4.1.1. Establishment of Thermodynamic Model
The equilibrium between copper-containing species and simple ions in the solution can be expressed by Eq. (4).
In the formula, i, j, k and p are respectively the number of Cu2+, NH3, OH- and SO42- in the coordinated ions, and their values are: i=1,2; j=0,1,2,3,4,5; k=0,1,2,3,4; p=0,1.
The stability constant expression of the complex is shown in Eq. (5) [
19].
The relationship between the available coordinated ions and their constituent simple ions is shown in Eq. (6).
Three solid phases (CuO, Cu(OH)
2 and Cu(OH)
1.5(SO
4)
0.25 )may exist in the CuO-NH
3-(NH
4)
2SO
4 system, because CuO and Cu(OH)
2 have the same solubility product form. The existence of two stable solid phases can be determined by comparing the solubility product (Ksp) of CuO and Cu(OH)
2. When CuO is assumed as the stable solid phase, the dissolution reaction is shown in Eq. (7). When Cu(OH)
2 is assumed as the stable solid phase, the solution equation is shown in Eq. (8).
By comparing equations (7) and (8), it can be seen that the Ksp of CuO is smaller than that of Cu(OH)2, indicating that CuO is more stable than Cu(OH)2. Even if solid Cu(OH)2 is formed during the dissolution process, it will be converted to CuO until the Cu(OH)2 disappears. This shows that there is no Cu(OH)2 solid phase in CuO-NH3-(NH4)2SO4 system with CuO.
According to Eq. (7), Eq. (9) is obtained.
By substituting Eq. (9) into Eq. (6), Eq. (10) can be obtained.
where A=ln(βi,j,k,p10-19.51i)-(k-2i)ln10
pKw; B=j; C=(k-2i)ln10; D=p.
When the stable solid is Cu(OH)
1.5(SO
4)
0.25, the dissolution reaction is shown in Eq. (11).
According to Eq. (11), Eq. (12) is obtained.
Substitute Eq. (12) into Eq. (6) to obtain Eq. (13).
where A=ln(βi,j,k,p10-16.86i)-(k-1.5i)ln10
pKw; B=j; C=(k-1.5i)ln10; D=p-0.25i.
The equilibrium equations of total copper, total ammonia and charge in solution can be expressed as equations (14), (15) and (16) respectively.
where c(Cu
2+)T and c(NH
3)T respectively represent the total concentration of copper and ammonia in the solution. The total concentration of ammonia in the solution is the sum of the concentration of ammonia added to the solution c(NH
3) and twice the concentration of ammonium sulfate c[(NH
4)
2SO
4]. ‘m’ and ‘[X]’ respectively represent the charge number and concentration of each ion in the solution.
When the stable solid phase is CuO, the mass balance equation of total sulfate root can be expressed as Eq. (17) respectively.
where c(SO
42-)T equal to the concentration of total ammonium sulfate added to the solution.
When the stable solid phase is Cu(OH)
1.5(SO
4)
0.25, the total sulfuric acid concentration in the solution comes from two parts: Per mole of ammonium sulfate added to the solution and per mole of Cu(OH)
1.5(SO
4)
0.25 dissolved in the solution contribute 1 mol and 0.25 mol of sulfate ions respectively, so the mass balance equation of total sulfuric acid concentration can be expressed as Eq. (18).
The expression [X]=exp(A+Bln[NH3]+CpH+Dln[SO42-]) respectively into equations (14), (15), (16) and (17) or (14), (15), (16), (18). It can be seen that there are six unknowns in the system of equations: c(Cu2+)T, c(NH3), c[(NH4)2SO4)], [NH3], pH and [SO42-]. During the experiment, the concentration of ammonia and ammonium sulfate is taken as the control condition to change between 0 mol/L and 5 mol/L. Only four variables [NH3], pH, [SO42-] and c(Cu2+) remain in the system of unknowns. It can be solved by calling the function “fsolve” in the Matlab software to obtain the stereogram of the changes of [NH3], pH, [SO42-] and c(Cu2+) with c(NH3) and c[(NH4)2SO4)].
Surface diagram of relation between variables c(NH3) and c[(NH4)2SO4]
The solution results of the thermodynamic model of CuO-NH
3-(NH
4)
2SO
4-H
2O system regarding [NH
3], pH, [SO
42-] and [c(Cu
2+)] are shown in
Figure 4.
As shown in (a), (b) and (c) in
Figure 4, [NH
3] is jointly controlled by c(NH
3) and c[(NH
4)
2SO
4], but it is mainly provided by c(NH
3), and c[(NH
4)
2SO
4] plays a very small role. c(NH
3) can promote the increase of pH, and c[(NH
4)
2SO
4] can inhibit the increase of pH. When c(NH
3) is between 0 and 1 mol/L, pH and c(NH
3) have a linear relationship. Free sulfate ions are provided by c[(NH
4)
2SO
4] and increase with the increase of c[(NH
4)
2SO
4]. The surface relation between c(Cu
2+)T, c[NH
3] and c[(NH
4)
2SO
42-] is shown by in
Figure 4(d). The variation relation between c(Cu
2+)T and c[NH
3] and c[(NH
4)
2SO
42-] can be obtained, but it cannot accurately reflect the magnitude of the numerical variation.
(1)Effects of c(NH3) and c[(NH4)2SO4] on the concentration of Cu2+
The surface relation between c(Cu
2+)T and c[NH
3] and between c(Cu
2+)T and c[(NH
4)
2SO
42-] are projected onto the c(NH
3)–c(Cu
2+)T plane and c[(NH
4)
2SO
4]–c(Cu
2+)T plane, respectively, as shown in
Figure 5.
As shown in
Figure 5(a), the concentration of ammonium sulfate has a great influence on c(Cu
2+)T. Increasing the concentration of ammonium sulfate is conducive to the improvement of c(Cu
2+)T. When c[(NH
4)
2SO
4] ≥ 3 mol/L, c(Cu
2+)T increases linearly with the increase of c(NH
3). According to
Figure 5(b), when the solution is pure ammonium sulfate solution, the solubility of copper oxide (CuO) is very small, that is, pure ammonium sulfate can hardly leach copper oxide (CuO), and c[(NH
4)
2SO
4] only has a great influence on c(Cu
2+)T when it changes between 0 and 3 mol/L. After 3 mol/L, the further increase of c[(NH
4)
2SO
4] decreases the increase of c(Cu
2+)T gradually.
(2) Stable region map
In this system, there may be a case where the equilibrium solid phase is Cu(OH)
1.5(SO
4)
0.25. As shown in
Figure 6, it is easy to compare the size of Jsp and Ksp for Cu(OH)
1.5(SO
4)
0.25.
As shown in
Figure 6, when ammonium sulfate concentration is large and ammonia concentration is small, the Jsp of Cu(OH)
1.5(SO
4)
0.25 is larger than that of Ksp, that is to say, CuO is transformed into Cu(OH)
1.5(SO
4)
0.25, and this region is the stable region of Cu(OH)
1.5(SO
4)
0.25. Other regions are stable regions of CuO.