Page 52 - Modern physical chemistry

3.3 Temperature 41

heat can no longer flow, on the average, and the bodies are said to have reached the same
temperature. This induction is embodied in the zeroth law of thermodynamics: When-
ever two or more bodies contact each other, heat flows between them until their tem-
peratures become equal. One then says that thermal equilibrium has been established.
But what is temperature? From our standpoint, temperature is a statistical property
to be defined in terms of the average behavior of molecules.
Consider a mixture of two gases A and B in which the molecules do not interact appre-
ciably except at collisions. Such a mixture is said to be ideal.
For gas A in the mixture, equation (3.15) yields

P A V 2(Etr )A
[3.16]
NA = 3N A '

while for gas B, we have
PBV = 2(Etr )B [ 3.17]

NB 3NB
Here P A and PB are the partial pressures of A and B in the mixture, while NA and NB
are the number of molecules of A and B and (Etr)A and (Etr)B are the translational kinetic
energies of gas A and gas B in the mixture.
At equilibrium, the average pressure exerted by a molecule of A equals the average pres-
sure exerted by a molecule of B:
PA = PB =~ [3.18]
NA NB N
Here P and N are the total pressure and the total number of molecules. Solving for the
partial pressures leads to

NB
PB=-P=XBP, [3.19]
N
where
NA -x NB -X [3.20]
N - A, N - B,

by definition. Quantities X A and X B give the mole fractions of A and B in the mixture.
Combining (3.18) with (3.16) and (3.17) yields the equality
(Etr ) (Etr )
__ A_= __ [3.21 ]
B
NA NB
At equilibrium, the average translational energy of a molecule of A equals the average
translational energy of a molecule of B.
Gases A and B are at the same temperature since they are intimately mixed. But there
is no reason why the translational energy should be distributed differently were the gases
separated by a heat conducting membrane. Consequently, two ideal gases are said to be
at the same temperature when they exhibit the same average translational energy per
molecule, Et/N. When they are at different temperatures, Et/N differs.
One can thus take Et/N as a measure of temperature T. By convention, a simple pro-
portionality is assumed; so
[3.22]