## 3 Results and Discussion
### Distribution of Components between Phases at Precipitation of Ammonium Alum
At any total concentration of sulphate in the solution, the values of the distribution coefficients correspond to the series: Cs\({}^{+}>>\) Rb\({}^{+}>>\) NH\({}_{4}\)\({}^{+}>\) K\({}^{+}>>\) Na\({}^{+}\), exactly corresponding to the series of ionic radii in octahedral coordination 1.70 >> 1.49 >> 1.40 > 1.38 >> 1.02 [24,25]:  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\] (1-4)  
\[k_{Me_{i}}=\frac{c_{Me_{i}^{+}}c_{SO_{4}{}^{2-}}}{c_{MeSO_{4}{}^{-}}}\] (5-8)  
\[k_{NH_{4}}=\frac{c_{NH_{4}^{+}}c_{SO_{4}{}^{2-}}}{c_{NH_{4}SO_{4}{}^{-}}} \tag{9}\]  
\[k_{H}=\frac{c_{H^{+}}c_{SO_{4}{}^{2-}}}{c_{HSO_{4}{}^{-}}} \tag{10}\]  
\[c_{Me_{i}^{+}}+c_{MeiSO_{4}{}^{-}}=c_{\Sigma\text{Mei}}\] (11-15)  
\[c_{NH_{4}^{+}}+c_{NH_{4}SO_{4}{}^{-}}=c_{\Sigma NH_{4}} \tag{16}\]  
\[c_{H^{+}}+c_{HSO_{4}{}^{-}}=c_{\Sigma H} \tag{17}\]  
\[c_{SO_{4}{}^{2-}}+\sum c_{MeiSO_{4}{}^{-}}+c_{NH_{4}SO_{4}{}^{-}}+c_{HSO_{4}{}^ {-}}=c_{\Sigma SO_{4}} \tag{18}\]  
Figure 6:0  
Figure 6:1  
\(Me_{i}^{+}\) is any of the alkali metal cations (Na\({}^{+}\), K\({}^{+}\), Rb\({}^{+}\), Cs\({}^{+}\)) contained in solution and solid album, \(q\) and \(c\) are the concentrations of components in the solid phase and solution in mol/kg and mol/L, respectively, \(q_{\Sigma}\) is the total concentration of cations in album (cation capacity), and \(c_{\Sigma^{+}}\) is the sum of all free cations in solution. Relationships (1-4) describe mass action law for ion substitutions in solid phase, and they are analogues of ion exchange equation with the equilibrium coefficients \(K_{NH_{4}}^{Me_{i}}\):  
Figure 6:2  
Figure 6:3 Distribution of Sr, Zr, and La between the sediment and liquor. Relations (5-10) are the equations of dissociation of sulfate complexes in solution with the constants \(k_{i}\); Relations (11)-(18) and (20) describe the material balances, and Relation (19) is the equation of electroneutrality in the solution. Of the 20 equations written, only 16 are independent. For example, one of the Equations (1)-(4) can be replaced using (20) without introducing the unknown concentration of any one component \(q_{M\epsilon_{i}}\). Similarly, it follows from the terms of balances that not all of the equations in (1)-(4) and (5)-(10) are independent. Taking into account the tabular values of the dissociation constants of the complexes and other parameters which are known or can be determined in analyses: \(c_{\Sigma M\epsilon_{i}}\), \(c_{\Sigma NH_{4}}\), \(c_{\Sigma H}\), \(c_{\Sigma SO_{4}}\), \(q_{M\epsilon_{i}}\), \(q_{\Sigma\epsilon}\), the unknowns are: \(K_{NH_{4}}^{M\epsilon_{i}}\), \(c_{M\epsilon_{i}^{+}}\), \(c_{M\epsilon_{i}SO_{4}}\)-, \(c_{NH_{4}+}\), \(c_{NH_{4}SO_{4}}\)-, \(c_{SO_{4}}\)- (in total 16 parameters). We are looking for the ratios of concentrations shown on the graphs and measured in experiments, namely, the total content of each component in solid alum to its total content in equilibrium solution: \(q_{M\epsilon_{i}}/c_{\Sigma\text{Me}}\) and \(q_{NH_{4}}/c_{\Sigma NH_{4}}\). Taking into account (11-17), the relation (18) can be written as follows:  
Figure 6:4  
When summing (18) and (19), we obtain:  
Figure 6:5  
For ammonium, as a macro-component, one can use a variety of approximations:  
Figure 6:6  
When using the last two expressions to make substitutions in (1-4), the equations of Langmuir type isotherm can be written for each metal cation:  
Figure 6:7  
Since the metal ions are minor components (\(c_{Me_{i}^{+}}/c_{\Sigma^{+}}\ll 1\)), relations for linear isotherms follow from (24):  
Figure 6:8  
Taking into account (1-4), (22), and (23)-(25), we obtain from (20):  
Figure 6:9  
In the same way, we obtain for ammonium:  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]0  
Equation (26) can be transformed as follows:  
\[\frac{q_{Me_{i}}}{c_{Me_{i}^{+}}}=K_{NH_{4}}^{Me_{i}}\frac{q_{NH_{4}}}{c_{NH_{4} ^{+}}}\]1  
Significant simplifications of (28) that do not affect the fundamental nature of the regularities can be made.