Physical Chemistry , 1st ed.

Equation 2.30 means we can substitute Cpin the previous equation, so we get

dHCpdT

and integrate to get the total change in enthalpy for the temperature change:

H

Ti

Tf

CpdTqp (2.31)

where again we use the fact that Hequals qfor a change that occurs under

constant pressure. Equation 2.31 must be used if the heat capacity varies with

temperature (see Example 2.10). IfCpis constant over the temperature range,

then equation 2.31 can be simplified to

HCpTqp (2.32)

The comments regarding units on CValso apply to Cp (that is, you should keep

track of whether a specific amount, in units of grams or moles, is specified or

if it is actually part of the calculation). We can also define a molar heat capac-

ity Cpfor a process that occurs under constant pressure conditions.

Do not confuse the heat capacity at constant volume for heat capacity at

constant pressure. For a gaseous system, they can be very different. For solids

and liquids, they are not so different, but for solids and liquids the heat capacity

can also vary with temperature. For a change in a gaseous system, you must

know whether the change is a constant pressure change (called an isobaric

change) or a constant volume change (called an isochoricchange) in order to

determine which heat capacity is the correct one for the calculation of heat,

U,H, or both.

Finally, it can also be shown that for an ideal gas,



H

p



T

 0 (2.33)

That is, the change in the enthalpy at constant temperature is also exactly zero.

This is analogous to the situation for U.

2.7 Joule-Thomson Coefficients

Although we have been working with a lot of equations, all of them are ulti-

mately based on two ideas: equations of state and the first law of thermody-

namics. These ideas are ultimately based on the definition of total energy and

various manipulations of that definition. In addition, we have seen several

cases in which the equations of thermodynamics are simplified by the specifi-

cation of certain conditions: adiabatic, free expansion, isobaric, and isochoric

conditions are all restrictions on a process that simplify the mathematics of

thermodynamics. Are there other useful restrictions?

Another useful restriction based on the first law of thermodynamics is de-

scribed by the Joule-Thomson experiment, illustrated in Figure 2.10. An adia-

batic system is set up and filled with a gas on one side of a porous barrier. This

gas has some temperature T 1 , a fixed pressure p 1 , and an initial volume V 1 .A

piston pushes on the gas and forces all of it through the porous barrier, so the

final volume on this side of the barrier is zero. On the other side of the bar-

rier, a second piston moves out as the gas diffuses to the other side, where it

will have a temperature T 2 , a fixed pressure p 2 , and a volume V 2. Initially, the

volume on the right side of the barrier is zero. Since the gas is being forced

through a barrier, it is understood that p 1

 p 2. Even though the pressures on

42 CHAPTER 2 The First Law of Thermodynamics