A Comparative Study of Gold Leaching Kinetics Using Alternative Reagents from Concentrates of Low-Grade Ores

Abstract

This study compares gold leaching using sodium cyanide (NaCN) with alternative YX500 and Jinchan reagents. The research object was a gold–sulfide polymetallic ore (Republic of Kazakhstan) with a gold content of 0.38 g/t. The closed flotation beneficiation experiment resulted in a concentrate with an 81.40% recovery and a 5.3 g/t gold grade. The resulting concentrate was subjected to oxidizing roasting to completely oxidize the sulfides and mineral grains of arsenopyrite, pyrite, and carbon. A comparative evaluation of leaching showed that the gold recovery from the roasted concentrate using alternative YX500 and Jinchan reagent solutions was comparable to that using NaCN, with the recoveries at approximately the same level of 86.5%. The differential analysis of the obtained multiplicative multifactor Protodyakonov–Malyshev model made it possible to determine the apparent activation energy of the process using the Arrhenius equation, which eliminates the widely used graphical model. In the proposed method of kinetic experiment planning, the time differentiation of the Kolmogorov–Erofeev equation is mandatory, determining it as a partial dependence on the duration and multiplicative equation for all transformations to determine the activation energy of the process at any given conversion value and other operative factors. The variation range of the apparent value of the activation energy of the gold leaching process, from 0.718 to 78.0 kJ/mol, indicates that the limiting stage of this process is the solid-phase diffusion of CN ions from the outside to the center of the grain material.

1. Introduction

The processing of gold-, copper-, and arsenic-containing ores can be effectively implemented using the cyanidation method after the preliminary mechanical (grinding), chemical (leaching), or thermochemical (roasting) gold recovery associated with the corresponding mineral components [1]. The primary recovery process for noble metals from ores and concentrates is cyanidation, through which up to 90% of gold and silver are recovered [1]. The basic reagent used in hydrometallurgy for gold and silver recovery from the ores of primary ore deposits is sodium cyanide, which has been used for more than 100 years. The cyanidation process recovers approximately 80%–90% of the gold and silver from the ores of primary ore deposits. The essence of this process is that the crushed ore material containing noble metals comes into contact with dilute alkaline solutions of sodium cyanide, under which gold and silver pass from the ore.

Gold and silver recovery from ores and ore concentrates using cyanide has significant technological and economic advantages over other metallurgical technologies. However, cyanide belongs to the category of highly toxic substances, and its use is accompanied by a large volume of complex measures:

• Ensuring the safety of operating personnel;
• Cyanide neutralization in the tailings;
• Environmental requirements for cyanide waste disposal.

Despite the widespread use of CN-leaching worldwide, high toxicity and environmental restrictions remain common disadvantages [2,3,4,5]. Recently, alternative gold-leaching reagents have received increasing attention [1,2,3]. Exploring new reagents that can provide efficient and economical gold leaching instead of highly toxic cyanide is an urgent task [5,6,7,8,9]. More than 40 known leaching systems can transform Au and Ag into soluble states. However, only a few of these systems are industrially important for the hydrometallurgy of noble metals. From the literature sources and the experience of gold processing plants, various chemical additives [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], such as oxygen [10,11,12], hydrogen peroxide [12,13], potassium permanganate [14,15,16], ammonium and potassium persulfates [17], and sodium hypochlorite [18], have been widely used for the intensification of cyanidation processes. In [26], alternative leaching systems based on sodium hypochlorite, bromine bromide solutions, ammonium thiosulfate stabilized with ammonia, and the sulfuric acid solutions of thiourea (oxidant iron is (III) sulfate) were compared with cyanide.

Non-cyanide reagents produced in the People’s Republic of China can successfully replace sodium cyanide in gold recovery. These reagents have advantages, such as a low toxicity, relative environmental safety, high recovery, stability, low consumption, convenient operation, storage, and transportation. A new high-tech product created by China’s largest enterprise, the Guangxi Senhe High Technology Co., Ltd., (Qingxiu District, Guangxi, China) is the patented Jinchan Environmental-friendly Gold Dressing Agent (JGD) (Patent Certificate ZL201010553418.7) [6]. The main chemical compounds in this reagent include SC(NH₂)₂, NaSiO₃, NaOH, and (NaPO₃)₆ [27]. Foreign large-scale industrial gold mining enterprises, including the China National Gold Mine Group, Shandong Gold Group, Zhaoyuan Gold Group, and Zijin Mine Group, have successfully tested and introduced this product [27].

In comparison with the known leaching schemes, in the absence of advantages, reagents produced by the People’s Republic of China should be directed to solve the problems of ecology and labor safety and with the increase in production cost, which determines the efficiency of the process.

