Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.5 APPLICATIONS OFEXPANSIONWORK 77

(^200) K
(^400) K
(^600) K
(^800) K
(^00) 0:10 0:20 0:30
2
4
6
8
0:40
V=dm^3
p
/ bar
Figure 3.6 An adiabat (solid curve) and four isotherms (dashed curves) for an ideal
gas (nD0:0120mol,CV;mD1:5R).
To express the finalpressureas a function of the initial and final volumes, we make the
substitutionsT 1 Dp 1 V 1 =nRandT 2 Dp 2 V 2 =nRin Eq.3.5.11and obtain
p 2 V 2
nR

D

p 1 V 1

nR



V 1

V 2

nR=CV

(3.5.13)

Solving this equation forp 2 , we obtain finally

p 2 Dp 1



V 1

V 2

 1 CnR=CV

(3.5.14)

(reversible adiabatic

expansion, ideal gas)

The solid curve in Fig.3.6shows how the pressure of an ideal gas varies with volume
during a reversible adiabatic expansion or compression. This curve is anadiabat. The
dashed curves in the figure areisothermsshowing how pressure changes with volume at
constant temperature according to the equation of statepDnRT=V. In the direction of
increasingV (expansion), the adiabat crosses isotherms of progressively lower tempera-
tures. This cooling effect, of course, is due to the loss of energy by the gas as it does work
on the surroundings without a compensating flow of heat into the system.

3.5.4 Indicator diagrams

Anindicator diagram(or pressure–volume diagram) is usually a plot ofpas a function
ofV. The curve describes the path of an expansion or compression process of a fluid that
is essentially uniform. The area under the curve has the same value as the integral

R

pdV,
which is the negative of the reversible expansion work given bywD

R

pdV. For example,