Concise Physical Chemistry

c03 JWBS043-Rogers September 13, 2010 11:24 Printer Name: Yet to Come

THE JOULE AND JOULE–THOMSON EXPERIMENTS 47

Not surprisingly,μJTis related to the van der Waalsaandb, most importantly the

van der Waals parameter of attractiona. For gases having attractive interactions (most

of them at room temperature) expansion against their attractive forces does internal

work to separate the gaseous particles, which is why the gas cools andμJT>0. For

helium, neon, hydrogen, and so on, the dominant forces are repulsive henceμJT< 0

at room temperature. At low temperatures, attractive forces become dominant for all

gases, soμJTchanges sign.

The Joule–Thomson inversion temperatureTican be related to van der Waalsa

andbby the equation

2 a

RTi

−

3 abp

R^2 Ti^2

−b= 0

This equation is a quadratic inTi; hence double roots are possible. Indeed, two

inversion temperatures, upper and lower, are found at some pressures. At very high

pressures, the two roots approach each other and become identical as shown in

Fig. 3.5. On the high branch of Fig. 3.5, the upper inversion temperature, the second

term in the inversion temperature equation becomes unimportant because it hasTi^2

in the denominator. Now

2 a

RTi

∼=b

Ti∼=

2 a

bR

Hence the upperTican be calculated ifaandbare known. The second equation above

is used to estimate the upper inversion temperature of real gases fromaandb(which

are themselves estimates). Though approximate, this value is important in practical

problems such as production of liquefied gas for cooling certain low-temperature

experimental instruments.

Ti

p

Root 1

Root 2

FIGURE 3.5 Inversion temperatureTias a function of pressure. The temperature extremum

dTi/dpfor nitrogen is at about 370 atm.