Physical Chemistry Third Edition

64 2 Work, Heat, and Energy: The First Law of Thermodynamics

would differ from the initial temperature. If the heat capacity of the apparatus and the

heat capacity of the gas are known, the change in temperature of the gas could be

calculated.

The Joule experiment was carried out several times with various volumes for the

second chamber. The ratio∆T/∆Vwould be determined for each experiment and

extrapolated to zero value of∆V, where∆Vis the final volume of the gas minus its

initial volume. This extrapolation is equivalent to taking the mathematical limit, so the

result is a partial derivative, called theJoule coefficientand denoted byμJ:

μJ lim

∆V→ 0

(

∆T

∆V

)



(

∂T

∂V

)

U,n

(2.4-13)

The Joule coefficient is related to (∂U/∂V)T,nby use of the cycle rule, Eq. (B-15), and

the reciprocal identity, Eq. (B-8):

(

∂U

∂V

)

T,n

−

(

∂T

∂V

)

U,n

(

∂U

∂T

)

V,n

−μJCV (2.4-14)

Exercise 2.10

Verify Eq. (2.4-14).

Joule was unsuccessful in his attempt to measure the Joule coefficient because

the changes in temperature that occurred were too small to be measured by his ther-

mometers, even though he used pressures up to 22 atm. Later versions of the exper-

iment with better apparatus have given nonzero values of (∂U/∂V)T,n for real

gases.

There are better ways than the Joule experiment to determine values of (∂U/∂V)T,n,

and we will discuss them in Chapter 4. Once values forCVand for (∂U/∂V)T,nare

obtained,∆Ucan be calculated for any process that begins with one equilibrium state

and ends with another equilibrium state.

EXAMPLE2.15

If the virial equation of state, Eq. (1.3-3), is truncated at the second term it can be shown

that

(

∂U

∂V

)

T,n



(

∂Um

∂Vm

)

T,n



RT^2

Vm^2

dB 2

dT

(2.4-15)

whereRis the gas constant and whereVmis the molar volume. The derivation of this

equation is found in Example 4.8. For argon gas at 298.15 K,B 2 is approximately equal to

− 15 .8cm^3 mol−^1 anddB 2 /dTis approximately equal to 0.20 cm^3 mol−^1 K−^1. Assume that

CV, m 3 R/2.

a.Find∆U,q, andwfor a reversible isothermal expansion of 1.000 mol of argon at 298.15 K

from a volume of 2.000 L to a volume of 20.00 L. Compare with values obtained assuming

ideal gas behavior.

b.Find the value of the Joule coefficient for 1.000 mol of argon at 298.15 K and a volume

of 20.000 L.