A significant portion of the refractory gold ore reserves includes finely disseminated gold composed of sulfides and rock-forming minerals. Gold was mainly associated with pyrite and arsenopyrite. To recover this gold, it is necessary to break the sulfide aggregates and the chemical bond of substitution. Studies [28,29,30] describe gold recovery from refractory arsenopyrite concentrates after preliminary preparation using a hydrometallurgical destruction of gold-containing minerals, a fine grinding of concentrates followed by hydrometallurgical oxidation of sulfides, an autoclave and nitric acid loosening of refractory concentrates, and the chemical destruction of sulfides [31,32].

In [33,34,35], a method of the oxidizing-sulfiding roasting of gold-arsenic concentrate in shaft-type furnaces was developed based on the sulfidation process of arsenopyrite by pyrite with the release of arsenic sulfides into the gas phase.

The oxidative liberation of raw materials aims to transform sulfides into oxides or sulfates [36], destroying the sulfide matrix and allowing the leaching solution to penetrate previously inaccessible Au particles and easily dissolve them [37,38].

Table 1. The results of gold recovery by NaCN and Jinchan leaching.

Process and Basic Principles		Au and Ag Recovery, %
Oxidizing roasting
Oxidation of S, As, and C in the concentrate; the destruction of the mineral structure that includes gold.	Leaching using Jinchan and NaCN for 4 h	88.5 [39]; 88.7 [35,39].
Roasting in a multiple hearth furnace at 550–600 °C.	NaCN—at a concentration of 2 g/L and a specific consumption of 0.83 kg/t of pyrite concentrate; L:S = 2:1, for 2 h.	78.05; throughout recovery into the marketable product—66% [40].
Roasting in two stages: (1) At 300–400 °C with restricted access to air (dearsenization); (2) At 450–550 °C with access to air, with a stirring of the material (desulfurization).	Cinder cyanidation at a concentration of 1.5 g/L for 24 h at a NaCN consumption of 1.7 kg/t.	92.6 [41].
Hydrometallurgical methods
Leaching of Kaimaz ore under the conditions: L:S = 0.4, concentration −300 mg/L, Jinchan consumption—1.27 kg/t and NaCN—0.95 kg/t.	Jinchan NaCN	81.55; 82.86 [42].
Leaching of 150 μm grain size ore at 80% yield.	NaCN Jinchan	97.31; 90.30 [7].
Ag leaching from waste copper tailings with Ag content of 1.7 g/t (specific consumption—0.9 kg/t tailings).	NaCN Jinchan	Ag −72.416; Ag −72.372 [43].

shows the results of gold recovery by leaching with NaCN/Jinchan under different conditions, using different materials.

A gold recovery of approximately 87%–92% in the cyanidation process after oxidizing roasting, as described in [35], is associated with an incomplete destruction of the minerals that may contain gold (pyrite, arsenopyrite, and quartz) during roasting. According to [42], liberated gold is partially covered with films of easily fusible compounds during roasting.

For a comparative evaluation of the studied reagents (NaCN, Jinchan, and Yx500) in the leaching of gold from concentrates, it is necessary to determine the kinetics of the process. To obtain this characteristic on the basis of the multiplicative model, it is necessary to use partial equations that allow transformations in the form of the reversal of variables, differentiation, and others. Such a transformation is possible using two modern and rigorous methods of technological processes running in time: a multifactor study of technological processes, which are described by the Protodyakonov–Malyshev correlation equations in technological processes with the addition of the Kolmogorov–Erofeev equation of topochemical kinetics. Such a coupling is a feature of our direction, which simplifies the procedures of transformation of the multiplicative, Kolmogorov–Erofeev, and Arrhenius equations for kinetic analyses with the determination of the activation energy.

2. Materials and Methods

The specific relevance, complexity, and necessity of the safe implementation of cyanide processing technology for refractory gold-containing ores stipulate the use of repeatedly tested methods of chemical kinetics and the multifactor mathematical representation of their formalized relationship to identify and select alternative reagents according to the scientific indicators of their chemical activity. To the greatest extent, this applies to the activation energy of the chemical and/or diffusion processes occurring on the surfaces of solids. Such processes are most widely covered by the generalized equation of topochemical kinetics in the range from zero to an infinitely long duration τ with the expression of substance conversion or the passage (completeness) of the process from zero to one:

$$\alpha =1-e^{-k·{\tau }^n}$$

(1)

Here, the duration value receives the dimension of time in degree n, which necessitates giving the rate a constant value of the dimensionality inverse to the dimensionality of time τ⁻ⁿ. When processing the data according to (1), the value of n can be any value, including n = 1. In the latter case, it provides a strict time dimensionality (e.g., in min) and the rate constant dimension as the frequency of events (particle collisions) in the inverse time dimensionality, that is, min⁻¹. Similar restrictions for the exponent n are necessary for theoretical regularities. In this case, the dimensionality of the factors in the exponent of the degree must correspond to the physical and mathematical sense of these factors, even though they are, in any case, mutually canceled by multiplication. Here, the forced constraint (1) in the condition n = 1 is based on the standard requirements of theoretical regularities using fundamental reference data.

