Physical Chemistry Third Edition

80 2 Work, Heat, and Energy: The First Law of Thermodynamics

We can use the cycle rule, Eq. (B-15) of Appendix B, to write

(

∂H

∂P

)

T,n

−

(

∂H

∂T

)

P,n

(

∂T

∂P

)

H,n

−CPμJT (2.5-30)

The Joule–Thomson coefficient of an ideal gas vanishes because (∂H/∂P)T,nvan-

ishes. Joule and Thomson found that the Joule–Thomson coefficient is measurably

different from zero for ordinary gases at ordinary pressures. It depends on tempera-

ture and is positive at room temperature for most common gases except for hydrogen

and helium. Even for these gases it is positive at some range of temperatures below

room temperature. This means that for some range of temperature any gas cools on

expansion through a porous plug. Expansion of a gas can be used to cool the gas

enough to liquefy part of it, and the final step in the production of liquid nitrogen or

liquid helium is ordinarily carried out in this way.

EXAMPLE2.23

For air at 300 K and 25 atm,μJT0.173 K atm−^1. If a Joule–Thomson expansion is carried

out from a pressure of 50.00 atm to a pressure of 1.00 atm, estimate the final temperature if

the initial temperature is equal to 300 K.

Solution

∆T≈

(

∂T

∂P

)

T,n

∆P

(

0 .173 K atm−^1

)

(49 atm)8K

T 2 ≈292 K

The molecular explanation for the fact that the Joule–Thomson coefficient is pos-

itive at sufficiently low temperature is that at low temperatures the attractive inter-

molecular forces are more important than the repulsive intermolecular forces. When

the gas expands, work must be done to overcome the attractions and the potential

energy increases. If no heat is added, the kinetic energy decreases and the temperature

decreases.

PROBLEMS

Section 2.5: Enthalpy

2.38 Show that ifdUdq+dw,ifdUis exact, and ifdqis
inexact, thendwmust be inexact.

2.39 The work done on a nonsimple system such as a spring or a
rubber band is given by
dwP(transferred)dV+τdL

whereτis the tension force andLis the length of the

spring or rubber band. One must specify whether a heat

capacity is measured at constantτor at constantL,in

addition to specifying constantPor constantV. Find

a relation analogous to Eq. (2.5-11) relatingCP,τ

andCP, L.

2.40 a.The Joule–Thomson coefficient of nitrogen gas at

50 atm and 0◦C equals .044 K atm−^1. Estimate

the final temperature if nitrogen gas is expanded

through a porous plug from a pressure of 60.0 atm

to a pressure of 1.00 atm at 0◦C.

b. Estimate the value of (∂Hm/∂P)Tfor nitrogen gas at

50 atm and 0◦C. State any assumptions.