1.1.9: Activity of Water - Two Solutes
A given solution contains two neutral (i.e. non-ionic) solutes, solute-\(i\) and solute-\(j\). We anticipate, for example, activity coefficient \(\gamma_{i}\) for solute –\(i\) is a function of the molalities of both solutes, \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\). The thermodynamic properties of this class of solutions are discussed by Bower and Robinson [1] and by Ellerton and Dunlop [2]. Because the analysis discussed by these authors concerns the properties of solvent, water in aqueous solutions, the starting point is isopiestic vapour pressure measurements [1-3]. Analysis of the thermodynamic properties of these mixed aqueous solutions has four themes which we develop separately, drawing the analysis together in a final section.
Theme A
Solution I is prepared by dissolving ni moles of solute-\(i\) in water, mass \(\mathrm{w}_{1}(\mathrm{I})\) at temperature \(\mathrm{T}\) and pressure \(p\)( which is close to the standard pressure \(p^{0}\)); \(\mathrm{M}_{1}\) is the molar mass of water and \(\phi(I)\) is the practical osmotic coefficient of the solvent, water, in solution(I). The contribution \(\mathrm{G}_{1}(\mathrm{I})\) of the solvent to the Gibbs energy of the solution is given by equation(a).
\[\mathrm{G}_{1}(\mathrm{I})=\left[\mathrm{w}_{1}(\mathrm{I}) / \mathrm{M}_{1}\right] \,\left\{\mu_{1}^{*}(\lambda)-\left[\phi(\mathrm{I}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})\right]\right\} \nonumber \]
Solution (II) is similarly prepared using \(n_{j}\) moles of solute-\(j\) in water, mass \(\mathrm{w}_{1}(\mathrm{II})\).
\[\mathrm{G}_{1}(\mathrm{II})=\left[\mathrm{w}_{1}(\mathrm{II}) / \mathrm{M}_{1}\right] \,\left\{\mu_{1}^{*}(\lambda)-\left[\phi(\mathrm{II}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\right\} \nonumber \]
We add a sample of solution (I) containing \(1 \mathrm{~kg}\) of water to a sample of solution (II) also prepared using \(1 \mathrm{~kg}\) of water. The resulting solution contains \(2 \mathrm{~kg}\) of water and the initial molalities \(\mathrm{m}_{i}(\mathrm{I})\) and \(\mathrm{m}_{j}(\mathrm{II})\) will be reduced by a half. Then we imagine that \(1 \mathrm{~kg}\) of water is withdrawn from the solution. This concentration process restores the original molalities of solutes \(i\) and \(j\). The letter ‘F’ identifies the new solution.
\[\mathrm{G}_{1}\left(\text { total } ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right] \nonumber \]
\(\phi(\mathrm{F}\)) is the practical osmotic coefficient of the solution prepared using solutions I and II from which \(1 \mathrm{~kg}\) of solvent has been removed. The results of the analysis given above can be summarised in three equations describing the activities of water in the three solutions.
\[\ln \left[\mathrm{a}_{1}(\mathrm{I})\right]=-\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I}) \, \mathrm{M}_{1} \nonumber \]
\[\ln \left[\mathrm{a}_{1}(\mathrm{II})\right]=-\phi(\text { II }) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{M}_{1} \nonumber \]
\[\ln \left[\mathrm{a}_{1}(\mathrm{~F})\right]=-\phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right] \, \mathrm{M}_{1} \nonumber \]
The molalities remain the same as in the original solutions; i.e. \(\mathrm{m}_{\mathrm{i}}(\mathrm{F})=\mathrm{m}_{\mathrm{i}}(\mathrm{I})\) and \(\mathrm{m}_{\mathrm{j}}(\mathrm{F})=\mathrm{m}_{\mathrm{j}}(\mathrm{II})\).
\[\text { By definition, } \Delta \equiv \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})+\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right] \nonumber \]
Experiments based on isopiestic measurements using the equilibrium between reference and mixed solutions and independently determined \(\phi(\mathrm{I})\) and \(\phi(\mathrm{II})\) yield the quantity \(\Delta\).