Because it is necessary to fulfill the aforementioned requirements, the Kolmogorov–Erofeev Equation (1) simultaneously obtains a stricter and simpler form (for n = 1):

$$\alpha =1-e^{-k\tau }$$

(2)

The identity transformation (2) makes it possible to represent it in the following form:

$$\begin{gathered} \mathrm{l}\mathrm{n}(1-\alpha )=-k\tau , \\ -\mathrm{l}\mathrm{n}(1-\alpha )=k\tau \end{gathered}$$

(3)

The kinetic analysis procedure for determining the activation energy was reduced to the second logarithm of the obtained refined equation:

$$\ln \left[-\ln \left(1-\alpha \right)\right]=lnk+ln\tau$$

(4)

The linearization of this equation in the form y = a + bx is possible when x = lnτ, b = 1, a = lnk, and y = ln[−ln(1 − α)]. It is characteristic that the proportionality factor b = 1; therefore, the constraint n = 1 is considered automatically.

The processing of the experimental data on τ and α by the least squares method makes it possible to obtain an array of new variables at n = 1 related to the rate of the topochemical process in a general form:

$${\displaystyle \frac{d\alpha }{d\tau }}=kF{C}^n$$

(5)

where k is some function of the rate from the temperature, which is revealed according to the Arrhenius equation $k=A^{/}{e}^{-{\displaystyle \frac{Ea}{RT}}}$; F is the fraction of the unused surface; and C is the current concentration of the reagent. Considering only the qualitative influence of these variables, their dimensionalities were not considered and a definition was not required.

It is possible to transform the variables F and C into constants if the process is conducted at a given value α. This can be achieved by transforming variable τ into α in the refined Kolmogorov–Erofeev equation. For this purpose, in Equation (2), it is necessary to express the variable τ through α in the sequence $\ln \left(1-\alpha \right)=-k\tau$. Therefore, $\tau =-{\displaystyle \frac{1}{k}}\ln \left(1-\alpha \right)$, which, with substitution into the rate equation ${\displaystyle \frac{d\alpha }{d\tau }}=k{e}^{-k[{\displaystyle \frac{1}{k}}·\mathrm{l}\mathrm{n}(1-\alpha \left)\right]}=k\left(1-\alpha \right),$ provides the possibility of linearizing the new data set in the rate equation:

$$\mathrm{l}\mathrm{n}{\displaystyle \frac{d\alpha }{d\tau }}=\mathrm{l}\mathrm{n}k+\ln \left(1-\alpha \right).$$

The resulting expression opens the possibility of setting any value for the degree of reaction α to determine the rate of the process and the rate constant, instead of defining this by additional experimentation or a graphical processing of the curves. In this equation, according to the Arrhenius equation, it is necessary to disclose the expression for the rate constant and other variables that are considered constants at a given value of α, which ensures the justification for taking the partial derivative (5):

$$\mathrm{l}\mathrm{n}{\displaystyle \frac{d\alpha }{d\tau }}=\mathrm{l}\mathrm{n}{A}^{/}+{\displaystyle \frac{Ea}{R}}·{\displaystyle \frac{1}{T}}+\mathrm{l}\mathrm{n}F+\mathrm{l}\mathrm{n}C+\ln \left(1-\alpha \right).$$

It is necessary to combine the constants of this expression in the form:

$$A^{\Vert }=A^{/}+\mathrm{l}\mathrm{n}F+\mathrm{l}\mathrm{n}C+\ln \left(1-\alpha \right)$$

and obtain an equation expressed by a straight-line correlation:

$$\mathrm{l}\mathrm{n}{\displaystyle \frac{d\alpha }{d\tau }}=A^{\Vert }+{\displaystyle \frac{Ea}{R}}·{\displaystyle \frac{1}{T}}.$$

For processing the straight-line equation, its variables and constants are denoted as x = 1/T = E_a/R, a = $A^{/}$, y = ln(dα/dτ). The resulting rate equation can be used to determine the activation energy under temperature variation using process rate data. When analyzing a multifactorial process, the possibility of using the Kolmogorov–Erofeev equation refers only to the partial dependence of the process duration. The multifactor equation was multiplicative. Therefore, it relates to a cumulative constant for all other factors with a single dependent generalizing variable. Thus, the possibility of taking partial derivatives is ensured by combining the remaining factors into a cumulative constant, simplifying the subsequent transformation procedures of the multiplicative and Kolmogorov–Erofeev equations for kinetic analysis with the determination of the activation energy.

