Page 207 - Theory and Problems of BEGINNING CHEMISTRY

196 KINETIC MOLECULAR THEORY [CHAP. 13

friction in molecular collisions. The molecules have the same total kinetic energy after each collision as before.
Postulate 4 concerns the volume of the molecule themselves versus the volume of the container they occupy.
The individual particles do not occupy the entire container. If the molecules of gas had zero volumes and zero
intermolecular attractions and repulsions, the gas would obey the ideal gas law exactly. Postulate 5 means that
if two gases are at the same temperature, their molecules will have the same average kinetic energies.

EXAMPLE 13.1. Calculate the value for k, the Boltzmann constant, using the following value for R:
R = 8.31 J/(mol·K)

R 8.31 J/(mol·K) 1.38 × 10 −23 J
Ans.
k = = =
23
N 6.02 × 10 molecules/mol molecule·K
EXAMPLE 13.2. Calculate the average kinetic energy of H 2 molecules at 1.00 atm and 300 K.
3
Ans. KE = kT = 1.5[1.38 × 10 −23 J/(molecule·K)](300 K) = 6.21 × 10 −21 J
2

13.3. EXPLANATION OF GAS PRESSURE, BOYLE’S LAW, AND CHARLES’ LAW
Kinetic molecular theory explains why gases exert pressure. The constant bombardment of the walls of the
vessel by the gas molecules, like the hitting of a target by machine gun bullets, causes a constant force to be
applied to the wall. The force applied, divided by the area of the wall, is the pressure of the gas.
Boyle’s law may be explained using the kinetic molecular theory by considering the box illustrated in
Fig. 13-1. If a sample of gas is placed in the left half of the box shown in the figure, it will exert a certain pressure.
If the volume is doubled by extending the right wall to include the entire box shown in the figure, the pressure
should fall to one-half its original value. Why should that happen? In an oversimplified picture, the molecules
bouncing back and forth between the right and left walls now have twice as far to travel, and thus they hit each
wall only one-half as often in a given time. Therefore, the pressure is only one-half what it was. How about the
molecules that are traveling up and down or in and out? There are as many such molecules as there were before,
and they hit the walls as often; but they are now striking an area twice as large, and so the pressure is one-half
what it was originally. Thus, doubling the volume halves the pressure. This can be shown to be true no matter
what the shape of the container.

Fig. 13-1. Explanation of Boyle’slaw

Charles’ law governs the volume of a gas at constant pressure when its temperature is changed. When the
absolute temperature of a gas is multiplied by 4, for example, the average kinetic energy of its molecules is also
1
2
multiplied by 4 (postulate 5). The kinetic energy of any particle is given by KE = mv , where m is the mass
2
of the particle and v is its velocity. When the kinetic energy is multiplied by 4, what happens to the velocity?
It is doubled. (Since the v term is being squared, to effect a fourfold increase, you need only to double the
2
velocity: 2 = 4.)
1 2
KE 1 = mv 1
2
2
1
1
KE 2 = mv 2 = 4KE 1 = m(2v 1 ) 2
2 2
The velocity v 2 is equal to 2v 1 . On average, in a sample of gas the molecules are going twice as fast at the higher
temperature. They therefore hit the walls (1) twice as often per unit time and (2) twice as hard each time they
hit, for a combined effect of 4 times the pressure (in a given volume). If we want a constant pressure, we have
to expand the volume to 4 times what it was before, and we see that multiplying the absolute temperature by 4
must be accompanied by a fourfold increase in volume if the pressure is to be kept constant.