j = the chemical potential of species j,
ponent i in the system. For the electroneutrality
w = the chemical potential of water,
equation bi = 0 and νij = zj , where zj is the charge
0 = the standard chemical potential of a
of the jth component. In matrix notation, eq 2 may
k
one-component solid-phase k, and
be written as
R = the universal gas constant.
r
r
Νn T = b
The chemical potential of aqueous solution spe-
r
cies j in terms of molality is defined by
where N is the stoichiometric matrix, n is the vec-
r
tor of numbers of moles of species, and
b is the
j = 0 + ln aj = 0 + ln(mj γ j )
vector of bulk chemical composition of the sys-
j
j
tem.
where 0 is the standard chemical potential,
It is convenient to solve the system of linear
equations (eq 2) with respect to P components,
including M (M ≤ P) solids,
nj
mj =
55.51
nw
J +1
nk = Bk - ∑ akj nj
k = 1, M
(2a)
is the moles of the jth species per 1 kg of water
j =1
(molality), and γ is an activity coefficient.
The chemical potential of water may be writ-
J +1
fj = Bj - ∑ aijnj = 0
i = M + 1, P
(2b)
ten as
j =1
w = 0 + ln aw
and in this way to switch to new independent
w
components. In such a manner the stoichiometry
r
where water activity aw, according to Pitzer
of other J + 1 Pr
components (vectors aj ) and the
(1987), is defined through the osmotic coefficient
matter balance B are now defined through these
of the solution φ and molalities of species by
independent components. This operation allows
the number of active constraints to be reduced up
W
to P M. The thermodynamic meaning of this lies
∑
ln aw = -φ
mj
in the fact that the chemical potential of a one-
1000 j
component solid phase is equal to the standard
Gibbs energy of formation and does not depend
where W is the molecular weight of water
on its amount, until this phase is present. This is
(18.0153).
why the system can be considered to be open
Accordingly, the free energy function of the
with respect to this component.
system is as follows:
It is obvious that
Σnj
r
M
g(n) = ∑ 0 nk + nw ( 0 - φ
nw > 0 and nj > 0.
(3)
)
k
w
nw
k =1
Minimization of the function in eq 1 under the
constraints of eq 2 and eq 3 can be replaced by a
nj 55.51
J
search of the extremum of the Lagrangian func-
+ ∑ nj 0 + ln
γ j .
(1)
tion, which may be written as
j
nw
j =1
rr
J +1
r
Mass balance constraints, including the electro-
M
∑ 0 (Bk -
∑ akjnj )
Φ(n, λ) = g(n) +
neutrality equation if necessary, may be written
k
k =1
j =1
as a system of linear equations:
M +1+ J
J +1
P
∑ νijnj = bi ,
i = 1, P
∑ λ i (Bi - ∑ aijnj )
+
(2)
j =1
i = m+1
j =1
where λ is a Lagrangian multiplier. It can be
where P is the number of independent chemical
shown (Karpov et al. 1976) that λ is the chemical
components in the system, and νij is the number
potential of the corresponding independent com-
of moles (stoichiometric units) of independent
ponent of the system. In particular for solids,
component i in one mole of component j. bi repre-
λ k = 0 .
sents the number of moles of independent com-
k
2
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