In all cases, the nonlinear multiple correlation coefficient R and its significance t_R, expressed by known formulas [44,45], were used in the derivation of the equation to check its adequacy, as follows:

$$R=\sqrt{1-{\displaystyle \frac{\left(n-1\right){\sum }_{i=1}^n{\left(y_{e,i}-y_{c,i}\right)}^2}{\left(n-k-1\right){\sum }_{i=1}^n{\left(y_{e,i}-{\overline{y}}_{e,m}\right)}^2}}}$$

(6)

$$t_R={\displaystyle \frac{R\sqrt{n-k-1}}{1-R^2}}>2$$

(7)

where y_e,i is the experimental value; y_c,i is the calculated value; y_e,m is the mean experimental value; n is the number of independent (not repeated) experimental data; k is the number of operating factors; (n − 1) is the number of degrees of freedom of the repeatability variance; and (n − k − 1) is the number of degrees of freedom of the adequacy variance.

The overall order of presentation of the multifactor multiplicative model of the leaching process and the use of this model to obtain the activation energy is as follows:

• Factors are selected in the experiment;
• Their partial dependencies and R(6) and t_R(7) are found;
• They are entered into the multiplicative model;
• Under the experimental conditions for obtaining each factor, the dependence of this factor is calculated as a part of the multiplicative model, and is directly compared with the experimental results for R and t_R:
• From the topochemical kinetics of the Kolmogorov–Erofeev equation, the dependence of the duration on all other factors was released (excluded), with the subsequent introduction of the rate (the partial derivative of τ) into the equation instead of the duration. From these analytical procedures, the dependence of the process rate on the degree of transformation was determined.
• The apparent activation energy is calculated under the temperature variation using the Arrhenius equation.

An essential feature of the multiplicative multifactor Protodyakonov–Malyshev (MMPM) model is the mandatory normalization of partial dependencies by their arithmetic mean values. Therefore, the natural dimensionalities of the partial dependencies are reduced, and they are considered in the form of fractions. In this case, the generalized multiplicative function y is strictly equal to the geometric mean.

Figure 1 shows a graphical representation (algorithm) of the multifactor multiplicative model of the leaching process.

The objective of this research (unlike the known ones) is to compare the efficiency of gold leaching with different reagents, such as NaCN, YX500, and Jinchan, and to determine the apparent activation energy of the process during leaching from the roasted concentrate after the beneficiation of low-grade refractory gold-containing ore.

The scientific novelty of this study lies in combining the multiplicative multifactor model with the generalized Kolmogorov–Erofeev equation of topochemical kinetics. It is recommended when leaching gold with different reagents NaCN, YX500, and Jinchan to use the strict dimensionality of duration (time) and rate constant in their usual units of measurement—min or min⁻¹.

This study aimed to determine the kinetic regularities of gold leaching using various reagents from the products of low-grade refractory ore beneficiation.

The reagents produced by the People’s Republic of China and domestic sodium cyanide were used in this study. Jinchan and YX500 are environment-friendly leaching agents currently used in metallurgical plants and mining and processing enterprises [5,7]. Jinchan (Golden Cicada) reagent is a chemical mixture of sodium salt, polymerized sodium cyanamide, alkaline thiocarbamide, and stabilizer. This reagent is manufactured from chemical materials such as urea, caustic soda, sodium carbonate, and sodium sulfide mixed with a suitable catalyst [11]. YX500 is a new leaching reagent that is readily soluble in water and consists of 1,3,5-triazine-2,4,6(1H,3H,5H)-trione, sodium thiourea, and cementite.

Planning experiments for the sequential study of operating factors involved the method described in [44,45,46] to obtain the multiplicative multifactor Protodyakonov–Malyshev model. This Protodyakonov–Malyshev model consists of the fact that the form of a multiplicative union, which relates a multivariate function to partial (univariate) functions, is obtained by the statistical treatment of some sets of data in the form of point dependencies that refer to experimental values. The influence of the initial concentrations of NaCN, YX500, and Jinchan reagents in the range of 0.5–10.0 g/dm³ and the leaching duration (τ) from 2 h to 8 h at the solution temperature of 20–40 °C at the sample grinding fineness of 80% of −0.045 + 0 mm grain size class on gold recovery into the solution was studied. The central experiment corresponded to the following conditions: sample mass, 50 g; concentration of NaCN, YX500, and Jinchan, 6 g/dm³; and temperature, 20 °C. The use of three points for the correlation analysis of any technology is necessary [44,45].

Figure 2 shows the flowchart of the research methodology.

During the leaching process, the concentration of sodium cyanide and pH of the medium were monitored and corrected by adding reagents. After leaching, the solution and cake were analyzed to determine the content of the primary elements.

3. Results and Discussion

The diffraction and phase compositions of the YX500 and Jinchan reagents compared to sodium cyanide were determined by phase analysis using a D2 Phaser diffractometer (

Table 2. Diffractometric (material) analysis of reagents.

YX 500		Jinchan
Chemical Formula	Content, %	Chemical Formula	Content, %
Na2CO3	16.3	Na2CO3	45.1
NaCN	66.6	NaCN	41.7
Na2SO4	7.0	NaCNO	13.2
NaCNO	10.1

and

Table 3. The elemental (atomic) composition of reagents.

