Page 83 - Physical chemistry eng

60 CHAPTER 3 The Importance of State Functions: Internal Energy and Enthalpy

Solution
Because H is a state function, any path between the initial and final states will give the
same ¢H . We choose the path methanol (l, 1.00 bar, 298 K) : methanol (l, 1.00 bar,
425 K) : methanol (l, 2.50 bar, 425 K). The first step is isothermal, and the second
step is isobaric. The total change in H is

T f P f
¢H = n C P,m dT + VdP L nC P,m (T - T ) + V(P - P )
i
f
f
i
3 3
T i P i
124 g
-1
-1
= 81.1 J K mol * -1 * (425 K - 298 K)
32.04 g mol
-6
5
124 g 10 m 3 10 Pa
+ * * (2.50 bar - 1.00 bar) *
0.791 g cm -3 cm 3 bar
3
= 39.9 * 10 J + 23.5 J L 39.9 kJ
Note that the contribution to ¢H from the change in T is far greater than that from the
change in P.
Example Problem 3.9 shows that because molar volumes of liquids and solids are
small, H changes much more rapidly with T than with P. Under most conditions, H can
be assumed to be a function of T only for solids and liquids. Exceptions to this rule are
encountered in geophysical or astrophysical applications, for which extremely large
pressure changes can occur.
The following conclusion can be drawn from this section: under most conditions
encountered by chemists in the laboratory, H can be regarded as a function of T alone
for liquids and solids. It is a good approximation to write

T 2 T 2
H(T ,P ) - H(T ,P ) =¢H = C dT = n C P,m dT (3.47)
f
f
i
P
i
3 3
T 1 T 1
even if P is not constant in the process under consideration. The dependence of H on P
for real gases is discussed in Section 3.8 and Section 3.9 in the context of the Joule-
Thomson experiment.
Note that Equation (3.47) 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 chemical reactions. Changes in H that arise from chemical reactions and changes
in phase will be discussed in Chapters 4 and 8.
Having dealt with solids, liquids, and ideal gases, we are left with real gases. For
real gases, (0H>0P) T and (0U>0V) T are small, but still have a considerable effect on
the properties of the gases upon expansion or compression. Conventional technology
for the liquefaction of gases and for the operation of refrigerators is based on the fact
that (0H>0P) T and (0U>0V) T are not zero for real gases. To derive a useful formula for
calculating (0H>0P) T for a real gas, the Joule-Thomson experiment is discussed first in
the next section.

3.7 The Joule-Thomson Experiment

If the valve on a cylinder of compressed N 2 at 298 K is opened fully, it will become
covered with frost, demonstrating that the temperature of the valve is lowered below
the freezing point of H O . A similar experiment with a cylinder of H 2 leads to a consid-
2
erable increase in temperature and, potentially, an explosion. How can these effects be
understood? To explain them, we discuss the Joule-Thomson experiment.