163
Ionic Interactions
γ T (i) = ([i] F /[i] T ) γ F (i)
(4.60)
This total activity coefficient is the one desired to obtain activities from total concentrations (i.e., using Equation 4.60). The term α F = [i] F /[i] T is the fraction of free ions in a solution
of fixed composition and ionic strength. If a series of one- to- one complexes is formed,
M i + X i → M i X i
(4.61)
the ion- pairing constant for the formation of M i X i
o is given by
K* MX = [M i X 0 ]/[M +
i ][X i ]
(4.62)
K MX * = K MX [γ F (M) γ F (X)/γ F [M i X 0 ]
(4.63)
where K MX is the thermodynamic constant in pure water, K MX
* is the stoichiometric constant, and γ F (i) is the activity coefficient of species i. The total concentration of M i and X i is
given by
[M i ] T = [M i ] F + ∑ [M i X i
0 ]
(4.64)
[X i ] T = [X i ] F + ∑ [M i X i
0 ]
(4.65)
where ∑ [M i X i
o ] is the sum of all the various ion pairs in the solution. By combining these
equations with Equation 4.62, we have
α M = [M] F /[M] T = (1 + ∑ K MX *[X i ] F ) –1
(4.66)
α X = [X] F /[X] T = (1 + ∑ K MX *[M i ] F ) –1
(4.67)
These equations can be solved by a series of iterations if K MX * is known. Several computer
programs are available to aid in these iterations. The results, however, are dependent on
the quality of the values of K MX
* , which are functions of ionic strength and, to a degree, the
composition of the solutions. The fraction of a given ion pair can be obtained from
[MX i ]/[M] T = K MX *[X i ] F α M
(4.68)
[M i X]/[X] T = K MX *[M i ] F α X
(4.69)
It should be pointed out that the form of the given complex does not affect the thermodynamic activity of M or X.
γ T (M) = α M γ F (M)
(4.70)
γ T (X) = α X γ F (X)
(4.71)
For the major ionic components of seawater, Millero and Schreiber (1982) have given the
ionic strength function for K MX
* and γ F (i) for a number of ions. These equations can be used
to calculate the speciation and activity coefficients of the major components of natural
waters using a personal computer.
Ionic Interactions
γ T (i) = ([i] F /[i] T ) γ F (i)
(4.60)
This total activity coefficient is the one desired to obtain activities from total concentrations (i.e., using Equation 4.60). The term α F = [i] F /[i] T is the fraction of free ions in a solution
of fixed composition and ionic strength. If a series of one- to- one complexes is formed,
M i + X i → M i X i
(4.61)
the ion- pairing constant for the formation of M i X i
o is given by
K* MX = [M i X 0 ]/[M +
i ][X i ]
(4.62)
K MX * = K MX [γ F (M) γ F (X)/γ F [M i X 0 ]
(4.63)
where K MX is the thermodynamic constant in pure water, K MX
* is the stoichiometric constant, and γ F (i) is the activity coefficient of species i. The total concentration of M i and X i is
given by
[M i ] T = [M i ] F + ∑ [M i X i
0 ]
(4.64)
[X i ] T = [X i ] F + ∑ [M i X i
0 ]
(4.65)
where ∑ [M i X i
o ] is the sum of all the various ion pairs in the solution. By combining these
equations with Equation 4.62, we have
α M = [M] F /[M] T = (1 + ∑ K MX *[X i ] F ) –1
(4.66)
α X = [X] F /[X] T = (1 + ∑ K MX *[M i ] F ) –1
(4.67)
These equations can be solved by a series of iterations if K MX * is known. Several computer
programs are available to aid in these iterations. The results, however, are dependent on
the quality of the values of K MX
* , which are functions of ionic strength and, to a degree, the
composition of the solutions. The fraction of a given ion pair can be obtained from
[MX i ]/[M] T = K MX *[X i ] F α M
(4.68)
[M i X]/[X] T = K MX *[M i ] F α X
(4.69)
It should be pointed out that the form of the given complex does not affect the thermodynamic activity of M or X.
γ T (M) = α M γ F (M)
(4.70)
γ T (X) = α X γ F (X)
(4.71)
For the major ionic components of seawater, Millero and Schreiber (1982) have given the
ionic strength function for K MX
* and γ F (i) for a number of ions. These equations can be used
to calculate the speciation and activity coefficients of the major components of natural
waters using a personal computer.