Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.7 SURFACEWORK 143

Note thatCV is a state function whose value depends on the state of the system—that
is, onT,V, and any additional independent variables. CV is anextensiveproperty: the
combination of two identical phases has twice the value ofCV that one of the phases has
by itself.
For a phase containing a pure substance, themolar heat capacity at constant volume

is defined byCV;m
def
D CV=n.CV;mis anintensiveproperty.
If the system is an ideal gas, its internal energy depends only onT, regardless of whether
Vis constant, and Eq.5.6.1can be simplified to

CV D

dU

dT

(5.6.2)

(closed system, ideal gas)

Thus the internal energy change of an ideal gas is given by dU DCVdT, as mentioned
earlier in Sec.3.5.3.
Theheat capacity at constant pressure,Cp, is the ratio∂q=dT for a process in a
closed system with a constant, uniform pressure and with expansion work only. Under
these conditions, the heat∂qis equal to the enthalpy change dH(Eq.5.3.7), and we obtain
a relation analogous to Eq.5.6.1:

CpD



@H

@T



p

(5.6.3)

(closed system)

Cpis an extensive state function. For a phase containing a pure substance, themolar heat
capacity at constant pressureisCp;mDCp=n, an intensive property.
Since the enthalpy of a fixed amount of an ideal gas depends only onT (Prob. 5. 1 ), we
can write a relation analogous to Eq.5.6.2:

CpD

dH

dT

(5.6.4)

(closed system, ideal gas)

5.7 Surface Work

Sometimes we need more than the usual two independent variables to describe an equi-
librium state of a closed system of one substance in one phase. This is the case when,
in addition to expansion work, another kind of work is possible. The total differential of
Uis then given by dUDTdSpdV CYdX(Eq.5.2.7), whereYdXrepresents the
nonexpansion work∂w^0.
A good example of this situation is surface work in a system in which surface area is
relevant to the description of the state.
A liquid–gas interface behaves somewhat like a stretched membrane. The upper and
lower surfaces of the liquid film in the device depicted in Fig.5.1on the next page exert a
forceF on the sliding rod, tending to pull it in the direction that reduces the surface area.
We can measure the force by determining the opposing forceFextneeded to prevent the rod
from moving. This force is found to be proportional to the length of the rod and independent
of the rod positionx. The force also depends on the temperature and pressure.