Chapter 14

The Ideal Gas Law

& Kinetic Theory

AP Learning ObjectivesKinetic theory Ideal gases

Students should understand the kinetic theory model of an ideal gas, so they can:

State the assumptions of the model. State the connection between temperature and mean

translational kinetic energy, and apply it to determine the mean speed of gas molecules as a function of their mass and the temperature of the gas.

State the relationship among Avogadro’s number, Boltzmann’s constant, and the gas constant R, and express the energy of a mole of a monatomic ideal gas as a function of its temperature.

Explain qualitatively how the model explains the pressure of a gas in terms of collisions with the container walls, and explain how the model predicts that, for fixed volume, pressure must be proportional to temperature.

Table of Contents

1. Molecular Mass, the Mole, Avogadro’s Number

2. Ideal Gas law

3. Kinetic Theory of Gases

4. Diffusion (AP?)

Chapter 14:Ideal Gas Law & Kinetic Theory

Section 1:

Molecular Mass, the Mole, and Avogadro’s Number

Atomic Mass

To facilitate comparison of the mass of one atom with another, a mass scale know as the atomic mass scale has been established.

The unit is called the atomic mass unit (symbol u). The reference element is chosen to be the most abundant isotope of carbon, which is called carbon-12

Carbon-12 is defined at 12 u The atomic mass is given in atomic mass units. For

example, a Li atom has a mass of 6.941u.

kg106605.1u 1 27

123 mol10022.6 AN

AN

Nn

number ofmoles

number ofatoms

The Mole & Avogadro’s Number One mole of a substance contains as many particles as

there are atoms in 12 grams of the isotope cabron-12. The number of atoms per mole is known as

Avogadro’s number, NA.

M

AN

Nn

Molar Mass

μ is mass of the molecule M is molar mass The molar mass (in g/mol) of a

substance has the same numerical value as the atomic or molecular mass of the substance (in atomic mass units).

For example Hydrogen has an atomic mass of 1.00794 g/mol, while the mass of a single hydrogen atom is 1.00794 u.

Example 1 The Hope Diamond and the Rosser Reeves Ruby

The Hope diamond (44.5 carats) is almost pure carbon. The RosserReeves ruby (138 carats) is primarily aluminum oxide (Al2O3). Onecarat is equivalent to a mass of 0.200 g. Determine (a) the number ofcarbon atoms in the Hope diamond and (b) the number of Al2O3 molecules in the ruby.

M

n(a)

(b) molg96.101

carat 1g 200.0carats 138

AnNN

AnNN

molg011.12

carat 1g 200.0carats 5.44 mol741.0

123 mol10022.6mol 741.0 atoms1046.4 23

M

n

99.15398.262

mol 271.0

123 mol10022.6mol 271.0 atoms1063.1 23

14.1.1. In 1865, Loschmidt calculated the number of molecules in a cubic centimeter of a gas under standard temperature and pressure conditions. He later used this number to estimate the size of an individual gas molecule. Calculate Loschmidt’s number for helium using the density, the atomic mass, and Avogadro’s number.

a) 2.7 1019 atoms/cm3

b) 3.5 1020 atoms/cm3

c) 4.1 1021 atoms/cm3

d) 5.4 1022 atoms/cm3

e) 6.2 1023 atoms/cm3

14.1.2. Suppose that molecules of water (molecular mass = 0.01802 kg/m3) completely fill a container so that there is no empty space within the container. Using the density of water and Avogadro’s number, estimate the size of the water molecule. Hint: assume the water molecule fits within a cube and that these cubes are stacked to fill the volume of the container.

a) 4 1011 m

b) 2 109 m

c) 3 1010 m

d) 5 109 m

e) 6 1010 m

14.1.3. The standard for determining atomic masses is the carbon-12 atom, so that the mass of one mole of carbon-12 is exactly twelve grams. What would Avogadro’s number and the atomic mass of oxygen-16 be if the standard were that one mole of hydrogen is exactly one gram?

a) 6.020 1023 mol1, 15.9898 grams

b) 6.069 1023 mol1, 16.1200 grams

c) 5.844 1023 mol1, 15.7845 grams

d) 5.975 1023 mol1, 15.8707 grams

e) 6.122 1023 mol1, 16.3749 grams

14.1.4. Under which of the following circumstances does a real gas behave like an ideal gas?

a) The gas particles move very slowly.

b) The gas particles do not collide with each other very often.

c) The interaction between the gas particles is negligible.

d) The interaction between the gas particles and the walls of the container is negligible.

e) There are only one kind of particles in the container.

Chapter 14:Ideal Gas Law & Kinetic Theory

Section 2:

The Ideal Gas Law

TP

The Ideal Gas Law

An ideal gas is an idealized model for real gases that have sufficiently low densities.

The condition of low density means that the molecules are so far apart that they do not interact except during collisions, which are effectively elastic.

