The Derivation of the Partial Differential Equation of the Activity Coefficients between the Solutes of Mixed Electrolyte Solutions and Its Numerical Solution

From the basic thermodynamic formule, the partial differential equation of activity coefficients between two solutes in mixed electrolyte solutions is derived by the substitution method of variables. For the ternary system of electrolyte（1）-electrolyte（2）-solvent（s）, the eq. is m\left(\frac\partial \ln \gamma_2\partial m\right)_y-y\left(\frac\partial \ln \gamma_2\partial y\right)_m=\frac\nu_1 k_2\nu_2 k_1\leftm\left(\frac\partial \ln \gamma_1\partial m\right)_y+(1-y)\left(\frac\partial \ln \gamma_1\partial y\right)_m\right（1） where γ_i is the mean activity coefficient of electrolyte i in mixture, v_i is the number of ions per its molecule and v_i=1 for nonelectrolyte. m and y are defined as m=k₁m₁+k₂m₂;y=k₂m₂/m （2） Here k₁ and k₂ are constants. Putting k_i=v_i, m denotes the total ionic concentration and eq. （1） is converted into m\left(\frac\partial \ln \gamma_2\partial m\right)y-y\left(\frac\partial \ln \gamma_2\partial y\right)_m=m\left(\frac\partial \ln \gamma_1\partial m\right)_y+(1-y)\left(\frac\partial \ln \gamma_1\partial y\right)_m（3） As putting k_i=v_i\left(v_i^(+) v_i^(-)\right) / 2, m denotes ionic strength I and eq. （1） is transformed to I\left(\frac\partial \ln \gamma_2\partial I\right)_y-y\left(\frac\partial \ln \gamma_2\partial y\right)_I=\fracv_2^(+) v_2^(-)v_1^(+) v_1^(-)\leftI\left(\frac\partial \ln \gamma_1\partial I\right)_y+(1-y)\left(\frac\partial \ln \gamma_1\partial y\right)_I\right（4） Where v_i⁽⁺⁾ and v_i^(-) denotes the numbers of cation and anion per molecule of electrolyte i. As the experimental data are commonly measuremed under the conditions, the eq. （4） is used more than （3）. As two electrolytes have the same charge type eq.（4） is simplified to eq.（3）.
The solution of partial differential equation are substituted into the opimization parameters over the experimental data by the least square method, which belongs to one of the direct methods for solution the differential equations in the calculus variations. The suitable trail function is chosen by the electrolyte solution theory. Less time is spended in solving the equation and better results are obtained with it than with common trial function of power series. The computed values are consistent with those of Pitzer’s equation. This consistency indicates that the method is reliable and applicable.