Physical Chemistry , 1st ed.

can be thought of as a measure of the number of possible places objects can

occupy. This is one way of defining. And when such a number of places are

possible, statistics shows that they will be occupied: a high disorder content is

directly related to a high absolute entropy. The qualitative relationship is very

useful for predictive purposes. But be aware that the relationship also has di-

rect quantitative consequences. We will be able to determine those quantities

shortly—when we find q, the partition function.

The second and third law of thermodynamics can be understood in terms

of the disorder concept of entropy. For an isolated system in which there is no

transfer of mass or energy, a spontaneous change can be thought of as a change

in which the particles of the system access more possible arrangements. That

is, for a spontaneous process from state 1 to state 2,

(state 2) (state 1)

From equation 17.40, which is Boltzmann’s postulate for the definition ofS,

we get for the Sof a spontaneous process:

Skln [ (state 2)] kln [ (state 1)]

Skln ^

(

(

s

s

t

t

a

a

t

t

e

e

2

1

)

)



which is always a positive number: the fraction [ (state 2)]/[ (state 1)] is

always 1, and the logarithm of a number greater than 1 is positive. Thus, a spon-

taneous change occurs with an increase in the total entropy—or “disorder”—of

the system. See Figure 17.11.

For the third law, we can substitute the expression for qinto equation 17.42

and take the derivative ofqwith respect to temperature. We get

Skln ( gie   i/kT) 

T

(^1) 
(^) i
g



i

g



i

e





e



i/k

T

i/kT



If we take the limit of this expression as T→0,* we would find that

limT→ 0 Skln g 0

where g 0 is the degeneracy of the ground state. In the limit ofT→0, the low-

est possible energy states are the only states that are populated.

If the ground state is nondegenerate, then g 0 1 and Sis exactly zero, in

exact agreement with the third law of thermodynamics. This would be strictly

true if there was only a single particle in the system. In most systems, there are

usually enough atoms and molecules that we can speak of their quantities in

molar amounts, that is, on the order of 10^20 and greater. Therefore,g 0 can be

at least 10^20 in real systems. Does this lead to a violation of the third law?

Not really. The logarithm of 10^20 is about 46, and multiplying ln (10^20 ) by

Boltzmann’s constant, 1.381  10 ^23 J/K, gives about 6  10 ^22 J/K—an im-

measurably small amount of entropy (especially considering that molar en-

tropies, which aremeasured, are on the order of dozens or hundreds of J/K,

25 or more orders of magnitude higher). We would need something on the or-

der of 10^10

19

atoms before the entropy at absolute zero would be noticeable,

and to give you an idea of how big that number is, there isn’t room in the vis-

ible universe for that many atoms! Therefore, for all practical purposes, we can

indeed say that Sapproaches zero as the temperature approaches absolute zero,

17.5 Thermodynamic Properties from Statistical Thermodynamics 603

*The limit can be determined by applying L’Hôpital’s rule from calculus.

??

Figure 17.11 Which direction is the sponta-
neous one from a strict disorder perspective? It is
also the spontaneous direction from a strict en-
tropy perspective.