Concise Physical Chemistry

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

ADIABATIC WORK 51

andp=RT

/

Vso

CVdT+

RT

V

dV=CV

dT

T

+R

dV

V

= 0

SinceCp−CV=R, we obtain

CV

dT

T

+R

dV

V

=

dT

T

+

Cp−CV

CV

dV

V

=

dT

T

+(γ− 1 )

dV

V

= 0

Integrating between limits, with a little algebraic manipulation, yields

TVγ−^1 =k and pVγ=k′

wherekandk′are constant.

The expressionpVγ=k′looks like Boyle’s law except for the parameterγ,

which is always greater than 1. The presence ofγcauses the pressurep=k′/Vγto

be lower at any point during the expansion than it is during the isothermal (Boyle’s

law) expansion (upper curve in Fig. 3.7).

The difference between the two expansions is in the heat that flows or does not flow

into the system to maintainT=const. In the isothermal case heat transfer is allowed.

Heat is not allowed into the system in the adiabatic case wheredq=0 by definition.

Without a compensating heat flow, the adiabatic system cools during expansion and

the pressure is always lower at any specified volume than it is in the isothermal

case (p=k′/Vγ,γ> 1 .0). The entirepVcurve falls below the isothermal curve in

Fig. 3.7.

p

V

V 1 V 2

FIGURE 3.7 Two expansions of an ideal gas. The upper curve is isothermal and the lower

curve is adiabatic. The adiabatic expansion does less work because there is no heat flow into

the system.