latter names. The corresponding molar heat capacities are denoted by c V and c p ; the
formulas of which are
c V ¼
1
N
dQ
dT
V
¼
@u
@T
V
ð32BÞ
c p ¼
1
N
dQ
dT
p
¼
@h
@T
p
ð33BÞ
In general, heat capacities and molar heat capacities are functions of temperature
and pressure for a given substance. In application to processes that take place within
narrow ranges of temperature and pressure, it is often an acceptable approximation
to treat their values as constants.
3.8 Joule’s Law (Joule Free Expansion): The Caloric
Equation of State for Ideal Gases
We call Eqs. (3), (4), and (5) the thermal equation of state for ideal gases. Another
important property relation for an ideal gas is Joule’s law, which shall be called the
caloric equation of state for ideal gases. Joule’s law states that u is a function of
T alone, i.e., the change in internal energy of an ideal gas depends on the temperature change only.
@u
@v
T
¼ 0 and uðT; vÞ ¼ uðTÞ
i.e.,
dU ¼ Nc V dT
ð34Þ
DU ¼ N
Z
T 2
T 1
c V ðTÞdT ¼ Nc V DT
ð34AÞ
U ¼ U 0 þ Nc V T À T 0
ð
Þ
ð34BÞ
This result was originally discovered through experimental method by Joule. It
turns out that Joule’s law, Eq. (34), is an inference of Eq. (3), as shown in
Sect. 9.5.2, Eq. (175A): any gas that satisfies the thermal equation of state obeys the
caloric equation of state.
From the thermal equation of state, we obtain h = u + RT. The enthalpy of an
ideal gas, thus like the internal energy in accordance with its caloric equation of
state, is also a function of temperature alone,
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3 The First Law: The Production of Heat …