NaCN		YX500		Jinchan
Symbol	Weight %	Symbol	Weight %	Symbol	Weight %
Na	46.9	Na	44.18	Na	43.82
C	24.5	C	20.02	C	17.75
N	28.6	N	21.20	N	14.76
		O	13.02	O	23.67
		S	1.58

).

A gold–sulfide polymetallic ore (Republic of Kazakhstan) was used for the comparative studies of reagents.

Table 4. The chemical composition of the initial ore sample.

Component	Au, g/t	Ag, g/t	Cu	Pb	Zn	Fe	P	Mo	As
Weight %	0.38	0.15	0.012	0.006	0.01	4.60	0.087	0.0001	0.08
Component	Sn	Sb	Bi	Cd	SiO2	Al2O3	CaO	C	S total
Weight %	-	0.0015	-	0.0002	62.80	15.70	3.64	4.17	0.51

The chemical composition of the samples was determined using an inductively coupled plasma optical emission spectrometer (Avio 500 ICP-OES, PerkinElmer, Waltham, MA, USA).

lists the chemical composition of the samples.

Flotation beneficiation has been proposed to solve the problem of gold recovery from sulfide ore [47,48,49,50]. The laboratory test was conducted in a closed cycle according to the developed technology and reagent regime, until the gold quality and recovery indicators stabilized. Figure 3 shows the schematic of the laboratory tests. The conditions and results of the laboratory tests are listed in

Table 5. Experimental conditions of the flotation beneficiation.

Operation	Content of −0.071 mm Grain Size Class, %	Time, min	pH	Consumption of Reagents, g/t
				CuSO4	Kx	T-92
Grinding	90	-	-	-	-	-
1st main flotation	-	10	8.5	80	65	20
Control flotation	-	10	-	60	25	5
1st recleaning	-	4	-	-	-	-

and

Table 6. Results of the experiment conducted in a closed cycle.

Products	Output, %	Au Content, g/t	Au Recovery, %
Concentrate of the 1st recleaning	5.82	5.3	81.40
Tailings	94.18	0.075	18.60
Initial ore sample	100.0	0.38	100.0

.

The presence of organic carbon may indicate the need to apply roasting, which makes it possible to eliminate sorption-active substances that absorb gold during the cyanidation stage. The mechanism by which gold is lost during cyanidation also differs, depending on the nature of the material causing this effect (Table 4).

Oxidizing roasting of the obtained concentrate (Table 6) occurred at 550–600 °C for 90 min. The complete oxidation of sulfides and a sharp change in the physical structure of arsenopyrite and pyrite mineral grains were observed under these conditions.

The phase analysis results obtained using a D2 Phaser diffractometer revealed that the sample consisted of hematite, calcium sulfate, albite, and quartz (

Table 7. Results of the diffractometric analysis of cinder after the oxidizing roasting of the concentrate.

Material	Chemical Formula	Content, %
Quartz	SiO2	37.1
Albite	K[AlSi3O8]	41
Hematite	Fe2O3	6.7
Calcium sulfate	CaSO4	15.2
Total		100

).

The roasted material after ore beneficiation with various reagents, NaCN, YX500, and Jinchan, was used for the leaching studies.

In cyanidation, gold is oxidized by air oxygen to Au⁺¹ and forms [Au(CN)₂]⁻ anions in a solution. In general, the chemistry of the process is described by the following reaction [1,4]:

2Au + 4NaCN + H₂O + ½O₂ = 2Na[Au (CN)₂] + 2NaOH

As a result of the research, dot diagrams with an approximation of partial dependencies were constructed (Figure 4).

As seen in Figure 4, the concentration of the reagents (NaCN, YX500, and Jinchan) in the solution significantly affected the rate of the gold recovery process. The decrease in the rate at high reagent concentrations can be explained by an increase in the pH of the solution as a result of hydrolysis.

At low cyanide concentrations (approximately 0.1–0.6 g/dm³), the dissolution rate of gold depended only on the concentration. At high cyanide concentrations, the dissolution rate did not depend on the cyanide. In this case, another factor, the partial pressure of oxygen, began to have an effect.

The dissolution rate of gold increases with temperature, but simultaneously, the solubility and, consequently, the oxygen concentration in the solution decrease. Therefore, an optimal temperature is necessary to achieve the highest dissolution rate. However, maintaining a high temperature during the production process is economically inexpedient because heating vast volumes of ore pulp requires significant energy, which can significantly increase the cost of gold production. Therefore, cyanidation is usually conducted at temperatures not exceeding 15–25 °C.

Partial equations were obtained from the experimental data (

Table 8. The partial functions of gold recovery into the solution with the determination of R and t_R.