At constant volume the pressure is proportional to the temperature.

At constant temperature, the pressure is inversely proportional to the volume.

VP 1

The pressure is also proportionalto the amount of gas.

nP

The Ideal Gas Law

The absolute pressure of an ideal gas is directly proportional to the Kelvintemperature and the number of moles of the gas and is inversely proportionalto the volume of the gas.

V

nRTP

nRTPV

KmolJ31.8 R

The Ideal Gas Law

V

nTP

TNkB

AN

Nn

KJ1038.1

mol106.022

KmolJ31.8 23123

A

B N

Rk

The Ideal Gas Law

nRTPV TN

RN

A

Boltzmann’s Constant

TNknRTPV B

Example 2 Oxygen in the LungsIn the lungs, the respiratory membrane separates tiny sacs of air(pressure 1.00x105Pa) from the blood in the capillaries. These sacsare called alveoli. The average radius of the alveoli is 0.125 mm, andthe air inside contains 14% oxygen. Assuming that the air behaves asan ideal gas at 310K, find the number of oxygen molecules in one ofthese sacs.

TNkPV B

Tk

PVN

B

21314 O of molecules107.214.0109.1 N

K 310KJ1038.1

m10125.0Pa1000.123

33345

air of molecules109.1 14N

Conceptual Example 3 Soda Bubbles on the Rise

Watch the bubbles rise in a glass of soda. If you look carefully, you’llsee them grow in size as they move upward, often doubling in volumeby the time they reach the surface. Why does the bubble grow as itascends?

Consider a sample of an ideal gas that is taken from an initial to a finalstate, with the amount of the gas also changing.

nRTPV ii

ii

ff

ff

Tn

VP

Tn

VPconstant R

nT

PV

Other Gas Laws

Constant T, constant n: iiff VPVP Boyle’s law

Constant P, constant n:i

i

f

f

T

V

T

V Charles’ law

Constant P, constant T:i

i

f

f

n

V

n

V Avogadro’s law

Constant V, constant n:i

i

f

f

T

P

T

P Gay-Lussac’s law

14.2.1. Using the ideal gas law, estimate the approximate number of air particles within an otherwise empty room that has a height of 2.5 m, a width of 4.0 m, and a length of 5.0 m.

a) 1.2 1027

b) 6.8 1025

c) 3.0 1026

d) 2.5 1028

e) 9.1 1024

14.2.2. An ideal gas is enclosed within a container by a moveable piston. If the final temperature is two times the initial temperature and the volume is reduced to one-fourth of its initial value, what will the final pressure of the gas be relative to its initial pressure, P1?

a) 8P1

b) 4P1

c) 2P1

d) P1/2

e) P1/4

14.2.3. Consider a commercial sightseeing hot air balloon that carries a basket with more than 20 passengers. Assume that balloon contains 1.5 × 104 m3 of air. Estimate the order of magnitude of the number of air molecules inside the balloon.

a) 1023

b) 1029

c) 1035

d) 1018

e) 104

Chapter 14:Ideal Gas Law & Kinetic Theory

Section 3:

The Kinetic Theory of Gases

Postulates of Kinetic(-Molecular) Theory All gases are made up of particles

Usually molecules The particles are in constant, random

motion, colliding with each other and with the walls of the container.

All collisions are perfectly elastic Volume of the particles is insignificant There are no interactions between

particles (attraction/repulsion) The average kinetic energy of the

particles is a function of only absolute temperature

THE DISTRIBUTION OF MOLECULAR SPEEDS

collisions successivebetween Time

momentum Initial-momentum Final F

t

mv

t

vmmaF

Kinetic Theory

L

mv2

vL

mvmv

2F

L

mvF

2

For a single molecule, the average force is:

For N molecules in 3 dimensions, the average force is:

L

vmNF

2

3 root-mean-squarespeed

A

FP

volume

Kinetic Theory

2L

F

3

2

3 L

vmN

V

vmNP

2

3

231

rmsmvNPV

TNkB K

TkB23K

Kinetic Theory

221

32

rmsmvN

Tk

M

RTv B

rms

33

Molar mass

Conceptual Example 5 Does a Single Particle Have a Temperature?

Each particle in a gas has kinetic energy. On the previous page, we haveestablished the relationship between the average kinetic energy per particleand the temperature of an ideal gas.

Is it valid, then, to conclude that a single particle has a temperature?

No, the temperature relates to the average of the whole sample, as there is one temperaturefor the sample.

Example 6 The Speed of Molecules in AirAir is primarily a mixture of nitrogen N2 molecules (molecular mass 28.0 u) and oxygen O2 molecules (molecular mass 32.0 u). Assumethat each behaves as an ideal gas and determine the rms speedsof the nitrogen and oxygen molecules when the temperature of the airis 293K.

