Page 208 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 13] KINETIC MOLECULAR THEORY 197

13.4. GRAHAM’SLAW
An experimental law not yet discussed is Graham’s law, which states that the rate of effusion or diffusion
of a gas is inversely proportional to the square root of its molar mass. Effusion is the passage of a gas through
small holes in its container, such as the pores in a porous cup. The deflation of a helium-filled party balloon
over several days results from the helium atoms effusing through the tiny pores of the balloon wall. Diffusion
is the passage of a gas through another gas. For example, if a bottle of ammonia water is spilled in one corner
of a room, the odor of ammonia is soon apparent throughout the room. The ammonia molecules have diffused
through the air molecules. Consider two gases with molar masses MM 1 and MM 2 . The ratio of their rates of
effusion or diffusion is given by

r 1 MM 2
=
r 2 MM 1
That is, the heavier a molecule of the gas, the more slowly it effuses or diffuses.
The rate of effusion or diffusion of a gas is directly proportional to the “average” velocity of its molecules.

EXAMPLE 13.3. A sample of nitrogen and a sample of neon are both at the same temperature. What is the ratio of the
“average” velocities of their molecules?
Ans. Since the temperatures are the same, so are the average kinetic energies of their molecules. From Graham’slaw,

v Ne MM nitrogen 28.0
= = = 1.18
v nitrogen MM Ne 20.2
The neon atoms are moving 1.18 times as fast as the (heavier) nitrogen molecules.
Graham’s law may be explained in terms of the kinetic molecular theory as follows: Since two gases are at
the same temperature, their average kinetic energies are the same:
1 2 1 2
KE 1 = KE 2 = m 1 v 1 = m 2 v 2
2 2
Multiplying the last of these equations by 2 yields
2
m 1 v 2
2 2
m 1 v 1 = m 2 v 2 or =
m 2 v 1 2
Since the masses of the molecules are proportional to their molar masses, and the average velocity of the molecules
is a measure of the rate of effusion or diffusion, all we have to do to this equation to get Graham’s law is to take
2
its square root. (The square root of v is not quite equal to the average velocity, but is a quantity called the root
mean square velocity. See Problem 13.17.)

Solved Problems

POSTULATES OF THE KINETIC MOLECULAR THEORY
◦
13.1. (a) Calculate the volume at 100 C of 18.0 g of liquid water, assuming the density to be 1.00 g/mL.
◦
(b) Calculate the volume of 18.0 g of water vapor at 100 C and 1.00-atm pressure, using the ideal gas
law. (c) Assuming that the volume of the liquid is the total volume of the molecules themselves, calculate
the percentage of the gas volume occupied by molecules.
1.00 mL

Ans. (a)18.0g = 18.0mL = 0.0180 L
1.00 g
nRT (1.00 mol)[0.0821 L·atm/(mol·K)](373 K)
(b) V = = = 30.6L
P 1.00 atm
(c) The percentage occupied by the molecules is
0.0180 L
× 100% = 0.0588%
30.6L