## 3 Results and Discussion
### Distribution of Components between Phases at Precipitation of Ammonium Alum
Taking into account the fact that ammonium sulfate is a major component in the solution, we can write approximate expressions for electrical neutrality: \(c_{NH_{4}^{+}}\approx c_{NH_{4}SO_{4}^{-}}+2c_{SO_{4}2^{-}}\) and material balance: \(c_{\Sigma SO_{4}}\approx c_{NH_{4}SO_{4}^{-}}+2c_{SO_{4}2^{-}}\). If we subtract the second one from the first, using Equation (9) and supposing that the inequality \(c_{SO_{4}2^{-}}\ll c_{\Sigma SO_{4}}\) takes place, we can obtain the following expression:  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]2  
Inasmuch as \(k_{NH_{4}}\ll 1\) (0.085 was found in [26]), it follows from (29) that, for concentrations \(c_{\Sigma SO_{4}}\geq 0.5\), which are typical for real experiments, the said inequality is indeed satisfied, and the value of the expression in the first bracket in the denominator (28) can be equated to unity. It also follows from (29):  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]3  
Thereafter, with using (30), the expression (28) can be written as follows:  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]4  
For ammonium: \(q_{NH_{4}}/c_{\Sigma NH_{4}}\approx\ q_{\Sigma}/2c_{\Sigma SO_{4}}\approx 1/c_{ \Sigma SO_{4}}\) at \(q_{\Sigma}=2\)  
Taking into account the large number of approximations made above, this dependence may not strictly coincide with the experimental curve. Let us use correction parameter \(\alpha\) to fit the theoretical dependence, and then, using the same coefficient, draw the closest dependences for metal ions to the corresponding curves shown in \[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]5 For such a case:  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]6  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]7  
It can be seen that the constants \(\text{K}_{\text{NH}_{4}}^{\text{Me}_{1}}\) correspond to the series: Cs\({}^{+}>>\) Rb\({}^{+}>>\)NH\({}_{4}+>\) K\({}^{+}>>\) Na\({}^{+}\) more definitely than the dissociation constants of complexes. Theoretical lines shown in Figure 6 are drawn using the same dimensionless parameter: \(\alpha=-0.5\). The constants of hetero-phasic equilibrium of \(\text{K}_{\text{NH}_{4}}^{\text{Mo}_{1}}\) corresponding to these linear dependences are given in Table 3 in comparison with the dissociation constants of sulfate complexes. Figure 7 shows the dependences for the distribution coefficients of other polycharged ions that differ in their chemical nature from aluminum: zirconium, strontium, and lanthanum. The results of tests of three solution samples for each element show that the distribution of these micro-components is more random, and the figures obtained in different experiments may not coincide with each other. It can be seen that, with an increase in the concentration of ammonium bisulfite, the distribution coefficients for lanthanum and strontium can increase (in different ways) in different concentration regions. Unlike these elements, zirconium always remains in solution. In all likelihood, this is due to the fact that zirconium compounds do not co-crystallize with any alum. Considering that lanthanum is a micro-component and strontium is not precipitated by ammonium, analysis of the totality of the obtained data shows that, in order to obtain pure \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Constants** & **Na\({}^{\star}\)** & **K\({}^{\star}\)** & **NH\({}_{4}\)\({}^{\star}\)** & **Rb\({}^{\star}\)** & **Cs\({}^{\star}\)** \\ \hline \(Lgk\) & \(-1.00\) & \(-1.03\) & \(-1.07\) & \(-1.12\) & \(-1.08\) \\ \(Lg\text{K}_{\text{NH}_{4}}^{\text{Me}_{1}}\) & \(-0.86\) & \(-0.14\) & \(0\) & \(1.20\) & \(1.75\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dissociation constants for monocharged sulfate complexes [26] and the obtained constants of hetero-phasic exchange.