Multicomponent Equilibrium Thermodynamics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 59

4.585Torr. The boiling point of the liquid T b at a given pressure is the temperature
at which P P l while the normal boiling point is that at which P l 1 atm. BA

is the solid-vapor equilibrium line. Similar to the normal boiling point for liquids,
the normal melting point is the temperature at which P s

= =

1 atm. Beyond point C,
liquid and vapor phases cannot co-exist in equilibrium; this is called the critical
point. The critical temperature, T c , critical pressure, P c , and critical volume, V c , are
unique to a compound. For water T c is 647 K and P c is 218 atm. The phase above
the critical point of a compound is called super - critical state . Extending the line
BA beyond A to B A, one obtains the hypothetical sub-cooled liquid state , which
is important in estimating the solubility of a solid in a liquid. The ratio of the
sub-cooled liquid vapor pressure to the solid vapor pressure is the fugacity of the
solid.
The equilibrium at any point along the equilibrium P - T line in Figure 3.6 can
be described in terms of chemical potentials. If I and II represent two phases in
equilibrium, then we have the following criteria:

=

S I d T V I d P S II d T V II d P . − + =− + (3.32)

Hence

d T = Δ V = Δ

d P S H

V ,

(3.33)

Δ T Δ

where V are the entropy and volume changes for the phase transitions I

and II. We also used the definition of entropy change,

Δ S and Δ

Δ S = Δ H/T .

For molar changes we then have the following equation:

d P

d T = Δ

H m
T Δ V m

. (3.34)

This is the Clausius-Clapeyron equation . For both vapor-liquid and vapor-solid
equilibrium it gives the change in vapor pressure with temperature (Table 3.8). Since
Δ vapor transitions, d P /d T

is always positive, that is, both liquid and solid vapor pressure increase with temper-
ature. For both liquid

H m and Δ V m are positive for liquid → vapor and solid →

vapor transitions, the molar volume of
the gas is much larger than that of the liquid or solid. Hence,

→ vapor and solid →

V m =

Δ V m ≈ RT/P .

Thus

dln P

d T = Δ

H m
RT 2 .

(3.35)

Δ

The quantity H m is not independent of T . However, over small ranges of

temperature, it can be assumed to be constant. Then we can integrate the above
equation to obtain

ln P 2
P 1

1
T 2 −

.

=− Δ

H m
R

1
T 1

(3.36)