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Chapter 2 of the experiment and at the pressure of the experiment. C of a gas is found from the
P
The First Law of Thermodynamics temperature increase produced by electrically heating the gas flowing at a known rate.
The thermodynamic state of an equilibrium system at rest in the absence of ap-
plied fields is specified by its composition (the number of moles of each component
present in each phase) and by any two of the three variables P, V, and T. Commonly,
P and T are used. For a closed system of fixed composition, the state is specified by P
and T. Any state function has a definite value once the system’s state is specified.
Therefore any state function of a closed equilibrium system of fixed composition is a
function of T and P. For example, for such a system, H H(T, P). The partial deriv-
ative ( H(T, P)/ T) is also a function of T and P. Hence C is a function of T and P
P
P
and is therefore a state function. Similarly, U can be taken as a function of T and V,
and C is a state function.
V
For a pure substance, the molar heat capacities at constant P and at constant V
are C P,m C /n and C V,m C /n. Some C P,m values at 25°C and 1 atm are plotted in
P
V
Fig. 2.4. The Appendix gives further values. Clearly, C P,m increases with increasing
size of the molecules. See Sec. 2.11 for discussion of C P,m values.
For a one-phase system of mass m, the specific heat capacity c is c C /m.
P
P
P
The adjective specific means “divided by mass.” Thus, the specific volume v and spe-
cific enthalpy h of a phase of mass m are v V/m 1/r and h H/m.
Do not confuse the heat capacity C (which is an extensive property) with the
P
molar heat capacity C P,m or the specific heat capacity c (which are intensive proper-
P
ties). We have
C P,m C >n pure substance (2.54)*
P
c C >m one-phase system (2.55)*
P
P
C P,m and c are functions of T and P. Figure 2.5 plots some data for H O(g). These
P
2
curves are discussed in Sec. 8.6.
One can prove from the laws of thermodynamics that for a closed system, C and
P
C must both be positive. (See Münster, sec. 40.)
V
C 7 0, C 7 0 (2.56)
P
V
Exceptions to (2.56) are systems where gravitational effects are important. Such sys-
Figure 2.4 tems (for example, black holes, stars, and star clusters) can have negative heat capac-
ities [D. Lynden-Bell, Physica A, 263, 293 (1999)].
Molar heat capacities C P,m at 25°C What is the relation between C and C ? We have
V
P
and 1 bar. The scale is
logarithmic. 0H 0U 01U PV2 0U
C C a b a b a b a b
V
P
0T P 0T V 0T P 0T V
0U 0V 0U
C C a b Pa b a b (2.57)
P
V
0T P 0T P 0T V
We expect ( U/ T) and ( U/ T) in (2.57) to be related to each other. In
V
P
( U/ T) , the internal energy is taken as a function of T and V; U U(T, V). The total
V
differential of U(T, V) is [Eq. (1.30)]
0U 0U
dU a b dT a b dV (2.58)
0T V 0V T
Equation (2.58) is valid for any infinitesimal process, but since we want to relate
( U/ T) to ( U/ T) , we impose the restriction of constant P on (2.58) to give
P
V
0U 0U
dU a b dT a b dV P (2.59)
P
P
0T V 0V T