Page 115 - Physical chemistry eng

92 CHAPTER 5 Entropy and the Second and Third Laws of Thermodynamics

Next consider ¢S for the reversible isothermal expansion or compression of an
T
V
ideal gas, described by ,T : V f , . Because ¢U = 0 for this case,
i
i
i
V f
q reversible =-w reversible = nRT ln and (5.14)
V i
dq reversible 1 V f
¢S = = * q reversible = nR ln (5.15)
T T V i
L
Note that ¢S 7 0 for an expansion (V > V ) and ¢S 6 0 for a compression (V < V ).
i
f
i
f
Although the preceding calculation is for a reversible process, ¢S has exactly the same
value for any reversible or irreversible isothermal path that goes between the same ini-
tial and final volumes and satisfies the condition T = T . This is the case because S is a
f
i
state function.
Why does the entropy increase with increasing V at constant T if the system is viewed
at a microscopic level? As discussed in Section 15.2, the translational energy levels for
atoms and molecules are all shifted to lower energies as the volume of the system
increases. Therefore, more states of the system can be accessed at constant T as V
increases. This is a qualitative argument that does not give the functional form shown in
Equation (5.15). The logarithmic dependence arises because S is proportional to the loga-
rithm of the number of states accessible to the system rather than to the number of states.
Consider next ¢S for an ideal gas that undergoes a reversible change in T at constant
V or P. For a reversible process described by , T : VV i i i , T f , dq reversible = C dT , and
V
dq reversible nC V,m dT T f
¢S = = L nC V,m ln (5.16)
T T T i
L L
For a constant pressure process described by , T : PP i i i , T f , dq reversible = C dT , and
P
dq reversible nC P,m dT T f
¢S = = L nC P,m ln (5.17)
T T T i
L L
The last expressions in Equations (5.16) and (5.17) are valid if the temperature interval
and C can be neglected.
is small enough that the temperature dependence of C V,m P,m
Again, although ¢S has been calculated for a reversible process, Equations (5.16) and
(5.17) hold for any reversible or irreversible process between the same initial and final
states for an ideal gas.
We again ask what a microscopic model would predict for the dependence of S on T.
As discussed in Chapter 30, the probability of a molecule accessing a state with energy E i
is proportional to exp(-E >k T) . This quantity increases exponentially as T increases, so
B
i
that more states become accessible to the system as T increases. Because S is a measure
of the number of states the system can access, it increases with increasing T. Again, the
logarithmic dependence arises because S is proportional to the logarithm of the number of
states accessible to the system rather than to the number of states.
The results of the last two calculations can be combined in the following way.
Because the macroscopic variables V,T or P,T completely define the state of an ideal
V
gas, any change , T : V f , T f can be separated into two segments, ,T : V f ,T i
V
i
i
i
i
and V f , T : V f , T f . A similar statement can be made about P and T. Because S is a
i
state function, ¢S is independent of the path. Therefore, any reversible or irreversible
process for an ideal gas described by , T : VV i i f , T f can be treated as consisting of
two segments, one of which occurs at constant volume and the other of which occurs at
constant temperature. For this two-step process, ¢S is given by
V f T f
¢S = nR ln + nC V,m ln (5.18)
V i T i
Similarly, for any reversible or irreversible process for an ideal gas described by
P i , T : P f , T f
i
T
P f f
¢S =-nR ln + nC P,m ln (5.19)
P i T i