Theme B
The starting points are general equations for the activity coefficients \(\gamma_{i}\) and \(\gamma_{j}\) for solutes \(i\) and \(j\) respectively as a function of the molalities \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\) in the mixed solutions. Two equations based on Taylor series are used.
\[\ln \left(\gamma_{i}\right)=\sum_{k=0}^{k=\infty} \sum_{\lambda=0}^{\lambda=\infty} A_{k \lambda} \,\left(m_{i} / m^{0}\right)^{k} \,\left(m_{j} / m^{0}\right)^{\lambda} \nonumber \]
\[\ln \left(\gamma_{k}\right)=\sum_{k=0}^{k=\infty} \sum_{\lambda=0}^{\lambda=\infty} B_{k \lambda} \,\left(m_{i} / m^{0}\right)^{k} \,\left(m_{j} / m^{0}\right)^{\lambda} \nonumber \]
With reference to equations (h) and (i), both \(\mathrm{A}_{00}\) and \(\mathrm{B}_{00}\) are zero. The dimensionless coefficients \(\mathrm{A}_{k} \lambda\) and \(\mathrm{B}_{k} \lambda\) are interlinked by the Gibbs-Duhem equation . It also turns out that the series up to and including ‘\(k = 4\)’ and ‘\(\lambda = 4\)’ are sufficient in the analysis of experimental results.
According to equation (h), a description of the properties of solute-\(i\) is given by equation (j).
\[\begin{aligned}
\ln \left(\gamma_{\mathrm{i}}\right)=& \mathrm{A}_{10} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{20} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \\
&+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\
&+\mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{A}_{30} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\
&+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{40} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \\
&+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\
&+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{4}
\end{aligned} \nonumber \]
In the event that \(\mathrm{m}_{j}\) is zero,
\[\begin{aligned}
\ln \left[\gamma_{\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}=0\right)\right]=\mathrm{A}_{10} \, &\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{A}_{20} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \\
&+\mathrm{A}_{30} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{40} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4}
\end{aligned} \nonumber \]
Moreover \(\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]\) can be calculated from the measured properties of aqueous solutions containing only solute-\(i\). Therefore the dependence of \(\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]\) on \(\mathrm{m}_{i}\) can be analysed using a linear least squares procedure to yield the coefficients \(\mathrm{A}_{k 0}\) for \(k=1- 4\). Hence \(\ln \left(\gamma_{i}\right)\) for the mixed solute system is given by a combination of equations (j) and (k) to yield equation (\(\lambda\)).
\[\begin{aligned}
&\ln \left(\gamma_{\mathrm{i}}\right)=\ln \left[\gamma_{\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}=0\right)\right] \\
&+\mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\
&+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\
&\quad+\mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\
&\quad+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\
&\quad+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{4}
\end{aligned} \nonumber \]
According to equation (\(\lambda\)), the dependence of \(\gamma_{i}\) on \(\mathrm{m}_{j}\) at fixed \(\mathrm{m}_{i}\) is given by equation (m).
\[\begin{aligned}
{\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})} } &=\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{02} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\
&+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\
&+\mathrm{A}_{03} \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-1} \\
&+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{2}+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\
&+\mathrm{A}_{04} \, 4 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4}
\end{aligned} \nonumber \]
A cross-differential link yields the following interesting equation.
\[\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{i}}}\right]_{\mathrm{m}(\mathrm{j})} \nonumber \]
We combine equations (m) and (n).
\[\begin{aligned}
{\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{i}}}\right]_{\mathrm{m}(\mathrm{j})} } &=\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{02} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\
&+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\
&+\mathrm{A}_{03} \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-1} \\
&+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\
&+\mathrm{A}_{04} \, 4 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4}
\end{aligned} \nonumber \]
We integrate the latter equation to yield an equation for \(\gamma_{j}\left(m_{i}=0\right)\) where at ‘\(\mathrm{m}_{i} = 0\)’ , \(\gamma_{j}\) is represented as \(\gamma_{j}\left(\mathrm{~m}_{\mathrm{i}}=0\right)\). The outcome is an equation for \(\ln \left(\gamma_{j}\right)\) in terms of the \(\mathrm{A}_{i}\)-variables, making the \(\mathrm{B}_{i}\) variables somewhat redundant.
