Page 116 - Physical chemistry eng

5.4 CALCULATING CHANGES IN ENTROPY 93

In writing Equations (5.18) and (5.19), it has been assumed that the temperature
dependence of C V,m and C P,m can be neglected over the temperature range of interest.

EXAMPLE PROBLEM 5.3
Using the equation of state and the relationship between C P,m and C V,m for an ideal gas,
show that Equation (5.18) can be transformed into Equation (5.19).

Solution
V f T f T P T f
f i
¢S = nR ln + nC V,m ln = nR ln + nC V,m ln
V i T i T P T i
i f
P f T f P f T f
=-nR ln + n(C V,m + R)ln =-nR ln + nC P,m ln
P i T i P i T i

Next consider ¢S for phase changes. Experience shows that a liquid is con-
verted to a gas at a constant boiling temperature through heat input if the process is
carried out at constant pressure. Because q =¢H , ¢S for the reversible process
P
is given by
dq reversible q reversible ¢H vaporization
¢S vaporization = = = (5.20)
T T T
L vaporization vaporization
Similarly, for the phase change solid : liquid,

dq reversible q reversible ¢H fusion
¢S fusion = = = (5.21)
T T T
L fusion fusion
Finally, consider ¢S for an arbitrary process involving real gases, solids, and liq-
b
uids for which the isobaric volumetric thermal expansion coefficient and the isother-
k
mal compressibility , but not the equation of state, are known. The calculation of ¢S
for such processes is described in Supplemental Sections 5.12 and 5.13, in which the
properties of S as a state function are fully exploited. The results are stated here. For the
V
system undergoing the change , T : V f , T f ,
i
i
T f V f
C V b T f b
¢S = dT + dV = C V ln + (V f - V ) (5.22)
i
T k T k
3 3 i
T i V i
In deriving the last result, it has been assumed that and bk are constant over the tem-
perature and volume intervals of interest. For the system undergoing a change P , i
T : P f , T , f
i
T f P f
C P
¢S = dT - Vb dP (5.23)
T
3 3
T i P i
For a solid or liquid, the last equation can be simplified to
T f
¢S = C ln - Vb(P - P ) (5.24)
f
i
P
T i
if C P , V, and b are assumed constant over the temperature and pressure intervals of
interest. The integral forms of Equations (5.22) and (5.23) are valid for ideal and real
gases, liquids, and solids. Examples of calculations using these equations are given in
Example Problems 5.4 through 5.6.