a dozen eggs, but if we have 6.0 x 10 atoms, we say we have a mole of those atoms. In other words, a mole of a substance is Avogadro s number (6.0 x 10 ) of atoms or molecules of that substance. 14. he Ideal Gas Law All gases display similar behavior. When examining the behavior of gases under varying conditions of temperature and pressure, it is most convenient to treat them as ideal gases. An ideal gas represents a hypothetical gas whose molecules have no intermolecular forces, that is, they do not interact with each other, and occupy no volume. Although gases in reality deviate from this idealized behavior, at relatively low pressures and high temperatures many gases behave in nearly ideal fashion. herefore, the assumptions used for ideal gases can be applied to real gases with reasonable accuracy. 158

3 Chapter 14 he Ideal Gas Law and Kinetic heory he state of a gaseous sample is generally defined by four variables: pressure (p), volume (), temperature (), and number of moles (n) though as we will see, these are not all independent. he pressure of a gas is the force per unit area that the atoms or molecules exert on the walls of the container through collisions. he SI unit for pressure is the pascal (a), which is equal to one newton per meter squared. Sometimes gas pressures are expressed in atmospheres (atm). One atmosphere is equal to 10 5 a, and is approximately equal to the pressure the earth s atmosphere exerts on us each day. olume can be expressed in liters (L) or cubic meters (m ), and temperature is measured in Kelvins (K) for the purpose of the gas laws. Recall that we can find the temperature in K by adding 7 to the temperature in Celsius. Gases are often discussed in terms of standard temperature and pressure (S), which refers to the conditions of a temperature of 7 K (0 C) and a pressure of 1 atm. hese four variables are related to each other in the ideal gas law: nr where R is a constant known as the universal gas constant 8.1 J / (mol K). If the number of moles of a gas does not change during a process, then n and R are constants, and we can write the equation as the combined gas law: where a subscript of 1 indicates the state of the gas before something is changed, and the subscript indicates the state of the gas after something is changed. Example 1 A F Fig. I Fig. Fig. I A cylinder is closed at one end with a piston which can slide to change the closed volume of the cylinder. When the piston is at the end of the cylinder, as in Figure I, the volume of the cylinder is 1.0 liter. he area of the piston is 0.01 m. he piston is positioned at half the length of the cylinder in Fig. I, and the cylinder is filled with an ideal gas. he force F necessary to hold the piston in this position is 10 N, and the temperature of the gas is 50 C. 159

4 Chapter 14 he Ideal Gas Law and Kinetic heory (a) Determine the following for the gas in Fig. I: i. the volume of the gas ii. the pressure of the gas A force is applied to the piston so that it is now positioned at one-third the length of the cylinder from its closed end, but the temperature of the gas remains at 50 C. (b) Determine the following for the gas in Fig. : i. the volume of the gas ii. the pressure of the gas (c) he temperature of the gas is raised to 80 C between Fig. and Fig. I. Determine the pressure of the gas in Fig. I. Solution (a) i. he volume of the gas is half of the full cylinder, or 0.5 liter. ii. F N I 1000 a A m (b) i. he volume of the gas is one-third of the full cylinder, or 0. liter. ii. At a constant temperature, the pressure and volume of the gas are inversely proportional according to Boyle s law: ( 1000 a) (0.5l) ( 0.l) I I a (c) Converting Celsius degrees to Kelvins: ( a)( 0.l) ( 1.0 l) I I 50 C + 7 K 80 C K I K I I a I 5 K 14. Kinetic heory of Gases As indicated by the gas laws, all gases show similar physical characteristics and behavior. A theoretical model to explain why gases behave the say they do was developed during the second half of the 19 th century. he combined efforts of Boltzmann, Maxwell, and others led to the kinetic theory of gases, which gives us an understanding of gaseous behavior on a microscopic, molecular level. Like the gas laws, this theory was developed in reference to ideal gases, although it can be applied with reasonable accuracy to real gases as well. he assumptions of the kinetic theory of gases are as follows: 160

