Page 55 - Modern physical chemistry

44 Gases and Collective Properties

A solution in which this interaction is negligible, except insofar as it promotes ran-
domness, is said to be ideal. For the ith gas in such a solution, we have

[3.33]
where Pi is the partial pressure exerted by the ni moles of gas i in the volume Vat tem-
perature T.
Summing (3.33) over all gases in the solution yields

[3.34]

Since the sum of the partial pressures is the total pressure P and the sum of the partial
moles the total moles n, equation (3.34) is equivalent to
PV=nRT. [3.35]

But dividing (3.33) by (3.35) gives us
Pi = ni [3.36]
P n
whence
n·
Pi =_t P=XiP. [3.37]
n
In the last step, the ith mole fraction n/n is replaced by the symbol Xi'
This result checks (3.19), which was derived with the assumption that the average
pressure caused by a molecule of A in volume V equals the average pressure caused by
a molecule of B in that volume, under ideal conditions.

Example 3.6
Two moles N 2 , ten moles O 2 , and three moles CO 2 are mixed. Calculate the partial
pressure of each component when the total pressure is 720 torr.
Apply equation (3.37) to each constituent. For the nitrogen,

P N2 = ~ torr = 96 torr,
720
15.00
for the oxygen,
P 02 = 10.00 720 torr = 480 torr,
15.00
and for the carbon dioxide,
P C02 = 3.00 720 torr = 144 torr.
15.00

3.6 Translational, Rotational, Vibrational Energies

Each molecule in a gas executes various kinds of motion. These include translation
of its center of mass, rotation about the center of mass, vibration in one or more modes.
To describe translation in general requires three independent coordinates. To describe
rotation of a linear molecule requires two additional coordinates; for a nonlinear mole-
cule three independent additional coordinates are needed. The total number of inde-
pendent coordinates for a molecule equals three times the number of atoms a. The number
of vibrational modes equals 3a - 5 if the molecule is linear, 3a - 6 if it is not linear.