Physical Chemistry , 1st ed.

Equation 3.17 also implies

dS 

d

T

q

 not allowed

The last statement is particularly important: the infinitesimal change in Swill
not be less than dq/T. It may be equal to or greater than dq/T,but it will not
be less than that.
Consider, then, the following description. A process occurs in an isolated
system. Under what conditions will the process occur? If the system is truly
isolated (there is no transfer of energy or matter between system and sur-
roundings), then the process is adiabatic, since isolation implies that q0, and
by extension dq0. Therefore,dq/Tis equal to zero. We can therefore revise
the above statements:

dS  0 if the process is irreversible and spontaneous

dS 0 if the process is reversible

dS 0 is not allowed for a process in an isolated system

We conceptually collect the above three statements into one, which is the sec-
ond law of thermodynamics:

The second law of thermodynamics: For an isolated system, if a

spontaneous change occurs, it occurs with a concurrent increase in

the entropy of the system.

If a spontaneous change does occur, entropy is the sole driving force for that
change because both qand ware zero—and therefore Uis zero—under the
stated conditions.

3.5 More on Entropy

In Example 3.2, we calculated the entropy change for an isothermal process.
What if the process were not isothermal? For a given mass

dqC dT

where Cis the heat capacity, we can substitute for dqin the infinitesimal
change in entropy:

dS

dq

T

revC

T

dT



and then integrate:

SdS

C

T

dT

C

d

T

T

Cln TTTif

for a constant heat capacity. Evaluating at the temperature limits and using the
properties of logarithms:

SCln 

T

T

f

i

 (3.18)

For nmoles, this equation becomes SnCln(Tf/Ti) and Cwill have units
of J/molK. IfChas units of J/gK, then the mass of the system is necessary. If
the heat capacity is not constant over the specified temperature range, then the
temperature-dependent expression for Cmust be included explicitly inside the
integral and the function must be evaluated on a term-by-term basis.

3.5 More on Entropy 75