Physical Chemistry , 1st ed.

temperature, until the temperature drops so low that it condenses into a

liquid. Liquid nitrogen and oxygen are commonly prepared that way, on a vast

industrial scale. However, a gas must be below its inversion temperature in or-

der for the Joule-Thomson effect to work in the proper direction of decreas-

ing temperature! Gases that have very low inversion temperatures must be

cooled before using a sort of Joule-Thomson expansion to liquefy them. Before

this was widely realized, it was thought that some gases were “permanent

gases,” because they could not be liquefied by “ordinary” means. (Such gases

were first described by Michael Faraday in 1845, because he was unable to

liquefy them.) They included hydrogen, oxygen, nitrogen, nitric oxide, methane,

and the first four noble gases. Nitrogen and oxygen were easily liquefied by

performing a cyclic Joule-Thomson expansion on them, and the other gases

soon followed. However, the inversion temperatures of hydrogen and helium

are so low (about 202 K and 40 K, respectively) that they have to be precooled

substantially before any kind of Joule-Thomson expansion will cool them

further. Hydrogen was finally liquefied by the Scottish physicist James Dewar

in 1898, and helium in 1908 by the Dutch physicist Heike Kamerlingh-Onnes

(who used liquid helium to discover superconductivity).

2.8 More on Heat Capacities

Recall that we defined two different heat capacities, one for a change in a sys-

tem kept at constant volume, and one for a change in a system kept at constant

pressure. We labeled them CVand Cp. What is the relationship between the two?

We start with an equation that eventually yielded equation 2.22. The rele-

vant equation is

dq

U

T



V

dT+ 

U

V



T

• pdV (2.36)

where pis the external pressure. We have defined the derivative ( U/ T)Vas

CV, so we can rewrite the equation as

dqCVdT+ 

U

V



T

• pdV

So far, we have imposed no conditions on the system in deriving the above ex-

pression, other than the sample being an ideal gas. We now impose the addi-

tional condition that the pressure be kept constant. Nothing really changes,

since the infinitesimal change in heat dqis expressed in terms of a change in

temperature,dT, and a change in volume,dV. We can therefore write the above

equation as

dqpCVdT+ 

U

V



T

• pdV

where dqnow has the subscript p. If we divide both sides of the equation by

dT,we get



T

q



p

CV+ 

U

V



T

• p

V

T



p

Note that the derivative V/ Thas a psubscript, due to our specifying that

this is for constant-pressure conditions. Also note that the expression is a par-

tialderivative, because the quantities in the numerators depend on multiple

46 CHAPTER 2 The First Law of Thermodynamics