Physical Chemistry , 1st ed.

The Nand qterms cancel:



N

N

k

i

g

g

k

ie

e









i

k

and we can combine the two exponentials algebraically:



N

N

k

i

g

g

k

ie(^ i k)

This expression is usually written as



N

N

k

i

g

g

k

i e    (17.21)

where  is the difference in energies of the ith and kth states. Notice how the

degeneracies do not automatically cancel.

A fractional population is numerically equivalent to a probability. The prob-

ability that any individual particle selected at random will be in the ith energy

state is therefore

Pi

1

q

gie i (17.22)

The reason we point this out is that now we can use some statistical per-

spectives to understand thermodynamic properties. For example, with the

ideas from section 17.2, we can use

possible

u

because we have an expression for Pi. Suppose we want to know what the av-

erage energy values of the microstates are. We can rewrite the above equa-

tion as

E 

where E is the average energy and (^) iis the energy of each individual state.
Because qis a constant for a given set of conditions, it can be factored out of
every term in each sum and then canceled from both the numerator and de-
nominator. Therefore,

E  (17.23)

Thus, we have a way to calculate the average energy E from a statistical con-

sideration of the energies of the individual particles in the system. Further, we

postulate that the average energy E is equal to the thermodynamic energy E of

the system.

We need to determine what is. In order to do this, we will have to use

some equations from phenomenological thermodynamics. Recall that the first



i

(^) igie i


i
gie i

i
(^) i

1

q

gie i





i



1

q

gie i



values

i 1

uiPi





i

Pi

17.4 The Most Probable Distribution: Maxwell-Boltzmann Distribution 597