378 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th8-1.pm5

To derive the equation of state for a perfect gas let us consider a unit mass of a perfect gas
to change its state in the following two successive processes (Fig. 8.3)
(i) Process 1-2′ at constant pressure, and
(ii) Process 2′-2 at constant temperature.

p

v

p = Constant

T = Constant

(^12)
2
Fig. 8.3. Formulation of equation of state of a perfect gas.
For the process 1-2′, applying Charle’s law
v
T
v
T
1
1
2
2
= ′
′
and since T 2 ′ = T 2 , we may write
v
T
v
T
1
1
2
2
= ′ ...(i)
For the process 2′-2, using Boyle’s law
p 2 ′v 2 ′ = p 2 v 2
and since p 2 ′ = p 1
p 1 v 2 ′ = p 2 v 2
i.e., v 2 ′ = pv
p
22
1
...(ii)
Substituting the value of v 2 ′ from eqn. (ii) in eqn. (i), we get
v
T
1
1

pv
pT
22
12
or
pv
T
11
1

pv
T
22
2
i.e.,
pv
T = constant ...(8.5)