between them (e.g., ice and water), the total energy is
Properties of chemical potential
the sum of the energy of the various parts:
Equation 18 shows that the chemical potential of a
component is a function of temperature, pressure, and
(15)
dU = dUa + dUb + dUΨ .
amounts of other chemical species. This leads to inter-
esting system behavior. If two regions in the same sys-
The surface work term (see Table 1) is ΨdAr; there-
tem are at different pressures or temperatures, with all
fore,
other properties being held constant, then they will have
different chemical potentials. At constant temperature
dU = PadVa PbdVb + ΨdAr.
(16)
and pressure, the chemical potential of a component in
two regions may be different due to different concen-
At equilibrium, dU = 0, therefore
trations of it. Another property of chemical potential is
(Pa Pb)dVa = ΨdAr.
that, at constant concentration of a species, a pressure
(17)
difference and a temperature difference may compen-
sate each other, thereby maintaining a constant chemi-
Thus, at equilibrium there is a pressure difference across
cal potential.
the interface unless it is planar. If the interface is pla-
Other properties of chemical potential can be deduced.
nar, dAr = 0 and Pa = Pb. The difference in pressure
At constant temperature for a single-component sys-
across a curve interface is the physical reason for cap-
tem, the pressure dependence derived from eq 4a is dG
illary rise and depression of liquids in porous materials
= VdP, which can be integrated to obtain
(e.g., Castellan 1983).
P
∫ VdP
Chemical equilibrium
G = Go (T ) +
(21)
Po
Conditions for chemical equilibrium
From eq 4b,
where Go is the Gibbs free energy at one atmosphere
of pressure, Po. For liquids and solids (constant vol-
G
ume), this relation becomes
= i .
(18)
ηi T,P,η
G = Go(T) + V(P Po).
j≠i
(22)
The chemical potential of a substance, i, is the Gibbs
For ideal gases
free energy increase per mole of substance i added to a
system at constant temperature, pressure, and numbers
nRT
V=
of moles of other substances (j) present in the system.
P
For a system consisting of a pure substance i, dG =
and
SdT + VdP (eq 4a), and this can be divided by ηi so that
G = Go (T ) + nRT ln o
P
(23)
P
d i = -SdT + VdP
(19)
or
where S and V are the entropy and volume per mole
= o (T ) + RT ln o
P
(24)
P
of substance i, respectively.
For a system at constant temperature, pressure, and
where o is the chemical potential of a pure substance
numbers of moles, j, and divided into two regions, a
at one atmosphere. For solid, liquid, or gaseous mix-
and b:
tures of ideal solutions (i.e., Pi = xiPio, where xi is the
dG = dGa + dGb = ia dηi + ibdηi .
mole fraction of the substance and Pi is its partial pres-
(20)
sure)
If dηi moles go into b, and dηi moles leave a, then dGa
i = o (T, P) + RT lnxi .
(25)
= ia (dηi); dGb = ibdηi and dG = (ib ia)dηi.
i
The dependence of Gibbs free energy on temperature
A spontaneous reaction requires that dG be nega-
tive, therefore ib < ia, and matter flows from regions
at constant pressure can be expressed by using eq 4a,
which yields
of high chemical potential to low chemical potential.
G
potentials for substance i must have the same values
= -S.
(26)
T P
throughout a system in chemical equilibrium.
5
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