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3.4 THE VARIATION OF ENTHALPY WITH TEMPERATURE AT CONSTANT PRESSURE 55

is sufficiently accurate to consider U as a function of T only [U = U(T)] for real gases
in processes that do not involve unusually high gas densities.
Having discussed ideal and real gases, what can be said about the relative magni-
tude of ¢U = 1 V i V f (0U>0V) dV and ¢U = 1 T i T f C dT for processes involving liq-
T
V
V
T
uids and solids? From experiments, it is known that the density of liquids and solids
varies only slightly with the external pressure over the range in which these two forms
of matter are stable. This conclusion is not valid for extremely high pressure conditions
such as those in the interior of planets and stars. However, it is safe to say that dV for a
solid or liquid is very small in most processes. Therefore,
V 2
0U 0U
solid, liq
¢U T = a b dV L a b ¢V L 0 (3.24)
0V T 0V T
3
V 1
because ¢V L 0 . This result is valid even if (0U>0V) T is large.
The conclusion that can be drawn from this section is as follows. Under most condi-
tions encountered by chemists in the laboratory, U can be regarded as a function of T
alone for all substances. The following equations give a good approximation even if V
is not constant in the process under consideration:

T f
T f
U(T , V ) - U(T , V ) =¢U = C dT = n C V, m dT (3.25)
i
i
f
f
V
LT i
3
T i
Note that Equation (3.25) is only applicable to a process in which there is no change in
the phase of the system, such as vaporization or fusion, and in which there are no chem-
ical reactions. Changes in U that arise from these processes will be discussed in
Chapters 4 and 8.
The Variation of Enthalpy with
3.4 Temperature at Constant Pressure

As for U, H can be defined as a function of any two of the three variables P, V, and T.
It was convenient to choose U to be a function of T and V because this choice led to
the identity ¢U = q V . Using a similar reasoning, we choose H to be a function of T
and P. How does H vary with P and T? The variation of H with T at constant P is dis-
cussed next, and a discussion of the variation of H with P at constant T is deferred to
Section 3.6.
Consider the constant pressure process shown schematically in Figure 3.4. For this
process defined by P = P external ,
dU = dq - PdV (3.26) P external P
P
Although the integral of dq is in general path dependent, it has a unique value in this
Mass
case because the path is specified, namely, P = P external = constant . Integrating both
sides of Equation (3.26),
Mass Piston
f f f
Piston
dU = dq - PdV or U - U = q - P(V - V ) (3.27)
P
f
i
f
P
i
P, V,T P, V ,T f
f
3 3 3 i i
i i i
Because P = P = P i , this equation can be rewritten as
f
Initial state Final state
(U + P V ) - (U + P V ) = q or ¢H = q P (3.28)
f
i i
P
f f
i
FIGURE 3.4
The preceding equation shows that the value of ¢H can be determined for an arbitrary The initial and final states are shown for
process at constant P in a closed system in which only P–V work occurs by simply meas- an undefined process that takes place at
uring q , the heat transferred between the system and surroundings in a constant pressure constant pressure.
P