TITLE.PM5

THERMODYNAMIC RELATIONS 345

dharm
\M-therm\Th7-1.pm5

where cp = T

∂

∂

s

T p

F

HG

I

KJ ...(7.26)

Also

∂

∂

F

HG

I

KJ

s

pT = –

∂

∂

v

T p

F

HG

I

KJ [Maxwell’s eqn. (7.21)]

whence, substituting in eqn. (7.25)

Tds = cpdT – T

∂

∂

v

T p

F

HG

I

KJ^ dp ...(7.27)

This is known as the second form of entropy equation or the second Tds equation.

7.5. Equations for Internal Energy and Enthalpy

(i) Let u = f(T, v)

du =

∂

∂

u

T v

F

HG

I

KJ dT +

∂

∂

u

vT

F

HG

I

KJ dv = cv^ dT +

∂

∂

u

v T

F

HG

I

KJ^ dv ...(7.28)

To evaluate

∂

∂

u

vT

F

HG

I

KJ let u = f (s, v)

Then du =

∂

∂

u

s v

F

HG

I

KJ^ ds +

∂

∂

u

v s

F

HG

I

KJ^ dv

or

∂

∂

u

v T

F

HG

I

KJ =

∂

∂

∂

∂

u

s

s

v

u

vTv s

F

HG

I

KJ

F

HG

I

KJ +

∂

∂

F

HG

I

KJ

But

∂

∂

u

s v

F

HG

I

KJ = T,

∂

∂

s

vT

F

HG

I

KJ =

∂

∂

s

T

u

vsv

F

HG

I

KJ

∂

∂

F

HG

I

, KJ = – p

Hence

∂

∂

F

HG

I

KJ

u

v T = T^

∂

∂

p

T v

F

HG

I

KJ – p ...(7.29)

This is sometimes called the energy equation.

From equation (7.28), we get

du = cvdT + T

p

T

p

v

∂

∂

F

HG

I

KJ

−

R

S

|

T|

U

V

|

W|

dv ...(7.30)

(ii) To evaluate dh we can follow similar steps as under

h = f(T, p)

dh = ∂

∂

h

T p

F

HG

I

KJ

dT +

∂

∂

h

pT

F

HG

I

KJ

dp

= cpdT +

∂

∂

h

p

dp

T

F

HG

I

KJ

...(7.31)