Page 20 - Partition & Adsorption of Organic Contaminants in Environmental Systems

VAPOR–LIQUID AND VAPOR–SOLID EQUILIBRIA 11

DS = S v - S l = DH evap T (1.45)

where DH evap is the molar heat of evaporation of the liquid, one gets

(
dP dT = D H evap T V v - V l ) (1.46)
which is known as the Clapeyron equation. Because V v >> V l , Eq. (1.46) can
be reduced further to

dP dT =D H evap TV v (1.47)

If the ideal-gas law holds, PV v = RT, then

dln P dT = D H evap RT 2 or dln P d(1 T) =-D H evap R (1.48)

Integration of Eq. (1.48) on the assumption of constant DH evap gives

lnP =-D H evap RT + constant (1.49)

or

logP =-D H evap . 2 303 RT + constant (1.50)

For the solid–vapor equilibrium, one gets a similar expression:

log P =-D H sub . 2 303 RT + constant (1.51)

where DH sub is the molar heat of sublimation of the solid,

DH sub = DH evap + DH fus (1.52)

Equation (1.50) or (1.51) is called the Clausius–Clapeyron equation. It enables
one to determine the heat associated with liquid–vapor or solid–vapor transi-
tion from the P–T data more conveniently than by direct calorimetry. This heat
serves as a useful reference for comparison with the heat involved when the
vapor is transferred to a substrate or phase where it may either condense onto
the surface (as in adsorption) or disperse into the matrix (as in partition).
Given in Table 1.1 are the vapor pressures of some liquids as a function of
temperature (0 to 100°C). The vapor–pressure data at temperatures consider-
ably higher than the normal boiling points of the liquids are excluded. Over
this small-to-moderate temperature range, the P–T data of the liquids are rea-
sonably well represented by Eq. (1.50), as illustrated in Figure 1.2, showing
that DH evap is not very sensitive to temperature. A similar plot for solid com-
pounds with the vapor pressure data below the melting points would lead to
a similar conclusion that the DH sub is essentially constant over a small-to-