Kinetics of Hydrogen Reduction of Chalcopyrite Concentrate

Abstract

A Ghatshila chalcopyrite concentrate (average particle size, 50 μm) containing primarily CuFeS₂ and SiO₂ (Cu 16 pct) was reduced by a stream of hydrogen in a thermogravimetric analyzer (TGA) at selected temperatures [1173 K to 1323 K (900 °C to 1050 °C)], hydrogen flow rates, partial pressures of hydrogen (0.33 × 101.3 to 101.3 kPa), and sample bed heights. The product was a mixture of Cu (26 pct), SiO₂, CuFeO₂, and Fe. The rate equations for the three typical controlling mechanisms, namely, gas film diffusion (mass transfer), pore diffusion, and interfacial reaction, have been derived for the system geometry under study and applied to identify the rate-controlling steps. The first stage of the reduction, which extended up to the first 13 minutes, was rate controlled by the interfacial reaction. The last stage, which spanned over the last 60 to 120 minutes and accounted for a small percentage of reduction, was controlled by pore diffusion through the built-up Cu (and Fe) layer. The activation energy in the first stage was 101 kJ mol⁻¹ and that in the second stage was 76 kJ mol⁻¹. Subsequent acid leaching with 1 M HCl solution of the reduction product removed all soluble species, leaving a Cu (53.3 pct) + SiO₂ mixture, with a small concentration (2.7 pct) of Cu₂O in it. This result compares well with the predicted final mixture of Cu (59 pct)-SiO₂ based on a mass balance on the starting concentrate. A follow-up heating at 1523 K (1250 °C) produced a sintered Cu-SiO₂ composite with spherical copper particles of 400 µm diameter embedded in a silica matrix. Elemental chemical analyses were carried out by energy-dispersive X-ray spectroscopy/atomic absorption spectroscopy. The phase identification and microstructural characterization of Cu-SiO₂ mixtures were carried out by X-ray powder diffraction and optical microscopy.

Appendix

Derivation of the Rate Equations (Eqs. [5–7])

Gas film diffusion-controlled kinetics

According to the shell mass balance over a shell thickness Δz in the gas film inside the crucible shown in Figure 1,

$$ \begin{aligned} & \left( {{\text{Molar}}\;{\text{rate}}\;{\text{of}}\;{\text{A}}\;{\text{into}}\;{\text{the}}\;{\text{shell}}} \right)\, - \,\left( {{\text{Molar}}\;{\text{rate}}\;{\text{of}}\;{\text{A}}\;{\text{out}}\;{\text{of}}\;{\text{the}}\;{\text{shell}}} \right) \\ & \quad + \left( {{\text{Molar}}\;{\text{rate}}\;{\text{of}}\;{\text{formation}}\;{\text{of}}\;{\text{A}}\;{\text{through}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{within}}\;{\text{the}}\;{\text{shell}}} \right) \\ & \quad - \left( {{\text{Molar}}\;{\text{rate}}\;{\text{of}}\;{\text{removal}}\;{\text{of}}\;{\text{A}}\;{\text{through}}\;{\text{chemical}}\;{\text{reaction}}\;{\text{within}}\;{\text{the}}\;{\text{shell}}} \right) \\ & \quad = \left( {{\text{Molar}}\;{\text{rate}}\;{\text{of}}\;{\text{accumulation}}\;{\text{of}}\;{\text{A}}\;{\text{within}}\;{\text{the}}\;{\text{shell}}} \right) \\ \end{aligned}. $$

(A1)

The species A is the reactant gas (H₂, in the current study) and species B, which will be used subsequently, is the product gas (H₂S, in the current study) in the gas-solid reaction given by Eq. [4]. At the steady state, the accumulation term disappears and since no chemical reaction occurs within the gas film, the relation simplifies to

$$ N_{A( - z)} |_{{{\text{z}} + \Delta {\text{z}}}} \cdot S - N_{{A\left( { - z} \right)}} |_{\text{z}} \cdot S = 0 $$

(A2)

$$ \mathop {\text{Lt}}\limits_{\Delta z \to 0} \frac{{N_{A( - z)} |_{{{\text{z}} + \Delta {\text{z}}}} - N_{A( - z)} |_{\text{z}} }}{\Delta z} = 0 $$

$$ \frac{\text{d}}{{{\text{d}}z}}\left[ {N_{{A\left( { - z} \right)}} } \right] = 0. $$