Reagent	Functions	tR > 2	R
NaCN	${\epsilon }_{Au}=10.303C_{NaCN}-0.589C_{NaCN}^2+43.894$	36.280	0.9668
	${\epsilon }_{Au}=87.41(1-e^{-0.552·\tau })$	6.598	0.8985
	${\epsilon }_{Au}=(0.0458t+85.729)$	29.380	0.9372
YX500	${\epsilon }_{Au}=8.907C_{Yx500}-0.4876C_{Yx500}^2+46.867$	84.755	0.9856
	${\epsilon }_{Au}=86.5(1-e^{-0.75·\tau })$	2.160	0.7249
	${\epsilon }_{Au}=(0.0671t+84.719)$	4.865	0.7244
Jinchan	${\epsilon }_{Au}=9.638C_{Jin Chan}-0.545C_{Jin Chan}^2+44.701$	67.545	0.9820
	${\epsilon }_{Au}=87.51(1-e^{-0.706·\tau })$	2.298	0.7386
	${\epsilon }_{Au}=(0.135t+82.53)$	498.00	0.9960

) to derive a mathematical model for gold recovery into the solution.

According to [45], the partial equations are generalized as their product with normalization by mean values (in this case, the mean values for NaCN = 83.188%, YX500 = 83.609%, and Jinchan = 83.932%). Therefore, the generalized equations for gold recovery in solutions using NaCN, YX500, and Jinchan are expressed as follows:

$$\begin{gathered} {\epsilon }_{Au}=1.407·{10}^{-4}\left(10.303C_{NaCN}-0.589C_{NaCN}^2+43.894\right)\left[87.41\left(1-e^{-0.552·\tau }\right)\right]·(0.458t+85.729), \\ R=0.93, t_R=21.46>2 \end{gathered}$$

(8)

$$\begin{gathered} {\epsilon }_{Au}=1.403·{10}^{-4}\left(8.907C_{Yx500}-0.4876C_{Yx500}^2+46.867\right)\left[86.5\left(1-e^{-0.75·\tau }\right)\right]·(0.0671t+84.719), \\ R=0.942, t_R=27.50>2 \end{gathered}$$

(9)

$$\begin{gathered} {\epsilon }_{Au}=1.405·{10}^{-4}\left(9.638C_{Jin Chan}-0.545C_{Jin Chan}^2+44.701\right)\left[87.51\left(1-e^{-0.706·\tau }\right)\right]·(0.135\mathrm{t}+82.53), \\ R=0.960, t_R=31.12>2 \end{gathered}$$

(10)

The obtained generalized equations make it possible to identify the joint effect of the initial reagent concentration and leaching duration on gold recovery into the solution and can be used for the kinetic analysis of this process [44]. To check reliability, all partial functions and equations were tested using the nonlinear multiple correlation coefficient R and its significance t_R using Equations (6) and (7).

For kinetic analysis with the determination of the process activation energy, the obtained multifactor multiplicative models allow for possible variable transformations. To determine the process rate, we transform Equations (8)–(10) by denoting the following:

$$\begin{gathered} 1.407·{10}^{-4}\left(10.303C_{NaCN}-0.589C_{NaCN}^2+43.894\right)(0.0458t+85.729)·87.41=A \\ 1.403·{10}^{-4}\left(8.907C_{Yx500}-0.4876C_{Yx500}^2+46.867\right)(0.0671t+84.719)·86.5=B \\ 1.405·{10}^{-4}\left(9.638C_{JinChan}-0.545C_{JinChan}^2+44.701\right)\left(0.1346\mathrm{t}+82.53\right)·87.51=C, \end{gathered}$$

bringing them to the form:

$${\epsilon }_{Au\left(NaCN\right)}=A·(1-e^{-k·\tau })$$

(11)

For this purpose, it is necessary to determine the dependence of the reaction rate dε/dτ on some variables, for example, on the process duration. The multiplicative form of multifactor dependence makes it possible to transform the variables τ to ε and according to the algebraic transformation rules to obtain the following equality:

$$-{\displaystyle \frac{1}{k}}·\ln \left(1-{\displaystyle \frac{{\epsilon }_{Au}}{A}}\right)=\tau$$

(12)

Differentiating Equation (11), we obtain the expression for the rate:

$${\displaystyle \frac{d{\epsilon }_{Au}}{d\tau }}=A·k·e^{-k·\tau },$$

(13)

The substitution of (12) into (13) provides the dependence of the process rate on the reaction progression ε, which must be fixed to determine the rate constant using the Arrhenius equation:

$${\displaystyle \frac{d{\epsilon }_{Au}}{d\tau }}=A·k·e^{-k(-{\displaystyle \frac{1}{k}}\ln \left(1-{\displaystyle \frac{{\epsilon }_{Au}}{A}}\right))}$$

After mathematical transformations, the equation for the process rate takes the form:

$${\displaystyle \frac{d{\epsilon }_{Au}}{d\tau }}=k·\left(A-{\epsilon }_{Au}\right).$$

(14)

The transformation for gold recovery using YX500 and Jinchan reagents is similar.

