TITLE.PM5

354 ENGINEERING THERMODYNAMICS

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\M-therm\Th7-1.pm5

The Clausius-Claperyon equation can be derived in different ways. The method given below

involves the use of the Maxwell relation [eqn. (7.20)]

∂

∂

F

HG

I

KJ

p

T v =

∂

∂

F

HG

I

KJ

s

vT

Let us consider the change of state from saturated liquid to saturated vapour of a pure

substance which takes place at constant temperature. During the evaporation, the pressure and

temperature are independent of volume.

∴

dp

dT

F

HG

I

KJ =

ss

vv

g f

g f

−

−

where,sg = Specific entropy of saturated vapour,

sf = Specific entropy of saturated liquid,

vg = Specific volume of saturated vapour, and

vf = Specific volume of saturated liquid.

Also, sg – sf = sfg = h

T

fg

and vg – vf = vfg

where sfg = Increase in specific entropy,

vfg = Increase in specific volume, and

hfg = Latent heat added during evaporation at saturation temperature T.

∴ dp

dT

ss

vv

s

v

h

Tv

g f

g f

fg

fg

fg

fg

=

−

−

==

.

...(7.50)

This is known as Clausius-Claperyon or Claperyon equation for evaporation of liquids.

The derivative

dp

dT is the slope of vapour pressure versus temperature curve. Knowing this slope

and the specific volume vg and vf from experimental data, we can determine the enthalpy of

evaporation, (hg – hf) which is relatively difficult to measure accurately.

Eqn. (7.50) is also valid for the change from a solid to liquid, and from solid to a vapour.

At very low pressures, if we assume vg ~− vfg and the equation of the vapour is taken as

pv=RT, then eqn. (7.50) becomes

dp

dT =

h

Tv

fg

g

=

hp

RT

fg

2 ...(7.51)

or hfg = RT

p

dp

dT

2

...(7.52)

Eqn. (7.52) may be used to obtain the enthalpy of vapourisation. This equation can be

rearranged as follows :

dp

p =

h

R

dT

T

fg.

2

Integrating the above equation, we get

dp

z p =

h

R

dT

T

fg

z^2

ln

p

p

h

RT T

2 fg

112

=−L^11

N

M

O

Q

P ...(7.53)