Page 72 - Physical Chemistry

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Since U and V are extensive, H is extensive. The molar enthalpy of a pure sub- Section 2.6
stance is H H/n (U PV)/n U PV . Heat Capacities
m
m
m
Consider now a constant-volume process. If the closed system can do only P-V
work, then w must be zero, since no P-V work is done in a constant-volume process.
The first law U q w then becomes for a constant-volume process
¢U q closed syst., P-V work only, V const. (2.49)
V
where q is the heat absorbed at constant volume. Comparison of (2.49) and (2.46)
V
shows that in a constant-pressure process H plays a role analogous to that played by
U in a constant-volume process.
From Eq. (2.47), we have H U (PV). Because solids and liquids have
comparatively small volumes and undergo only small changes in volume, in nearly all
processes that involve only solids or liquids (condensed phases) at low or moderate
pressures, the (PV) term is negligible compared with the U term. (For example,
recall the example in Sec. 2.4 of heating liquid water, where we found U q .) For
P
condensed phases not at high pressures, the enthalpy change in a process is essentially
the same as the internal-energy change: H U.

2.6 HEAT CAPACITIES
The heat capacity C of a closed system for an infinitesimal process pr is defined as
pr
C dq >dT (2.50)*
pr
pr
where dq and dT are the heat flowing into the system and the temperature change of
pr
the system in the process. The subscript on C indicates that the heat capacity depends
on the nature of the process. For example, for a constant-pressure process we get C ,
P
the heat capacity at constant pressure (or isobaric heat capacity):
dq
C P (2.51)*
P
dT
Similarly, the heat capacity at constant volume (or isochoric heat capacity) C of a
V
closed system is
dq
C V (2.52)*
V
dT
where dq and dT are the heat added to the system and the system’s temperature change
V
in an infinitesimal constant-volume process. Strictly speaking, Eqs. (2.50) to (2.52)
apply only to reversible processes. In an irreversible heating, the system may develop
temperature gradients, and there will then be no single temperature assignable to the
system. If T is undefined, the infinitesimal change in temperature dT is undefined.
Equations (2.46) and (2.49) written for an infinitesimal process give dq dH at
P
constant pressure and dq dU at constant volume. Therefore (2.51) and (2.52) can
V
be written as
0H 0U
C a b , C a b closed syst. in equilib., P-V work only (2.53)*
P
V
0T P 0T V
C and C give the rates of change of H and U with temperature.
P V
To measure C of a solid or liquid, one holds it at constant pressure in an adiabat-
P
ically enclosed container and heats it with an electrical heating coil. For a current I
flowing for a time t through a wire with a voltage drop V across the wire, the heat gen-
erated by the coil is VIt. If the measured temperature increase T in the substance is
small, Eq. (2.51) gives C VIt/ T, where C is the value at the average temperature
P P