(A3)

The diffusion flux with respect to stationary coordinate, in mol area⁻¹ time⁻¹, is given by

$$ N_{A( - z)} = J^{*}_{A( - z)} + x_{A } \left[ {N_{{A\left( { - z} \right)}} + N_{{B\left( { - z} \right)}} } \right], $$

(A4)

where the first term in the right-hand side is the flux relative to the velocity(v*) of the gas mixture and the second term is the flux due to the velocity of the gas mixture. The second term is zero in the present case, because, according to Reaction [4], the number of moles of the reacting gas is equal to the number of moles of the produced gas. This represents a case of equimolar counterdiffusion for which N _A(−z) = −N _B(−z). Applying Fick’s first law of diffusion to the first term of the right-hand side of Eq. [A4], the relation reduces to

$$ N_{{A\left( { - z} \right)}} = C \cdot D_{\text{AB}} \frac{{{\text{d}}x_{A} }}{{{\text{d}}z}}. $$

(A5)

Equating [A3] and [A5]

$$ \frac{\text{d}}{{{\text{d}}z}}\left[ {C \cdot D_{\text{AB}} \frac{{{\text{d}}x_{A} }}{{{\text{d}}z}}} \right] = 0. $$

(A6)

In Eq. [A6], at a given temperature, C and D _AB are constants, because

$$ C = \frac{p}{{{\text{R}}T}} $$

(A7)

and $D_{\text{AB}}$, interdiffusion coefficient of A–B, is not a function of concentration for a low-density binary gas mixture. These give $\frac{\text{d}}{{{\text{d}}z}}\left( {\frac{{{\text{d}}x_{A} }}{{{\text{d}}z}}} \right) = 0$, which on two successive integration gives $\frac{{{\text{d}}x_{A} }}{{{\text{d}}z}} = C_{1}$ and $x_{A} = C_{1} z + C_{2}$, which is the equation of the concentration profile in terms of the two integration constants C ₁ and C ₂. The value of C ₁ can be evaluated with the help of the following two boundary conditions (a) at $z = z_{1}^{0}$, $x_{A} = x_{{A_{\text{eq}} }}$ and (b) at $z = z_{2}$, $x_{A} = x_{{A_{\text{in}} }}$ as $(x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} )/(z_{2} - z_{1}^{0} )$. Subscript ‘in’ stands for inlet gas and ‘eq’ stands for equilibrium. Substituting the value of C ₁

$$ N_{{A\left( { - z} \right)}} = C \cdot D_{\text{AB}} C_{1} = C \cdot D_{\text{AB}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{2} - z_{1}^{0} }}. $$

(A8)

Now, the molar rate of diffusion (in mol time⁻¹) of A, W _A(−z), is equal to the molar rate of removal of A through chemical reaction (Reaction [4]), $- {\text{d}}n_{{{\text{H}}_{2} }} /{\text{d}}t$, since the two steps (mass transfer and interfacial reaction) are in series, and the latter is equal to $2( - {\text{d}}n_{{{\text{CuFeS}}_{2} }} /{\text{d}}t)$from the stoichiometry of Reaction [4].

$$ W_{{A\left( { - z} \right)}} \equiv W_{{{\text{H}}_{2} \left( { - z} \right)}} = - \frac{{{\text{d}}n_{{{\text{H}}_{2} }} }}{{{\text{d}}t}} = - 2\frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}}. $$

Since W _A(−z) = S·N _A(−z) (S is the cross-sectional area of the crucible), and from Eq. [A8],

$$ - \frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = \frac{S}{2} \cdot C \cdot D_{\text{AB}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{2} - z_{1}^{0} }}. $$

(A9)

Letting $A_{1} = \frac{S}{2} \cdot C \cdot D_{\text{AB}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{2} - z_{1}^{0} }} = {\text{constant}}$, the integration of Eq. [A9] gives

$$ \mathop \smallint \limits_{{n_{{{\text{CuFeS}}_{2} }} = n^{0}_{{{\text{CuFeS}}_{2} }} }}^{{n_{{{\text{CuFeS}}_{2} }} = n_{{{\text{CuFeS}}_{2} }} }} {\text{d}}n_{{{\text{CuFeS}}_{2} }} = A_{1} \mathop \smallint \limits_{t = 0}^{t = t} {\text{d}}t $$

$$ n_{{{\text{CuFeS}}_{2} }} - n^{0}_{{{\text{CuFeS}}_{2} }} = A_{1} \cdot t. $$

(A10)

