Heat of Vaporization The Clausius-Clapeyron

Importance & Calculation

The heat of vaporization• The heat of vaporization of a liquid is a useful

thermodynamic quantity because it allows the calculation of the vapor pressure of the liquid at any temperature.Two practical applications of heats of vaporization are distillations and vapor pressure:

• Distillation is one of the most practical methods for separation and purification of chemical compounds. The heat of vaporization is the fundamental quantity that determines the experimental conditions at which an industrial or laboratory-scale distillation should be run.

• The concentration of a gas is given by its vapor pressure. Knowledge of the heat of vaporization permits the control of vapor pressure by setting the temperature of the liquid being vaporized.

Derivation of the Clausius-Clapeyron Equation

• Two phases in equilibrium at constant pressure and temperature have the same Gibb’s free energy. dG = Vdp - SdT yields

• The Clausius Clapeyron Equation is derived from the second law of thermodynamics. It applies to all types of transitions between phases, for example, melting, sublimation and solid-solid transitions between polymorphs.

• The Clapeyron equation is valid along any phase co-existence line. It explains, for example, that the solid-liquid phase co-existence line for water has a negative slope in a p vs. T phase diagram, since the change in volume is going from ice to water is negative (ice floats in water).

h Tessure witange of prRate of chdTdP

ke placechanges ta at which emperatureabsolute tTeightthe same wated with nge associvolume chaΔV

cet of subsiven weighion for agof transitt or heat latent heaΔH

where:VT

HdTdP

tan

• When the Clapeyron equation is applied to liquid/vapor phase co-existence, several simplifying assumptions can be made.

• First, since the volume of a gas is much greater than the volume of a condensed phase,

• Second, the gas is assumed to obey the ideal gas law

• which allows the Clapyron equation to be simplified as follows:

gasliquidgas VVVV

PnRTVgas

Third, ΔH is assumed to be independent of temperature and pressure, allowing indefinite integration over p and T

Choosing the constant of integration to equal ln p0, where p0 equals one pressure unit, e.g., 1 Torr if pressure is measured in Torr, allows the pressure units to cancel and yields:

22 e thereforTdT

nRH

PdP

nRTHP

dTdP

constant1ln

TnR

Hp

This equation is useful for determining ΔH from a plot of ln (p/p0) vs. 1/T.If definite integration from p1 to p2 and from T1 to T2 is performed, the result is

This equation is useful for determining the vapor pressure p2 at temperature T2 given ΔH and the vapor pressure at one temperature (p1, T1), e.g., the normal boiling point.

121

2

2

11ln

ln2

1

2

1

TTnRH

PPor

nRTHPd

T

T

P

P

• The Clausius-Clapeyron equation may also be treated graphically. Assuming ΔH to be constant in the range of temperature considered, integration of this equation may be written:

that is, of the form y = mx + c.• Experimental data for pressures and

temperatures may therefore be plotted as ln P against the reciprocal of temperature. The slope of the resulting graph is then (-ΔH/R) from which the heat change of the phase reaction may be calculated.

• Example To calculate the pressure under which water boils at 99°C.given that:

• Latent heat of water at 100°C = 540 x 18 cal/mole

• R = 1 .987 cal/mole • P2 = pressure required at T2 = 372°K

P2 = 733.6 mm pressure.

3731

3721

987.118 x 540

760ln 2P