TITLE.PM5

FIRST LAW OF THERMODYNAMICS 115

dharm

M-therm/th4-1.pm5

From Eqn. (4.31),

pv 11 γ = pv 22 γ or p

p

v

v

2

1

1

2

=FHG IKJ

γ

...(4.36)

From Eqn. (4.34),

Tv 11 γγ−−^1 =T v 22 1 or

T

T

v

v

2

1

1

2

1

=F

HG

I

KJ

−γ

...(4.37)

From Eqn. (4.35),

T

p

T

p

1

1

1

2

2

1

() ( )

γ

γ

γ

γ

−−= or

T

T

p

p

2

1

2

1

1

=FHG IKJ

−γ

γ

...(4.38)

From eqn. (4.30), the work done in an adiabatic process per kg of gas is given by W
= (u 1 – u 2 ). The gain in internal energy of a perfect gas is given by equation :
u 2 – u 1 = cv (T 2 – T 1 ) (for 1 kg)
∴ W = cv (T 1 – T 2 )
Also, we know that

cv = γ−R 1

Hence substituting, we get

W =

RT T() 12

1

−

−γ

Using equation, pv = RT

W =

pv 11 pv2 2

1

−

−γ

This is the same expression obtained before as eqn. (4.32).

5. Polytropic Reversible Process (pvn = constant) :
   It is found that many processes in practice approximate to a reversible law of form pvn
   = constant, where n is a constant. Both vapours and perfect gases obey this type of law closely in
   many non-flow processes. Such processes are internally reversible.
   We know that for any reversible process,
   W = zpdv
   For a process in pvn = constant, we have

p =

C

vn , where C is a constant

∴ WC

dv

v

C v

n

C vv

v n n

v nnn

==

−+

= −

−+

F

HG

I

z KJ

−+ −+ −+

1

2 1 2 1 1 1

11

i.e., WCvv

n

pv v p v v

n

nn nn nn

= −−

F

HG

I

KJ

= −−

−+ −+ −+ −+

1

1

2

1

11 1

1

22 2

1

11

(since the constant C, can be written as pv

n

11 or as p 2 v 2 n)