Physical Chemistry , 1st ed.

law of thermodynamics says that the change in energy of a system can be

partitioned into heat and work. Using the variables from this chapter, we can

write it as

dEdqdw

(Do not confuse qfor heat with qfor the partition function. Notice, also, that

in this chapter we are using Efor the total (that is, internal) energy and not U

as in Chapters 2 and forward. This is common in statistical thermodynamics.)

For an adiabatic change,dqequals 0, and the pressure-volume work performed

by the system allows us to rewrite the above equation as

dEpdV

If we wanted to bring the dVterm to the other side of the equation, we can

derive an expression for the pressure of a system:

p





V

E

For individual microstates in which the number of particles Niremains the

same, we can rewrite the above equation as

pi





V

(^) i
i

Ni

(17.24)

Since the energy of the microstate, (^) i, depends on the microstate, so does the
pressure pi. If pressure is defined like this, then we can take the derivative of
equation 17.23 with respect to Viand get the average pressure,p , just like we
determined the average energy:
p  (17.25)
However, increasing the number of microstates that have the same average
pressure doesn’t change the value for the overall pressure of the system. The
average pressure of the microstates equals the average pressure of the entire
system. That is,
p pthermo (17.26)
where pthermois the phenomenological, macroscopic,measurablepressure of
our system. Equation 17.26 is the first direct connection between statistical
thermodynamics and phenomenological thermodynamics.
In order to determine , we start by taking the derivative ofE (from equation
17.23) with respect to V, and the derivative ofp (equation 17.25) with respect to
. In both cases, we have to apply the chain rule of derivation, and certain sub-
stitutions can be made. Ultimately (the steps won’t be given here), we get




V

E

p (E p ) E p (17.27)







(^) p
E^ p^ (E^ p^ ) (17.28)
Substituting and rearranging:




V

E









p

p (17.29)



i 







V

(^) i
i

Ni
gie i


i
gie i
598 CHAPTER 17 Statistical Thermodynamics: Introduction