Physical Chemistry Third Edition

70 2 Work, Heat, and Energy: The First Law of Thermodynamics

If the molar heat capacity of a van der Waals gas is represented by

CV, mα+βT (2.4-25)

the equation analogous to Eqs. (2.4-21) and (2.4-24) is

T 2

T 1

exp

(

β(T 2 −T 1 )

α

)



(

Vm1−b

Vm2−b

)R/α

(2.4-26)

Exercise 2.16

Show that Eq. (2.4-26) is correct.

For a reversible adiabatic process, not only isTa function ofV, butPis also a

function ofV. For an ideal gas with constant heat capacity, we can substitute the ideal

gas equation,TPV /nR, into Eq. (2.4-21) to obtain

PP 1

(

V 1

V

) 1 +R/CV, m

(2.4-27)

The temperature can also be considered to be a function of the pressure:

TT 1

(

P

P 1

)R/(CV, m+R)

(2.4-28)

Exercise 2.17

Verify Eqs. (2.4-27) and (2.4-28).

EXAMPLE2.20

The “Santa Ana” winds of California are winds that begin in the mountains and drop to an

altitude near sea level. Assume that the air begins at a pressure of 0.81 atm (roughly the

barometric pressure at 6000 feet above sea level) and a temperature of 25◦C and that it is

adiabatically and reversibly compressed to a pressure of 1.00 atm as it moves to near sea

level. Assume that air is an ideal gas withCV, m 5 R/2. Find the final temperature. This

approximate treatment ignores other factors that raise the temperature, such as frictional

heating as the air passes along the ground.

T 2

T 1



(

P 2

P 1

)R/(CV, m+R)



(

1 .00 atm

0 .81 atm

) 2 / 7

 1. 062

T 2 (1.062)(298 K)316 K

tC,2 43 ◦C