\[\begin{aligned}
\ln \left(\gamma_{\mathrm{j}}\right)=& \ln \left[\gamma_{\mathrm{j}}\left(\mathrm{m}_{\mathrm{i}}=0\right)\right] \\
+& \mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\left(\mathrm{A}_{11} / 2\right) \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2}+2 \, \mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}}\right) \,\left(\mathrm{m}^{0}\right)^{-2} \\
&+\left(\mathrm{A}_{21} / 3\right) \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \\
&+3 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\
&+(1 / 4) \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \\
&+\left(2 \, \mathrm{A}_{22} / 3\right) \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4}+\left(3 \, \mathrm{A}_{13} / 2\right) \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \\
&+4 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4}
\end{aligned} \nonumber \]
Theme C Gibbs-Duhem Equations
A Single Solute
For an aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\) containing the single solute-\(i\), the Gibbs-Duhem equation yields the following relationship.
\[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{i}} \, \mathrm{d} \mu_{\mathrm{i}}(\mathrm{aq})=0 \nonumber \]
\[\text { Then, } \begin{aligned}
\frac{1}{\mathrm{M}_{1}} \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi_{\mathrm{i}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{i}}\right] \\
&+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right]=0
\end{aligned} \nonumber \]
The symbol \(\phi_{i}\) identifies the practical osmotic coefficient in a solution containing solute-\(i\).
\[\text { Hence }[4], \quad \mathrm{d}\left[\phi_{i} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{dm} \mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right) \nonumber \]
Similarly for an aqueous solutions containing solute-\(j\),
\[\mathrm{d}\left[\phi_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
Two Solutes
From the Gibbs-Duhem equation (at fixed \(\mathrm{T}\) and \(p\))
\[n_{1} \, d \mu_{1}(a q)+n_{i} \, d \mu_{i}(a q)+n_{j} \, d \mu_{j}(a q)=0 \nonumber \]
Then,
\[\begin{aligned}
&\frac{1}{\mathrm{M}_{1}} \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi_{\mathrm{ij}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\
&\quad+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \\
&\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0
\end{aligned} \nonumber \]
The practical osmotic coefficient \(\phi_{ij}\) identifies a solution containing two solutes, \(i\) and \(j\).
\[\text { Hence, } \mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right) \nonumber \]
\[\text { Therefore[5] } \mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right]=\mathrm{d}\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
According to equation (g)
\[\Delta \equiv \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})+\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right] \nonumber \]
\[\text { Then } \mathrm{d} \Delta=\mathrm{d}\left\{\phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\mathrm{d}\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})\right]-\mathrm{d}\left[\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\right. \nonumber \]
Labels (I) and (II) can be dropped when applied to solute molalities. Then using equations (s), (t) and (x),
\[\begin{array}{r}
\mathrm{d} \Delta=\mathrm{d}\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}(\mathrm{F})\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right] \\
\quad-\mathrm{dm}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{i}}(\mathrm{I})\right]-\mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right.
\end{array} \nonumber \]
\[\text { Or, } \mathrm{d} \Delta=\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left\{\ln \left(\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right]-\mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right\}\right.\right. \nonumber \]
In equation (\(\lambda\)) we identify \(\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]\) with \(\ln \left[\gamma_{i}(\mathrm{I})\right]\). Similarly \(\ln \left[\gamma_{\mathrm{j}}\left(\mathrm{m}_{\mathrm{i}}=0\right)\right]\) in equation (p) equals \(\ln \left[\gamma_{\mathrm{j}}(\mathrm{II})\right]\) in equation (za). Therefore [6]
\[\begin{aligned}
\Delta / & \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \\
=& \mathrm{A}_{01}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-1}+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \\
&+(3 / 2) \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \\
&+(4 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3}+2 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\
&+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3}
\end{aligned} \nonumber \]
Footnotes
[1] V. E. Bower and R. A. Robinson , J.Phys.Chem.,1963, 67 ,1524.