Now, fractional weight loss of CuFeS₂ sample is given as follows:

$$ \begin{aligned} X_{{{\text{CuFeS}}_{2} }} \, & = \frac{{n^{0}_{{{\text{CuFeS}}_{2} }} - n_{{{\text{CuFeS}}_{2} }} }}{{n^{0}_{{{\text{CuFeS}}_{2} }} }} = \frac{{A_{1} \cdot t}}{{n^{0}_{{{\text{CuFeS}}_{2} }} }} = \frac{{A_{1} \cdot t}}{{S \cdot z_{1}^{0} \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \\ & = \frac{{S \cdot C \cdot D_{\text{AB}} (x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} ) \cdot t}}{{2(z_{2} - z_{1}^{0} ) \cdot S \cdot z_{1}^{0} \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \\ \end{aligned}. $$

(A11)

In Eq. [A11], $\rho_{{m_{{{\text{CuFeS}}_{2} }} }}$is the molar density of CuFeS₂. Replacing A by H₂

$$ X_{{{\text{CuFeS}}_{2} }} = \frac{{C \cdot D_{\text{AB}} \left( {x_{{{\text{H}}_{{2_{\text{in}} }} }} - x_{{{\text{H}}_{{2_{\text{eq}} }} }} } \right)}}{{2(z_{2} - z_{1}^{0} )z_{1}^{0} \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \cdot t = \frac{{D_{\text{AB}} \left( {C_{{{\text{H}}_{{2_{\text{in}} }} }} - C_{{{\text{H}}_{{2_{\text{eq}} }} }} } \right)}}{{2(z_{2} - z_{1}^{0} )z_{1}^{0} \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \cdot t. $$

(A12)

The above equation is the rate equation, consistent with the system geometry under study, for gas film diffusion-controlled kinetics. Equation [A13] is reproduced as Eq. [5] in the text.

Pore diffusion-controlled kinetics

Unlike the gas film diffusion-controlled kinetics in which the gas film thickness is constant (steady), the pore diffusion involves a moving boundary (between porous product and CuFeS₂ powder, in Figure 1) in which the thickness of the diffusion zone (i.e., the thickness of the porous product) increases with time. The shell mass balance over a shell thickness Δz in the pore diffusion zone gives, as in Eq. [A1], (Molar rate of A in) − (Molar rate of A out) = (Molar rate of accumulation of A),

$$ N_{A( - z)} |_{z + \Delta z} \cdot S - N_{{A\left( { - z} \right)}} |_{z} \cdot S = 0. $$

(A13)

This is a case of pseudo-steady-state diffusion (since the product/reactant boundary is a moving one), for which also the accumulation term is zero.

$$ \begin{aligned} \mathop {\text{Lt}}\limits_{\Delta z \to 0} \frac{{N_{A( - z)} |_{z + \Delta z} - N_{A( - z)} |_{z} }}{\Delta z} = 0 \hfill \\ \frac{\text{d}}{{{\text{d}}z}}\left[ {N_{{A\left( { - z} \right)}} } \right] = 0 \hfill \\ \end{aligned} $$

(A14)

As considered previously, the diffusion flux with respect to stationary coordinate (in mol area⁻¹ time⁻¹) is given as follows:

$$ N_{A( - z)} = J^{*}_{A( - z)} + x_{A} \left[ {N_{{A\left( { - z} \right)}} + N_{{B\left( { - z} \right)}} } \right]. $$

(A15)

The second term is, again, zero due to the equimolar counter diffusion. Applying Fick’s first law to the first term of the right-hand side

$$ N_{{A\left( { - z} \right)}} = - C \cdot D_{\text{pore}} \frac{{{\text{d}}x_{A} }}{{{\text{d}}\left( { - z} \right)}}. $$

(A16)

Equating [A14] and [A16]

$$ \frac{\text{d}}{{{\text{d}}z}}\left[ {C \cdot D_{\text{pore}} \frac{{{\text{d}}x_{A} }}{{{\text{d}}z}}} \right] = 0 $$

(A17)