M

RTvrms

3

molM

RTvrms kg 8002.0

K293KmolJ31.833

For nitrogen…

molM

RTvrms kg 2003.0

K293KmolJ31.833

For oxygen…

sm511

sm478

Tkmv Brms 232

21K

nRTkTNU 23

23

Internal Energy of a Monatomic Ideal Gas

14.3.1. Two sealed containers, labeled A and B as shown, are at the same temperature and each contain the same number of moles of an ideal monatomic gas. Which one of the following statements concerning these containers is true?

a) The rms speed of the atoms in the gas is greater in B than in A.

b) The frequency of collisions of the atoms with the walls of container B are greater than that for container A.

c) The kinetic energy of the atoms in the gas is greater in B than in A.

d) The pressure within container B is less than the pressure inside container A.

e) The force that the atoms exert on the walls of container B are greater than in for those in container A.

14.3.2. Two identical, sealed containers have the same volume. Both containers are filled with the same number of moles of gas at the same temperature and pressure. One of the containers is filled with helium gas and the other is filled with neon gas. Which one of the following statements concerning this situation is true?

a) The speed of each of the helium atoms is the same value, but this speed is different than that of the neon atoms.

b) The average kinetic energy of the neon atoms is greater than that of the helium atoms.

c) The pressure within the container of helium is less than the pressure in the container of neon.

d) The internal energy of the neon gas is greater than the internal energy of the helium gas.

e) The rms speed of the neon atoms is less than that of the helium atoms.

14.3.3. A monatomic gas is stored in a container with a constant volume. When the temperature of the gas is T, the rms speed of the atoms is vrms. What is the rms speed when the gas temperature is increased to 3T?

a) vrms/9

b) c) 3vrms

d) vrms

e) 9vrms

rms / 3v

3

14.3.4. Closed containers A and B both contain helium gas at the same temperature. There are n atoms in container A and 2n atoms in container B. At time t = 0 s, all of the helium atoms have the same kinetic energy. The atoms have collisions with each other and with the walls of the container. After a long time has passed, which of the following statements will be true?

a) The atoms in both containers have the same kinetic energies they had at time t = 0 s.

b) The atoms in both containers have a wide range of speeds, but the distributions of speeds are the same for both A and B.

c) The average kinetic energy for atoms in container B is higher than that for container A.

d) The average kinetic energy for atoms in container A is higher than that for container B.

e) The atoms in both containers have a wide range of speeds, but the distributions of speeds has a greater range for container B than that for container A.

14.3.5. Assume that you have a container with 0.25 kg of helium gas at 20 C. How much energy must be added to the gas to increase its temperature to 70 C?

a) 4 × 104 J

b) 2 × 105 J

c) 5 × 106 J

d) 1 × 107 J

e) 3 × 108 J

Chapter 14:Ideal Gas Law & Kinetic Theory

Section 4:

Diffusion

The process in which molecules move from a region of higher concentrationto one of lower concentration is called diffusion.

Diffusion

Conceptual Example 7 Why Diffusion is Relatively Slow

A gas molecule has a translational rms speed of hundreds of metersper second at room temperature. At such speed, a molecule could travel across an ordinary room in just a fraction of a second. Yet, it often takes several seconds, and sometimes minutes, for the fragrance of a perfume to reach the other side of the room. Why does it take solong?

A Transdermal Patch

L

tCDAm

FICK’S LAW OF DIFFUSION

The mass m of solute that diffuses in a time t through a solvent containedin a channel of length L and cross sectional area A is

concentration gradientbetween ends

diffusion constant

SI Units for the Diffusion Constant: m2/s

Example 8 Water Given Off by Plant Leaves

Large amounts of water can be given off byplants. Inside the leaf, water passes from theliquid phase to the vapor phase at the wallsof the mesophyll cells.

The diffusion constant for water is 2.4x10-5m2/s.A stomatal pore has a cross sectional area of about 8.0x10-11m2 and a length of about 2.5x10-5m. The concentration on the interiorside of the pore is roughly 0.022 kg/m3, whilethat on the outside is approximately 0.011 kg/m3.

Determine the mass of water that passes throughthe stomatal pore in one hour.

kg100.3

m102.5

s 3600mkg011.0mkg022.0m100.8sm104.2

9

5-

3321125

L

tCDAm

14.4.1. Sealed containers with a valve in the middle contain equal amounts of two different monatomic gases at room temperature as shown. A few of the gas atoms are illustrated with arrows representing their velocities. Which of the following statements concerning the gases after the valve has been opened is correct?

a) Because there is no pressure difference, the two gases will remain separated for the most part. Only a few atoms of each gas will be exchanged.

b) As soon as the valve is opened, the two gases will mix completely.

c) The two gases will collide in the narrow tube and be scattered back into their original chambers, so no mixing will occur.

d) After a relatively long period of time, the two gases will be well mixed in both containers.

e) The lighter of the two gases will occupy both spherical containers, but the heavier gas atom will mostly remain in their original container.