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

process. Note the similarity between Equations (3.28) and (3.18). For an arbitrary process
in a closed system in which there is no work other than P–V work, ¢U = q V if the process
takes place at constant V, and ¢H = q P if the process takes place at constant P. These two
equations are the basis for the fundamental experimental techniques of bomb calorimetry
and constant pressure calorimetry discussed in Chapter 4.
A useful application of Equation (3.28) is in experimentally determining ¢H and
¢U of fusion and vaporization for a given substance. Fusion (solid : liquid) and
vaporization (liquid : gas) occur at a constant temperature if the system is held at a
constant pressure and heat flows across the system–surroundings boundary. In both of
these phase transitions, attractive interactions between the molecules of the system
must be overcome. Therefore, q 7 0 in both cases and C : q . Because ¢H = q P ,
P
¢H fusion and ¢H vaporization can be determined by measuring the heat needed to effect
the transition at constant pressure. Because ¢H =¢U +¢(PV) , at constant P,

¢H vaporization -¢U vaporization = P¢V vaporization 7 0 (3.29)
The change in volume upon vaporization is ¢V vaporization = V gas - V liquid W ; 0
therefore, ¢U vaporization 6¢H vaporization . An analogous expression to Equation (3.29)
can be written relating ¢U fusion and ¢H fusion . Note that ¢V fusion is much smaller than
¢V vaporization and can be either positive or negative. Therefore, ¢U fusion L¢H fusion .
The thermodynamics of fusion and vaporization will be discussed in more detail in
Chapter 8.
Because H is a state function, dH is an exact differential, allowing us to link
(0H>0T) P to a measurable quantity. In analogy to the preceding discussion for dU, dH
is written in the form

0H 0H
dH = a b dT + a b dP (3.30)
0T P 0P T
Because dP = 0 at constant P, and dH = dq P from Equation (3.28), Equation (3.30)
becomes
0H
dq = a b dT (3.31)
P
0T P
Equation (3.31) allows the heat capacity at constant pressure C to be defined in
P
a fashion analogous to C in Equation (3.15):
V
dq P 0H
= = a b
C P (3.32)
dT 0T P
Although this equation looks abstract, C is a readily measurable quantity. To measure
P
it, one need only measure the heat flow to or from the surroundings for a constant pres-
sure process together with the resulting temperature change in the limit in which dT
and dq approach zero and form the ratio lim (dq>dT) P .
dT:0
As was the case for C , C is an extensive property of the system and varies from
P
V
substance to substance. The temperature dependence of C must be known in order to
P
calculate the change in H with T. For a constant pressure process in which there is no
change in the phase of the system and no chemical reactions,
T f T f
¢H = C (T)dT = n C P, m (T)dT (3.33)
P
P
3 3
T i T i
If the temperature interval is small enough, it can usually be assumed that C is con-
P
stant. In that case,
¢H = C ¢T = nC P,m ¢T (3.34)
P
P
The calculation of ¢H for chemical reactions and changes in phase will be discussed in
Chapters 4 and 8.