The above transformation of the multifactor equation allows for calculations using the known Arrhenius kinetic equation and for determining the process activation energy with temperature variation in (14) using known data processing procedures at a fixed value of ε.

Table 9. Results of the calculation of the gold leaching process rate and the apparent activation energy in Arrhenius coordinates.

Given ${\mathit{\epsilon }}_{\mathit{A}\mathit{u}}$	Reagent	t, °C	$\frac{\mathit{d}{\mathit{\epsilon }}_{\mathit{A}\mathit{u}}}{\mathit{d}\mathit{\tau }}$	$\mathbf{ln}\frac{\mathit{d}{\mathit{\epsilon }}_{\mathit{A}\mathit{u}}}{\mathit{d}\mathit{\tau }}$	E, kJ/mol	Given ${\mathit{\epsilon }}_{\mathit{A}\mathit{u}}$	Reagent	t, °C	$\frac{\mathit{d}{\mathit{\epsilon }}_{\mathit{A}\mathit{u}}}{\mathit{d}\mathit{\tau }}$	$\mathbf{ln}\frac{\mathit{d}{\mathit{\epsilon }}_{\mathit{A}\mathit{u}}}{\mathit{d}\mathit{\tau }}$	E, kJ/mol
40	NaCN	20	27.628	3.319	0.718	80	NaCN	20	5.5402	1.712	3.454
		30	27.891	3.328				30	5.8030	1.758
		40	28.154	3.337				40	6.0658	1.802
	YX500	20	34.837	3.550	1.090		YX500	20	4.8451	1.578	7.225
		30	35.342	3.565				30	5.3506	1.677
		40	35.848	3.579				40	5.8560	1.767
	Jinchan	20	33.110	3.499	2.167		Jinchan	20	4.8784	1.585	12.757
		30	34.079	3.528				30	5.8473	1.766
		40	35.048	3.556				40	6.8161	1.919
50	NaCN	20	22.106	3.096	0.895	85	NaCN	20	2.7792	1.022	6.604
		30	22.369	3.107				30	3.0420	1.112
		40	22.632	3.119				40	3.3048	1.195
	YX500	20	27.339	3.308	1.383		YX500	20	1.0961	0.092	24.948
		30	27.844	3.326				30	1.6016	0.471
		40	28.350	3.345				40	2.1070	0.745
	Jinchan	20	26.052	3.260	2.734		Jinchan	20	1.3494	0.299	34.010
		30	27.021	3.297				30	2.3183	0.841
		40	27.990	3.332				40	3.2871	1.190
60	NaCN	20	16.584	2.808	1.189	86.5	NaCN	20	1.9509	0.668	9.097
		30	16.847	2.824				30	2.2137	0.795
		40	17.110	2.839				40	2.4765	0.907
	YX500	20	19.841	2.988	1.894		YX500	20	−0.028	-	-
		30	20.346	3.013				30	0.4769	−0.740	56.980
		40	20.852	3.037				40	0.9823	−0.018
	Jinchan	20	18.994	2.944	3.702		Jinchan	20	0.2907	−1.235	77.989
		30	19.963	2.994				30	1.2596	0.2308
		40	20.932	3.041				40	2.2284	0.801
70	NaCN	20	11.062	2.403	1.769
		30	11.325	2.427
		40	11.588	2.450
	YX500	20	12.343	2.513	3.000
		30	12.848	2.553
		40	13.354	2.592
	Jinchan	20	11.936	2.479	5.735
		30	12.905	2.557
		40	13.874	2.630

shows the rate values for the temperatures of 20, 30, and 40 °C and the values of the apparent activation energy at given values of gold recovery into the solution (${\epsilon }_{Au}$) in the range of 40% to 86.5% and a concentration of 6 g/dm³ in Arrhenius coordinates.

According to Equation (14), the obtained values of the process rate naturally increase with the increase in the initial reagent concentration and decrease with the increase in the given value of ${\epsilon }_{Au}$, which indicates the completeness of the liberation of almost all mineral associations. The involvement of oxygen in the cyanidation reaction in gold recovery has been established in many studies; therefore, our analysis of the technology is indirect, relying on a widely known fact.

The results indicated the possibility and feasibility of combining the generalized topochemical model with the multifactor multiplicative experimental planning method to strengthen the determination of partial dependencies and the multiplicative multifactor equation (by not exceeding the physical and analytical limits and other changes in the multifactor function).

The determination of the multiplicative Protodyakonov–Malyshev model by introducing the rate into the topochemical Kolmogorov–Erofeev equation for partial dependencies on the process duration makes it possible to represent the studied region of the factor space with a higher correlation coefficient and to increase the allowable extrapolation in the unstudied region of this space. In this process, only the analytical procedures for the differentiation of variables, substitution of dependent variables by independent ones, and operation of taking partial derivatives and linearization of the obtained dataset were used to determine the activation energy of the studied process based on the Arrhenius equation for the chemical reaction rate constant. The developed method eliminates the need to construct graphical dependencies for each factor and determine the slope angles of these curves and points of the intersection with the given value of the transformation degree, which, as a rule, introduces subjective errors in the determination of all graphical methods of kinetic characteristics, including the activation energy. For any technological process, it is possible to use a combined multiplicative and topochemical model based on its structure.