5 Chapter 14 he Ideal Gas Law and Kinetic heory Gases are made up of particles whose volumes are negligible compared to the container volume. Gas atoms or molecules exhibit no intermolecular attractions or repulsions. Gas particles are in continuous, random motion, undergoing collisions with other particles and the container walls. Collisions between any two gas particles are elastic, meaning that no energy is dissipated and kinetic energy is conserved. he average kinetic energy of gas particles is proportional to the absolute (Kelvin) temperature of the gas, and is the same for all gases at a given temperature. As listed in the list of equations, the average kinetic energy of each molecule is related to Kelvin temperature by the equation K k B, where k B is the Boltzmann constant, 1.8 x 10 - J/K. he root-mean k B square speed of each molecule can be found by vrms, where µ is the mass of µ each molecule. his equation is very seldom used on the A hysics B exam, and is provided on the exam if needed. Example he temperature of an ideal gas is 60 C. (a) Find the average kinetic energy of the molecules of the gas. (b) On the axes below, sketch a graph of i. average kinetic energy K vs Kelvin temperature ii. root-mean-square speed v rms of each molecule in the gas vs. Kelvin temperature. K v rms 161

6 Chapter 14 he Ideal Gas Law and Kinetic heory Solution K k B (a) K 6.89 x10 ( 1.8 x10 J / K )( 60 C + 7) 1 J (b) he average kinetic energy of each molecule is directly proportional to the Kelvin temperature of the gas, and v rms is proportional to the square root of the Kelvin temperature of the gas: K v rms CHAER 14 REIEW QUESIONS For each of the multiple-choice questions below, choose the best answer. 1. A mole is to Avogadro s number as (A) kinetic energy is to temperature (B) atomic mass unit is to kg (C) gas is to liquid (D) pressure is to volume (E) decade is to ten years. Which of the following is NO true of an ideal gas? (A) Gas molecules have no intermolecular forces. (B) Gas particles are in random motion. (C) Gas particles have no volume. (D) he collisions between any two gas particles are elastic. (E) he average kinetic energy of the gas molecules is proportional to the temperature in Celsius degrees.. A sample of argon occupies 50 liters at standard temperature. Assuming constant pressure, what volume will argon occupy if the temperature is doubled? (A) 5 liters (B) 50 liters (C) 100 liters (D) 00 liters (E) 500 liters 4. What is the final pressure of a gas that expands from 1 liter at 10 C to 10 liters at 100 C if the original pressure was atmospheres? (A) 0. atm (B) 0.4 atm (C) atm (D) 4 atm (E) 0 atm 16

7 Chapter 14 he Ideal Gas Law and Kinetic heory 5. Which of the following pressure vs. volume graphs best represents how pressure and volume change when temperature remains constant? (A) 6. Which of the following volume vs. temperature graphs best represents how volume changes with Kelvin temperature if the pressure remains constant? (A) (B) (B) (C) (C) (D) (D) (E) (E) 16

8 Chapter 14 he Ideal Gas Law and Kinetic heory Free Response Question Directions: Show all work in working the following question. he question is worth 10 points, and the suggested time for answering the question is about 10 minutes. he parts within a question may not have equal weight. 1. (10 points) A special balloon contains moles of an ideal gas and has an initial pressure of x 10 5 a and an initial volume of m. (a) Determine the initial temperature of the gas. (b) he pressure in the balloon is changed in such a way as to increase the volume of the balloon to m but the temperature is held constant. Determine the pressure of the gas in the balloon at this new volume. (c) If the balloon contracts to one-third of its initial volume, and the pressure is increased to twice its initial value, describe the change in temperature that would have to take place to achieve this result. ANSWERS AND EXLANAIONS O CHAER 14 REIEW QUESIONS Multiple Choice 1. E A mole is defined as a certain number of things (6 x 10 ), and a decade is a certain number of years (10).. E All of the statements are true, except that Celsius should be replaced with Kelvin.. C With constant pressure, volume and temperature are proportional to each other, so twice the temperature would result in twice the volume. 164

9 Chapter 14 he Ideal Gas Law and Kinetic heory 4. B First, we must convert Celsius to Kelvin: 1 10 C K 100 C K 1 atm 1 1 l 10 l Solving for and substituting, we get 0.4 atm. 5. E ressure and volume are inversely proportional to each other at constant temperature. 6. A olume and temperature are proportional to each other at constant temperature. Free Response Question Solution (a) 4 points nr nr (b) points 5 ( x10 a)( 0.015m ) ( moles)( 8.1 J / mol K ) For a constant temperature, 1 5 ( x10 a) (0.015m ) ( 0.045m ) 1 1x10 5 a K (c) points In order to keep the ratios constant on both sides, hus, the temperature would have to decrease to 1/6 its initial value for these changes in the pressure and volume to take place. 165

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