Page 209 - Theory and Problems of BEGINNING CHEMISTRY

198 KINETIC MOLECULAR THEORY [CHAP. 13

13.2. If two different gases are at the same temperature, which of the following must also be equal, (a) their
pressures, (b) their average molecular velocities, or (c) the average kinetic energies of their molecules?
Ans. (c) Their average kinetic energies must be equal since the temperatures are equal.

13.3. Does the kinetic molecular theory state that all the molecules of a given sample of gas have the same
velocity since they are all at one temperature?

Ans. No. The kinetic molecular theory states that the average kinetic energy is related to the temperature, not
the velocity or kinetic energy of any one molecule. The velocity of each individual molecule changes as it
strikes other molecules or the walls.

13.4. Calculatethe temperatureat whichCO 2 molecules have the same“average”velocity as nitrogen molecules
have at 273 K.
2
Ans. Let v = v 2 = v 2
N 2 CO 2
v 2 44.0
T CO 2 KE CO 2 m CO 2 m CO 2
= = = = = 1.57
v 2 28.0
T N 2 m N 2
KE N 2 m N 2
= 1.57(273 K) = 429 K
T CO 2 = 1.57T N 2
13.5. If the molecules of a gas are compressed so that their average distance of separation gets smaller, what
should happen to the forces between them? To their ideal behavior?
Ans. As the average distance gets smaller, the intermolecular forces increase. As the intermolecular forces increase
and the volume of the molecules themselves becomes a more significant fraction of the total gas volume,
their behavior becomes further from ideal.

EXPLANATION OF GAS PRESSURE, BOYLE’S LAW, AND CHARLES’ LAW
13.6. Suppose that we double the length of each side of a rectangular box containing a gas. (a) What will
happen to the volume? (b) What will happen to the pressure? (c) Explain the effect on the pressure on
the basis of the kinetic molecular theory.
3
Ans. (a) The volume will increase by a factor of (2) = 8. (b) The pressure will fall to one-eighth its original
value. (c) In each direction, the molecules will hit the wall only one-half as often, and the force on each
wall will drop to one-half of what it was originally because of this effect. Each wall has 4 times the area, and
so the pressure will be reduced to one-fourth its original value because of this effect. The total reduction in
1
pressure is 1 × 1 = , in agreement with Boyle’slaw.
2 4 8
GRAHAM’SLAW
13.7. (a) If the velocity of a single gas molecule doubles, what happens to its kinetic energy? (b) If the average
velocity of the molecules of a gas doubles, what happens to the temperature of the gas?
Ans. (a) v 2 = 2v 1
1
1
2
2
KE 2 = mv = m(2v 1 ) = 4 1 mv 2 = 4KE 1
2 2 2 2 1
The kinetic energy is increased by a factor of 4.
(b) The absolute temperature is increased by a factor of 4.
13.8. Would it be possible to separate isotopes by using the principle of Graham’s law? Explain what factors
would be important.
Ans. Since the molecules containing different isotopes have different masses, it is possible to separate them on the
basis of their different “average” molecular velocities. One would have to have gaseous molecules in which
the element being separated into its isotopes was the only element present in more than one isotope. For
example, if uranium is being separated, a gaseous compound of uranium is needed. The compound could
be made with chlorine perhaps, but chlorine exists naturally in two isotopes of its own, and many different