McKay, Donald. "Front matter" Multimedia Environmental Models ...

access to information about the stability of the crystalline state, i.e., does not
“know” its melting point. As a result, it behaves in a manner corresponding to the
liquid vapor pressure. A similar phenomenon occurs above the critical point where
a gas such as oxygen, when in solution in water, behaves as if it were a liquid at
25°C, not a gas. No liquid vapor pressure can be measured for either naphthalene
or oxygen at 25°C; it can only be calculated. Later, we term this liquid vapor
pressure the reference fugacity. We may need to know this fictitious vapor pressure
for several reasons.
The ratio of the solid vapor pressure to the supercooled liquid vapor pressure is
termed the fugacity ratio, F. To estimate F, we need to know how much energy is
involved in the solid-liquid transition, i.e., the enthalpy of melting or fusion. The
rigorous equation for estimating F at temperature T(K) is (Prausnitz et al., 1986)
©2001 CRC Press LLC
ln F = – DS(TM
– T)/RT + DCP(TM
– T)/RT – DCP
ln(TM/T)/R
where DS
(J/mol K) is the entropy of fusion at the melting point TM
(K), DCP
(J/mol
K) is the difference in heat capacities between the solid and liquid substances, and
R is the gas constant. The heat capacity terms are usually small, and they tend to
cancel, so the equation can be simplified to
ln F = – DS(TM
– T)/RT = –( DH/TM)(TM
– T)/RT = –( DH/R)(1/T
– 1/TM)
where DH
(J/mol) is the enthalpy of fusion and equals TMDS.
Note that, since TM
is greater than T, the right-hand side is negative, and F is
less than one, except at the melting point, when it is 1.0. F can never exceed 1.0. A
convenient method of estimating DH
is to exploit Walden’s rule that the entropy of
fusion at the melting point DS,
which is DH/TM,
is often about 56.5 J/mol K. It
follows that
The group
mated as
ln F = –(DS/R)(TM/T
– 1)
DS/R
is often assigned a value of 56/8.314 or 6.79. Thus, F is approxi-
F = exp[–6.79(TM/T
– 1)]
If base 10 logs are used and T is 298 K, this equation becomes
log F = –6.79(TM/298
– 1)/2.303 = –0.01(TM
– 298)
This is useful as a quick and easily remembered method of estimating F. If more
accurate data are available for DH
or DS,
they should be used, and if the substance
is a high melting point solid, it may be advisable to include the heat capacity terms.