A stricter method for determining the activation energy and kinetic analysis of the topochemical process increases the correlation coefficient and allows for a more allowable extrapolation of the multifactor equation to the unexplored regions of the factor space [51].

The variation range of the apparent activation energy of the gold leaching process from 0.718 kJ/mol to 78.0 kJ/mol indicates that the limiting stage of this process is the solid-phase diffusion of CN⁻ ions from the outside to the center of the grain material. The results of studies using multiplicative processes of the functional rigor equation obtained information on the activity and activation energy of CN⁻ ions directly indicates both the reasons for the high technological performance and increased environmental hazards.

From Table 9, it follows that the results are the most objective for solving this problem. These values correspond to internal diffusion, and sodium cyanide without additives differs greatly in this value, indicating the achievement of this process with almost the same degree of gold recovery. In this case, the absence of chemical-technological processes is objectively established when comparing the three reagents, but there remains the same reserve for continuing gold recovery, which indicates that the diffusion, external, and internal modes are actually mixed and, therefore, show a small difference, which can be assessed as related to the physical process of releasing gold particles from other components without chemical dissolution.

4. Conclusions

The analysis of comparative laboratory tests showed that the results of gold recovery from the roasted concentrate by Jinchan, YX500, and sodium cyanide reagent solutions were approximately at the same level of 86.5%.

According to the results of a series of experiments, the optimal leaching time was 4 h because a further increase in time did not significantly affect the process. This was confirmed by comparing the values of the activation energy for the direct use of cyanides and other reagents that complicate the process.

The obtained values of the apparent activation energy of the gold leaching process from 0.718 kJ/mol to 78.0 kJ/mol indicate that the limiting stage of this process is the solid-phase diffusion of CN⁻ ions from the outside to the center of the grain material and is the greatest obstacle to the leaching process, which refers to the large values for Jinchan and YX500 in contrast to NaCN.

Combining the advantages of the topochemical Kolmogorov–Erofeev equation and the multiplicative multifactor Protodyakonov–Malyshev model provides the analytical strict justification of all necessary procedures for determining the activation energy of technological processes, including gold leaching with a high correlation coefficient. The unification of the functional model of the topochemical process and the correlation multifactorial equation in multiplicative form for the first time allowed us to open the possibility of determining the activation energy by supplementing these two modern models of displaying complex technological processes and connecting them with the Arrhenius equation. Therefore, it solves the urgent problem of developing the directions of chemical processes into a single whole in which equal importance is occupied by the multiplicative Protodyakonov–Malyshev model, the functional Kolmogorov–Erofeev model, and the universal model of Arrhenius with the extraction of the most complete physical and chemical information from virtually any chemical and technological scientific and practical tasks and problems.

The Protodyakonov–Malyshev model allows the use of a narrow temperature range because it can be extrapolated to a previously unexplored area of space.

In future studies, we plan to conduct experiments with the tank leaching of special studies for more than 8 h per day and processing using at least five temperatures, to confirm the possibility of using a reserve of gold recovery up to 20%.

4.1. Implications and Explanation of the Findings

The dual determination of the most significant partial dependence on the process duration leads to more convincing results of the proposed technology analysis and can be extended to develop a multiplicative multifactor model.

A further development of the proposed method of substantiation of any technological process directly relates to the broader use of multifactor models based on the expansion of the extrapolation of multifactor equations for areas of actual processes with difficult access to experimentation, for example, in the processing of raw materials with the production of high-melting and high-boiling substances.

4.2. Strengths and Limitations of the Study

The strength of this study is that, for the first time, topochemical analysis was conducted within the framework of the probabilistic and deterministic planning of the experiment. In this experiment, the determination of partial dependencies in the form of the Kolmogorov–Erofeev equation is used as the limit, and the limit value of the multiplicative multifactor model restricted the experiment on the transformation of the substance to not more than one and not less than zero. Limitations owing to the incomplete process to the set limit of the degree of transformation caused by part of the extracted gold can restrict its recovery from other minerals contained in the refractory target minerals.

The procedures in this study aimed to determine the activation energy of the limiting stages of the chemical or diffusion processes, which is essential in theoretical and experimental terms because it is possible to determine the limiting values of the degree of recovery.

4.3. Recommendations and Next Steps

Further studies will relate to the complex processing of gold-containing ores according to the technological scheme for the final product, with further theoretical justification of the processes of thermochemical enrichment, leaching, and the multifactorial analysis of technological operations with the identification of zones of the process optimal modes.