Bringing the two constant terms C and D _pore out of the differentiation and integrating twice with the two boundary conditions: (a) at z = z ⁰₁ , $x_{A} = x_{{A_{\text{in}} }}$, and (b) at z = z ₁, $x_{A} = x_{{A_{\text{eq}} }}$, the diffusion flux of A becomes, as before,

$$ N_{{A\left( { - z} \right)}} = C \cdot D_{\text{pore}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{1}^{0} - z_{1} }}, $$

(A18)

where z ₁ is a function of t in accordance with the pseudo-steady state. The molar rate of diffusion (in mol time⁻¹) of A:

$$ W_{{A\left( { - z} \right)}} = S \cdot N_{{A\left( { - z} \right)}} = S \cdot C \cdot D_{\text{pore}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{1}^{0} - z_{1} }}. $$

(A19)

The molar rate of diffusion of A (H₂) at the product/reactant interface being equal to the molar rate of removal of H₂ by the interfacial reaction (Reaction [4]) because the two steps are in series and using the stoichiometry of Reaction [4], as before, and Eq. [A19], the following equalities are obtained:

$$ \begin{aligned} & W_{{A\left( { - z} \right)}} \equiv W_{{{\text{H}}_{2} \left( { - z} \right)}} = - \frac{{{\text{d}}n_{{{\text{H}}_{2} }} }}{{{\text{d}}t}} = - 2\frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} \\ & \quad - \frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = \frac{S}{2} \cdot C \cdot D_{\text{pore}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{1}^{0} - z_{1} }} \\ & \quad - \frac{\text{d}}{{{\text{d}}t}}\left[ {z_{1} \cdot S \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} } \right] = \frac{S}{2} \cdot C \cdot D_{\text{pore}} \frac{{x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} }}{{z_{1}^{0} - z_{1} }} \\ & \quad - \int_{{z_{1} = z_{1}^{0} }}^{{z_{1} = z_{1} }} {\left( {z_{1}^{0} - z_{1} } \right)} {\text{d}}z_{1} = \frac{{C \cdot D_{\text{pore}} (x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} )}}{{2 \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }}\int_{t = 0}^{t = t} {{\text{d}}t} \\ & \quad \frac{{z_{1}^{{0^{2} }} }}{2} - z_{1}^{0} \cdot z_{1} + \frac{{z_{1}^{2} }}{2} = {\text{A}}_{2} \cdot t \\ \end{aligned}. $$

(A20)

In Eq. [A20], ${\text{A}}_{2} = \frac{{C \cdot D_{\text{pore}} (x_{{A_{\text{in}} }} - x_{{A_{\text{eq}} }} )}}{{2 \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }}$ = constant. The fractional weight loss of CuFeS₂ sample is given as follows:

$$ \begin{aligned} X_{{{\text{CuFeS}}_{2} }} \, & = \frac{{n^{0}_{{{\text{CuFeS}}_{2} }} - n_{{{\text{CuFeS}}_{2} }} }}{{n^{0}_{{{\text{CuFeS}}_{2} }} }} = \frac{{Sz_{1}^{0} \rho_{{m_{{{\text{CuFeS}}_{2} }} }} - Sz_{1} \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }}{{Sz_{1}^{0} \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} = 1 - \frac{{z_{1} }}{{z_{1}^{0} }} \\ z_{1} \, & = z_{1}^{0} \left( {1 - X_{{{\text{CuFeS}}_{2} }} } \right) \\ \end{aligned}. $$

(A21)

Substitution of $z_{1}$ from Eqs. [A21] into [A20] gives

$$ \frac{{z_{1}^{{0^{2} }} }}{2} - z_{1}^{{0^{2} }} \left( {1 - X_{{{\text{CuFeS}}_{2} }} } \right) + \frac{{z_{1}^{{0^{2} }} \cdot \left( {1 - X_{{{\text{CuFeS}}_{2} }} } \right)^{2} }}{2} = A_{2} \cdot t. $$

