Chapter 14

The Ideal Gas Law and Kinetic Theory

14.1 Molecular Mass, the Mole, and Avogadro’s Number

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

The unit is called the atomic mass unit (symbol u).

kg106605.1u 1 27

The atomic mass is given in atomicmass units. For example, a Li atom has a mass of 6.941u.

14.1 Molecular Mass, the Mole, and Avogadro’s Number

One mole of a substance contains as manyparticles as there are atoms in 12 grams ofthe isotope cabron-12.

The number of atoms per mole is known asAvogadro’s number, NA.

123 mol10022.6 AN

AN

Nn

number ofmoles

number ofatoms

14.1 Molecular Mass, the Mole, and Avogadro’s Number

moleper Massparticle

particle m

Nm

Nmn

A

The mass per mole (in g/mol) of a substancehas the same numerical value as the atomic or molecular mass of the substance (in u).

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

14.1 Molecular Mass, the Mole, and Avogadro’s Number

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.

14.1 Molecular Mass, the Mole, and Avogadro’s Number

mol 741.0

molg011.12

carat 1g 200.0carats 5.44

moleper Mass

mn(a)

(b)

mol 271.0molg96.101

carat 1g 200.0carats 138

moleper Mass99.15398.262

mn

atoms1046.4mol10022.6mol 741.0 23123 AnNN

atoms1063.1mol10022.6mol 271.0 23123 AnNN

14.2 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.

TP

At constant volume the pressureis proportional to the temperature.

14.2 The Ideal Gas Law

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

VP 1

The pressure is also proportionalto the amount of gas.

nP

14.2 The Ideal Gas Law

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

14.2 The Ideal Gas Law

NkTTN

RNnRTPV

A

AN

Nn

KJ1038.1

mol106.022

KmolJ31.8 23123

AN

Rk

Boltzmann’s constant

14.2 The Ideal Gas Law

Example 2 Oxygen in the Lungs

In 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.

NkTPV

14.2 The Ideal Gas Law

14

23

33345

109.1

K 310KJ1038.1

m10125.0Pa1000.1

kT

PVN

1314 107.214.0109.1

14.2 The Ideal Gas Law

Conceptual Example 3 Beer Bubbles on the Rise

Watch the bubbles rise in a glass of beer. 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?

14.2 The Ideal Gas Law

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

nRTPV

i

ii

f

ff

T

VP

T

VP

constant nRT

PV

14.2 The Ideal Gas Law

i

ii

f

ff

T

VP

T

VP

Constant T, constant n:iiff VPVP Boyle’s law (At constant T, P and V

are inversely proportional

Constant P, constant n:

i

i

f

f

T

V

T

V

Charles’ law (At constant P, V and Tare proportional)

14.3 Kinetic Theory of Gases

The particles are in constant, randommotion, colliding with each otherand with the walls of the container.

Each collision changes the particle’s speed.

As a result, the atoms and molecules have different speeds.

14.3 Kinetic Theory of Gases

THE DISTRIBUTION OF MOLECULAR SPEEDS

14.3 Kinetic Theory of Gases

KINETIC THEORY

L

mv

vL

mvmv 2

2

collisions successivebetween Time

momentum Initial-momentum Final force Average

t

mv

t

vmmaF

14.3 Kinetic Theory of Gases

L

mvF

2

For a single molecule, the average force is:

For N molecules, the average force is:

L

vmNF

2

3 root-mean-squarespeed

3

2

2 3 L

vmN

L

F

A

FP

volume

14.3 Kinetic Theory of Gases

V

vmNP

2

3

221

322

31

rmsrms mvNmvNPV

NkT KE

kTmvrms 232

21KE

14.3 Kinetic Theory of Gases

Example 6 The Speed of Molecules in Air

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

kTmvrms 232

21

m

kTvrms

3

14.3 Kinetic Theory of Gases

sm511

kg1065.4

K293KJ1038.13326

23

m

kTvrms

For nitrogen…

kg1065.4g1065.4mol106.022

molg0.28 2623123

m

14.3 Kinetic Theory of Gases

kTmvrms 232

21KE

THE INTERNAL ENERGY OF A MONATOMIC IDEAL GAS

nRTkTNU 23

23