TITLE.PM5

344 ENGINEERING THERMODYNAMICS

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\M-therm\Th7-1.pm5

Also,

∂

∂

T

v

p

svs

F

HG

I

KJ

=−∂

∂

F

HG

I

KJ ...(7.18)

∂

∂

T

p

v

s s p

F

HG

I

KJ

= ∂

∂

F

HG

I

KJ ...(7.19)

∂

∂

p

T

s

vTv

F

HG

I

KJ

= ∂

∂

F

HG

I

KJ ...(7.20)

∂

∂

v

T

s

p pT

F

HG

I

KJ

=− ∂

∂

F

HG

I

KJ

...(7.21)

The equations (7.18) to (7.21) are known as Maxwell relations.

It must be emphasised that eqns. (7.14) to (7.21) do not refer to a process, but simply express

relations between properties which must be satisfied when any system is in a state of equilibrium.

Each partial differential co-efficient can itself be regarded as a property of state. The state may be

defined by a point on a three dimensional surface, the surface representing all possible states of

stable equilibrium.

7.4. Entropy Equations (Tds Equations)

Since entropy may be expressed as a function of any other two properties, e.g. temperature

T and specific volume v,

s = f(T, v)

i.e., ds =

∂

∂

s

T v

F

HG

I

KJ^ dT +

∂

∂

F

HG

I

KJ

s

v T^ dv

or Tds = T

∂

∂

s

T v

F

HG

I

KJ^ dT + T^

∂

∂

F

HG

I

KJ

s

vT^ dv ...(7.22)

But for a reversible constant volume change

dq = cv (dT)v = T(ds)v

or cv = T

∂

∂

s

T v

F

HG

I

KJ ...(7.23)

But,

∂

∂

F

HG

I

KJ

s

vT =

∂

∂

p

T v

F

HG

I

KJ [Maxwell’s eqn. (7.20)]

Hence, substituting in eqn. (7.22), we get

Tds = cvdT + T

∂

∂

p

T v

F

HG

I

KJ^ dv ...(7.24)

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

Similarly, writing

s = f(T, p)

Tds = T

∂

∂

s

T p

F

HG

I

KJ^ dT + T^

∂

∂

F

HG

I

KJ

s

pT^ dp ...(7.25)