TITLE.PM5

358 ENGINEERING THERMODYNAMICS

dharm
\M-therm\Th7-1.pm5

But,

∂

∂

F

H

I

K

p

T v =

∂

∂−−

RS

T

UV

W

L

NM

O

T QP

RT

vb

a

v^2 v =

R

vb−

∴ du c dT T

R

vb

v pdv

1

2

1

2

1

2

zz=+z −

F

HG

I

KJ

−

L

N

M

M

O

Q

P

P

= cdT T

R

vb

RT

vb

a

v

v + dv

−

F

HG

I

KJ

−

−

RS −

T

U

V

W

L

N

M

M

O

Q

P

zz (^12) P
2
1
2
= cdT
RT
vb
RT
vb
a
v
v + dv
−
−
−

• L
  N
  M
  O
  Q
  zz 1 2 P
  2
  1
  2
  = cdT
  a
  v
  v +zz 2 dv
  1
  2
  1
  2
  .
  ∴ u 2 – u 1 = cv(T 2 – T 1 ) + a 1
  v
  1
  12 v
  −
  F
  HG
  I
  KJ

. (Ans.)

(ii)Change in enthalpy :

The change in enthalpy is given by

dh = cpdT + vT

v

− T p

∂

∂

F

H

I

K

L

N

M

M

O

Q

P

P

dp

∂

∂

F

H

I

K

h

pT = 0 + v – T^

∂

∂

F

H

I

K

v

T p ...(1)

Let us consider p = f(v, T)

∴ dp = HF∂∂pvIK

T

dv + HF∂∂TpIK

v

dT

∴ (dp)T = FH∂∂pvIK

T

dv + 0 as dT = 0 ...(2)

From equation (1),

(dh)T = vTTv

p

− FH∂∂ IK

L

N

M

M

O

Q

P

P

(dp)T.

Substituting the value of (dp)T from eqn. (2), we get

(dh)T = vT

v

T

p

− p vT

∂

∂

F

H

I

K

L

N

M

M

O

Q

P

P

∂

∂

F

H

I

K^ dv

= v

p

v T

v

T

p

T p vT

∂

∂

F

H

I

K −

∂

∂

F

H

I

K

∂

∂

F

H

I

K

L

N

M

M

O

Q

P

P^ dv ...(3)

Using the cyclic relation for p, v, T which is

∂

∂

F

H

I

K

∂

∂

F

HG

I

KJ

∂

∂

F

HG

I

KJ =−

v

T

T

p

p

p vTv^1

∴

∂

∂

F

H

I

K

∂

∂

F

HG

I

KJ

=− ∂

∂

F

HG

I

KJ

v

T

p

v

p

p TvT