On simplification and substituting back the value of A₂ and replacing A by H₂

$$ \begin{aligned} & \frac{{z_{1}^{{0^{2} }} }}{2} \cdot X_{{{\text{CuFeS}}_{2} }}^{2} = \frac{{C \cdot D_{\text{pore}} (x_{{{\text{H}}_{{2_{\text{in}} }} }} - x_{{{\text{H}}_{{2_{\text{eq}} }} }} )}}{{2 \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \cdot t \\ & X_{{{\text{CuFeS}}_{2} }}^{2} = \frac{{C \cdot D_{\text{pore}} \left( {x_{{{\text{H}}_{{2_{\text{in}} }} }} - x_{{{\text{H}}_{{2_{\text{eq}} }} }} } \right)}}{{z_{1}^{{0^{2} }} \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \cdot t = \frac{{D_{\text{pore}} \left( {C_{{{\text{H}}_{{2_{\text{in}} }} }} - C_{{{\text{H}}_{{2_{\text{eq}} }} }} } \right)}}{{z_{1}^{{0^{2} }} \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }} \cdot t \\ \end{aligned}. $$

(A22)

The above equation is the rate equation, consistent with the system geometry under study, for pore diffusion-controlled kinetics. Eq. [A22] is reproduced as Eq. [6] in the text.

Interfacial reaction-controlled kinetics

To derive the rate equation for the interfacial reaction-controlled kinetics, Reaction [4] is considered as elementary as well as reversible and first order with respect to H₂. Accordingly, the reaction is rewritten as follows:

$$ 1/2{\text{CuFeS}}_{ 2} \left( {\text{s}} \right) \, + {\text{ H}}_{ 2} \left( {\text{g}} \right) \, \mathop{\rightleftarrows}\limits_{{\text{k}}_{\text{b}}}^{{\text{k}}_{\text{f}}} \, 1/2{\text{Cu }}\left( {\text{s}} \right) \, + \, 1/2{\text{ Fe }}\left( {\text{s}} \right) \, + {\text{ H}}_{ 2} {\text{S }}\left( {\text{g}} \right). $$

(A23)

In this case, there should be no concentration gradient across the gas film within the crucible as well as across the porous product layer because both gas film diffusion and pore diffusion steps are kinetically fast. Consequently, the hydrogen concentration at z = z ₁ (which is the interface of the product and the reactant CuFeS₂ and where the interfacial reaction takes place) is the same as that in the inlet gas, namely$C_{{{\text{H}}_{{2_{\text{in}} }} }}$.The rate of the reaction, in terms of removal of hydrogen, is

$$ \begin{aligned} - r_{{{\text{H}}_{2} }} \, & = - \frac{1}{S}\frac{{{\text{d}}n_{{{\text{H}}_{2} }} }}{{{\text{d}}t}} = k_{\text{f}} C_{{{\text{CuFeS}}_{2} }}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} C_{{{\text{H}}_{{2_{\text{in}} }} }} - k_{\text{b }} C_{\text{Cu}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} C_{\text{Fe}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} \\ & = k^{\prime}_{\text{f}} C_{{{\text{H}}_{{2_{\text{in}} }} }} - k^{\prime}_{\text{b}} C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} \\ \end{aligned}, $$

(A24)

where k _f is the forward rate constant, k _b is the backward rate constant, and $k^{\prime}_{\text{f}} = k_{\text{f}} C_{{{\text{CuFeS}}_{2} }}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ is the modified forward rate constant and $k^{\prime}_{\text{b}} = k_{\text{b }} C_{\text{Cu}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} C_{\text{Fe}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ is the modified backward rate constant. It is to be noted that the concentrations $C_{{{\text{CuFeS}}_{2} }}$, C _Cu, and C _Fe are constant with respect to time.

Using the stoichiometry, as before, the following equalities are obtained:

$$ \begin{aligned} - r_{{{\text{CuFeS}}_{2} }} \, & = - \frac{1}{S}\frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = - \frac{1}{2} \cdot \frac{1}{S}\frac{{{\text{d}}n_{{{\text{H}}_{2} }} }}{{{\text{d}}t}} = \frac{1}{2}\left( { - r_{{{\text{H}}_{2} }} } \right) \\ & \quad - \frac{1}{S}\frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = \frac{1}{2}\left[ {k^{\prime}_{\text{f}} C_{{{\text{H}}_{{2_{\text{in}} }} }} {-} k^{\prime}_{\text{b}} C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} } \right] \\ & = \frac{{k^{\prime}_{\text{f}} }}{2}\left[ {C_{{{\text{H}}_{{2_{\text{in}} }} }} - \frac{{C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} }}{{k^{\prime}_{\text{f}} /k^{\prime}_{\text{b}} }}} \right] \\ \end{aligned}. $$