[2] H. D. Ellerton and P. J. Dunlop, J. Phys Chem.1966, 70 ,1831.
[3] H. D. Ellerton, G. Reinfelds, D. E. Mulcahy and P. J. Dunlop, J. Phys. Chem., 1964, 68 ,398.
[4] Hence, \(-\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)=0\)
Then, \(-\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{i}}\right)+\ln \left(\gamma_{\mathrm{i}}\right)-\ln \left(\mathrm{m}^{0}\right)\right]=0\)
Or, \(\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{m}_{\mathrm{i}} \, \frac{1}{\mathrm{~m}_{\mathrm{i}}} \mathrm{dm} \mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)\)
Then, \(\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{dm}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)\)
[5] Or,
\[\begin{aligned}
&\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\
&\quad=\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Or
\[\begin{aligned}
&\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\
&\quad=\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}_{\mathrm{i}}\right) \, \mathrm{d}\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
&\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\
&\quad=\mathrm{d}\left(\mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{i}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
[6] Differentiation of equation (\(\lambda\)) yields
\[\begin{aligned}
&\mathrm{d}\left\{\ln \left[\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{i}}(\mathrm{I})\right]\right\}=\right. \\
&\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{j}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\
&+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}}+2 \, \mathrm{A}_{21} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\
&+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\
&+2 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{j}_{\mathrm{j}}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}} \\
&+3 \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\
&+2 \, \mathrm{A}_{22} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\
&+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+3 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\
&+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}}
\end{aligned} \nonumber \]
Similarly from equation (p),
\[\begin{aligned}
&\mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}=\right. \\
&\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}} \\
&+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\
&+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\
&+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\
&+6 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \\
&+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm} \mathrm{m}_{\mathrm{i}}+(2 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm} \\
&\mathrm{j} \\
&+3 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \\
&+3 \, \mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}}+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \\
&+12 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}}
\end{aligned} \nonumber \]
But according to equation (za),
\[\mathrm{d} \Delta=\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left\{\ln \left(\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right]-\mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right\}\right.\right. \nonumber \]
After rearranging one obtains the following equation.
\[\begin{aligned}
&\mathrm{d} \Delta= \\
&\mathrm{A}_{01} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{01} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{j}} \\
&+2 \, \mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\
&+2 \, \mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+4 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\
&+3 \, \mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \\
&+3 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+3 \, \mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \\
&+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+9 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \\
&+4 \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{4} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\
&+4 \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+(8 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\
&+4 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+6 \, \mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\
&+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{4} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+16 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}}
\end{aligned} \nonumber \]
Term by term integration of the latter equation yields
\[\Delta=\int \mathrm{d} \Delta \nonumber \]
As an example we cite the terms containing the coefficient \(\mathrm{A}_{11}\).
\[\begin{aligned}
&\int 2 \, A_{11} \, m_{i} \, m_{j} \,\left(m^{0}\right)^{-2} \, d m_{i}+\int A_{11} \,\left(m_{i}\right)^{2} \,\left(m^{0}\right)^{-2} \, d m_{j} \\
&=A_{11} \,\left(m^{0}\right)^{-2} \, d m_{i} \, \int\left[2 \, m_{i} \, m_{j} \, d m_{i}+\left(m_{i}\right)^{2} \, d m_{j}\right] \\
&=A_{11} \,\left(m^{0}\right)^{-2} \, \int d\left[\left(m_{i}\right)^{2} \, m_{j}\right] \\
&=A_{11} \,\left(m^{0}\right)^{-2} \,\left(m_{i}\right)^{2} \, m_{j}
\end{aligned} \nonumber \]