s
g
1
where Sm , Sm , and Sm are the molar entropies (J K1 mol1) of the gas, liquid, and solid
g
s
g
l
l
phases, respectively; Vm , Vm , and Vm are their molar volumes (m3 mol1); xB , xB , and
s
xB are the mole fractions of component B in the three phases; and g , B , and s are the
l
B
B
chemical potentials (J mol1) of component B in the three phases.
Kelvin equations. The Kelvin equations give the pressure gradients across the gas/solid
and liquid/solid interfaces in terms of the differential geometry of these interfaces:
sg
dAs,m
pg ps = γ
(59)
sg
g
dVm
dAssm
l
,
=
γ ls
(60)
pl
ps
l
dVm
where Assm and As,g (m2 mol1) are the molar areas of the liquid/solid and solid/gas inter-
s
l
,
m
faces, respectively, and γls and γsg (N m1) are the corresponding interfacial tensions. Capil-
lary pressures of liquid phases can be estimated by applying these equalities.
If the solid, liquid, and vapor phases are pure, then eq 67 can be rewritten:
*g
*g
g
0 = Sm,H 2OdT Vm,H 2Odpg + d* 2O
(61)
H
0 = Sml,H 2OdT Vml,H 2Odpl + d*l2O
*
*
(62)
H
0 = Sms,H 2OdT VmsH 2Odps + d*s2O
*
*,
(63)
H
where the superscript * indicates a pure phase. By taking the partial derivatives of eqs 68
and 69, dpg and dpl can be related to dps by
dAs,m
sg
=
+ d γ
dpg
dps
sg
g
(64)
dVm
and
ls dAssm
l
,
+ d γ
=
dpl
dps
dVm
.
(65)
l
By subtracting eq 67 from eq 69, and 69 from 68, then subtracting the latter difference from
the former, the following relation is obtained after rearrangement:
S*s
Sm,H 2O Sm,H 2O
*g
*s
*l
m,H 2O Sm,H 2O
dT =
V *s
Vml,H 2O VmsH 2O
*g
*,
*
m,H 2O Vm,H 2O
*g
dAs,m
sg
Vml,H 2O
*
ls dAssm
Vm,H 2O
l
,
d γ
d γ
.
(66)
sg
dVm
V *l
Vms,H 2O
*g
g
*
l
Vm,H 2O
dVm
*s
Vm,H 2O
m,H 2O
*g
By noting that Vm,H 2O >> Vml,H 2O and by assuming that
*
sg
dAs,m
γ
= 0,
sg
g
dVm
the following relation is obtained:
22
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