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Chapter 14

The Ideal Gas Law

and Kinetic Theory

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). The reference element is

chosen to be the most abundant isotope of carbon, which is called carbon-12.

kg

u

27

−

×

The atomic mass is given in atomicmass units.

For example, a Li atom

has a mass of 6.941u.

Molecular Mass, the Mole, and Avogadro’s Number

r

particle

particle

M

mole

per

Mass

sample

the

of

mass

m

N

m

N

m

n

A

=

=

=

The mass per mole (in g/mol) of a substancehas the same numerical value as the atomic ormolecular mass of the substance (in atomicmass units).For example Hydrogen has an atomic massof 1.00794 g/mol, while the mass of a singlehydrogen atom is 1.00794 u.Since one gram-mole of a substance contains Avogadro’s number of particles(atoms or molecules), The mass of particle (in grams) can be obtained bydividing the mass per mole (in g/mol) by Avogadro’s number.

A

N

m

mole

per

Mass

particle

=

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 (Al

2

O

3

). One

carat is equivalent to a mass of 0.200 g. Determine (a) the number ofcarbon atoms in the Hope diamond and (b) the number of Al

2

O

3

molecules in the ruby.

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 themolecules are so far apart that they do notinteract except during collisions, which areeffectively elastic. The ideal gas law expressesthe relationship between the absolute pressure,the Kelvin temperature, the volume, and thenumber of moles of the gas.

T

P

∝

At constant volume the pressureis proportional to the temperature.

The Ideal Gas Law

At constant temperature, the pressure isinversely proportional to the volume.

V

P

1

∝

The pressure is also proportionalto the amount of gas.

n

P

∝

The Ideal Gas Law

NkT

T

N

R

N

nRT

PV

A

=

  

  

=

=

A

N

N

n

=

(

)

K

J

10

38 .

1

mol

10

K

mol

J

31 .

8

,

tan

_

23

1

23

−

−

×

=

×

⋅

=

=

A

N

R

k

t

cons

Boltzman

The Ideal Gas Law^ Example 2

Oxygen in the Lungs

In the lungs, the respiratory membrane separates tiny sacs of air(pressure 1.00x

5

Pa) from the blood in the capillaries. These sacs

are 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.

NkT

PV

=

14.

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?

Conceptual Example 3 Beer Bubbles on the Rise Beer bubbles contain mostlycarbon dioxide (CO2), a gasin the beer because of thefermentation process. Thevolume of gas in the bubble isrelated to its temperature,pressure, and the number ofmole of CO2 by the ideal gaslaw. One or more of thesevariable must be responsiblefor the growth of a bubble.

Temperature is constantthroughout.

As the bubble rises, its depthdecreases, and so does thefluid pressure.

Each bubble acts as anucleation site for CO2molecules, so as a bubblemoves upward, itaccumulates carbon dioxidefrom the surrounding andgrow larger

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.

nRT

PV

i

i

i

f

f

f

T

V

P

T

V

P

constant

nR

T

PV

Kinetic Theory of Gases

The particles are in constant, randommotion, colliding with each otherand with the walls of the container.Each collision changes theparticle’s speed.As a result, the atoms andmolecules have differentspeeds.

Kinetic Theory of Gases

THE DISTRIBUTION OF MOLECULAR SPEEDS

Kinetic Theory of Gases

L

mv

F

2

For a single molecule, the average force is:

For N molecules, the average force is:

L

v

m

N

F

2

root-mean-squarespeed

3

2

2

L

v

m

N

F L

A

F

P

volume

Kinetic Theory of Gases

V

v

m

N

P

2

(

)

(

)

2

1 2

2 3

2

1 3

rms

rms

mv

N

mv

N

PV

NkT

KE

kT

mv

rms

3 2

2

1 2

KE

=

=