(A25)

At equilibrium,$- \frac{1}{S}\frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = 0,\;\frac{{k^{\prime}_{\text{f}} }}{{k^{\prime}_{\text{b}} }} = \left( {\frac{{C_{{{\text{H}}_{2} {\text{S}}}} }}{{C_{{{\text{H}}_{2} }} }}} \right)_{\text{eq}} = K_{\text{c }}$, where

$$ K_{\text{c}} = \left( {\frac{{C_{{{\text{H}}_{2} {\text{S}}}} }}{{C_{{{\text{H}}_{2} }} }}} \right)_{\text{eq}} = \left( {\frac{{p_{{{\text{H}}_{2} {\text{S}}}} /{\text{R}}T}}{{p_{{{\text{H}}_{2} }} /{\text{R}}T}}} \right)_{\text{eq}} = \left( {\frac{{p_{{{\text{H}}_{2} {\text{S}}}} }}{{p_{{{\text{H}}_{2} }} }}} \right)_{\text{eq}} = \left( {\frac{{a_{{{\text{H}}_{2} {\text{S}}}} }}{{a_{{{\text{H}}_{2} }} }}} \right)_{\text{eq}} = K, $$

where $K_{\text{c}}$ is the equilibrium constant in terms of concentration and K is the equilibrium constant in terms of activity. This allows Eq. [A25] to be rewritten as follows:

$$ - \frac{1}{S}\frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = \frac{{k^{\prime}_{\text{f}} }}{2}\left[ {C_{{{\text{H}}_{{2_{\text{in}} }} }} - \frac{{C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} }}{K}} \right]. $$

Letting $A_{3} = \frac{{k^{\prime}_{\text{f}} }}{2}\left[ {C_{{{\text{H}}_{{2_{\text{in}} }} }} - \frac{{C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} }}{K}} \right],$ where A ₃ is a constant that does not change with t,

$$ - \frac{{{\text{d}}n_{{{\text{CuFeS}}_{2} }} }}{{{\text{d}}t}} = S \cdot {\text{A}}_{3}, $$

$$ - \frac{\text{d}}{{{\text{d}}t}}\left[ {z_{1} \cdot S \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} } \right] = S \cdot {\text{A}}_{3}, $$

$$ - \int_{{z_{1} = z_{1}^{0} }}^{{z_{1} = z_{1} }} {{\text{d}}z_{1} } = \frac{{{\text{A}}_{3} }}{{\rho_{{m_{{{\text{CuFeS}}_{2} }} }} }}\int_{t = 0}^{t = t} {{\text{d}}t}, $$

$$ z_{1}^{0} - z_{1} = \frac{{{\text{A}}_{3} \cdot t}}{{\rho_{{m_{{{\text{CuFeS}}_{2} }} }} }}. $$

From Eq. [A21],

$$ X_{{{\text{CuFeS}}_{2} }} = \frac{{z_{1}^{0} - z_{1} }}{{z_{1}^{0} }} = \frac{{{\text{A}}_{3} \cdot t}}{{z_{1}^{0} \cdot \rho_{{m_{{{\text{CuFeS}}_{2} }} }} }}. $$

(A26)

Putting back the value of A₃ in Eq. [A26],

$$ X_{{{\text{CuFeS}}_{2} }} = \frac{{k^{\prime}_{\text{f}} }}{2}\frac{{\left[ {C_{{{\text{H}}_{{2_{\text{in}} }} }} - \frac{{C_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} }}{K}} \right]t}}{{\rho_{{m_{{{\text{CuFeS}}_{2} }} }} \cdot z_{1}^{0} }} = \frac{{C \cdot k^{\prime}_{\text{f}} }}{2}\frac{{\left[ {x_{{{\text{H}}_{{2_{\text{in}} }} }} - \frac{{x_{{{\text{H}}_{2} {\text{S}}_{\text{in}} }} }}{K}} \right]t}}{{\rho_{{m_{{{\text{CuFeS}}_{2} }} }} \cdot z_{1}^{0} }}. $$

(A27)

The above equation is the rate equation, consistent with the system geometry under study, for interfacial reaction-controlled kinetics. Eq. [A27] is reproduced as Eq. [7] in the text.