Heterogeneous Reactions/Processes PDF

− 𝑑[𝐴] 𝑑𝑡 =𝑘⃗ [𝐴]𝑛𝑎 [𝐵]𝑛𝑏 and ⃗ 𝑘 ⃖⃗ 𝑘 Combining both equations gives: 5.5 − 𝑑[𝐴] 𝑑𝑡 = =𝑘⃖⃗ [𝐶]𝑛𝑐 [𝐷]𝑛𝑑 [𝐴]𝑛𝑎 [𝐵]𝑛𝑏 [𝐶]𝑛𝑐 [𝐷]𝑛𝑑 =𝐾 The effect of temperature on rate of equation With few exceptions the rate of chemical reactions increases with increase in temperature. This is evident fr...

− 𝑑[𝐴] 𝑑𝑡 =𝑘⃗ [𝐴]𝑛𝑎 [𝐵]𝑛𝑏 and ⃗ 𝑘 ⃖⃗ 𝑘 Combining both equations gives: 5.5 − 𝑑[𝐴] 𝑑𝑡 = =𝑘⃖⃗ [𝐶]𝑛𝑐 [𝐷]𝑛𝑑 [𝐴]𝑛𝑎 [𝐵]𝑛𝑏 [𝐶]𝑛𝑐 [𝐷]𝑛𝑑 =𝐾 The effect of temperature on rate of equation With few exceptions the rate of chemical reactions increases with increase in temperature. This is evident from the Arrhenius equation. According to Arrhenius, 𝑑 𝑙𝑛𝑘 𝑑𝑇 = 𝐸𝐴 (This expression is similar to the van't Hoff's equation.) 𝑅𝑇 2 Rearranging and integrating, 𝑘 2 ∫𝑘 𝑑 ln 𝑘 = 1 𝐸𝐴 𝑘2 = 𝑘1 𝑒 𝑅 6 𝐸𝐴 𝑅 𝑇2 𝑑𝑇 ∫𝑇 1 𝑇2 1 1 − ) 𝑇1 𝑇2 ( ln or 𝑘2 𝑘1 = 𝐸𝐴 𝑅 1 1 𝑇1 𝑇2 ( − ) This equation is usually used in the determination of the activation energy using two temperatures HETEROGENEOUS REACTIONS/PROCESSES Homogeneous reactions occur in a single phase; thus, the reactants and products are either in gas phase or liquid phase, eg the neutralisation of acid by alkali. Heterogeneous reactions on the other hand are reactions that involve more than one phase. Usually only two phases take part in heterogeneous reactions even though more than two phases may be present because the rate of reaction is determined by only one step, the rate determining step, which may involve only two phases. For example, in gold cyanidation process involving gold (solid), aqueous NaCN (liquid) and oxygen gas, the rate of oxygen transfer into liquid phase is much faster than other reactions taking place. Therefore, the reaction is reduced to solid-liquid reaction. 6.1 Characteristics of Heterogeneous Reactions/Processes 6.1.1 Presence of interface In heterogeneous reactions, the presence of an interface between the reactants is important. Five categories of heterogeneous processes can be identified depending on the nature of the interface. (i) Solid-gas eg. physical adsorption, oxidation of metals, decomposition of carbonates or sulphates, oxidation of sulphides or gaseous reduction of oxides. (ii) Solid-liquid eg. physical melting, dissolution-crystallization, leaching, cementation. (iii) Solid-solid eg. sintering, phase transformation, reduction of oxides by carbon, reduction of oxides or halides by metals, high temperature solid-solid reactions, reactions in ceramic processing. (iv) Liquid-gas eg. distillation-condensation, absorption, steelmaking by pneumatic processes, absorption of gases in water. (v) Liquid-liquid eg. solvent extraction, metal-slag reactions, liquid metal-liquid metal extraction. 10 6.1.2 Nature of the interface For reactions involving solids, the nature of the interface plays an important role in determining the kinetics of these reaction. This is due to the presence of lattice defects and vacancies. These defects acts as points of initiation of reactions and act to speed up reactions. A crystal AB has equal number of A and B atoms at absolute zero temperature. With increase in temperature thermal vibrations of atoms can cause lattice defects. The various defects include: 1. Schottky type of lattice defect, in which vacant cation and anion sites are created due to the migration of atoms to the surface. 2. Frenkel type of lattice defect in which only one vacant lattice site results, that is either anion or cation). In both cases the number of A and B atoms remain the same. However, differences may exist in the number of atoms which may lead to a non-stoichiometric lattice (eg. FeXS, where 0.88 < x <1). 3. In some cases, also the atoms of nonmetal are removed from the crystal lattice leaving vacancies and resulting in metal atom excess in the lattice (or non-metal deficiency), with the electrons remaining trapped in the vacancies (KCl, NaCl, KBr, PbS, etc). 4. In other cases, atoms of the non-metal are also lost from the crystal lattice and the electrons previously associated with them remain behind, but the excess metal ions are forced to occupy interstitial cations (eg. ZnO, CdO, etc.). The electrons are extremely mobile and the compounds behave like semiconductors. 5. We also have cases where the lattice acquires additional atoms of the non-metal which become anions by gaining electrons as a result of the oxidation of some of the metal ions to a higher oxidation state leaving vacancies (non-metal excess). Conduction in these compounds is by the switch of electrons from an ion of lower oxidation state to one in higher oxidation state under a potential difference. The above deficiencies will result in differences in the behaviour of the solid during reactions. Impurities that cause changes in the lattice have substantial influence on the kinetics of heterogeneous reactions (sometimes this is intentionally done, DOPING). The lattice structure of the solid may also influence the kinetics (eg. in the leaching of gibbsitic, boehmitic and diasporic bauxites). 6.1.3 Area of the interface The rate of heterogeneous reactions depends also on the area of the interface. Reactions involving fine particles are faster than those of coarse ones. This can be seen from the general rate expression The higher the value of A the higher the rate. 6.1.4 Geometry of the interface This is also an important factor. For plates or discs the surface area remains constant throughout the reaction and so the rate will be constant. This is not so for spheres, pellets because the surface area changes as the reaction proceeds, therefore the rate changes. For example, for solid-liquid reaction and assuming constant concentration, 𝑟= 𝑑𝑀 𝑑𝑡 = 𝑘𝐴𝐶 where M is mass of solid at time t, kg A is surface area of solid, m² (m2/kg) C is the concentration of fluid reactant, mol/m3 k is the rate constant For a flat plate (constant A), 11 M -M 𝑡 o 𝑀 ∫ 𝑑𝑀 = 𝑘𝐴𝐶 ∫ 𝑑𝑡 𝑀𝑜 𝑜 giving the solution 𝑀𝑜 − 𝑀 = 𝑘𝐴𝐶𝑡 t For a sphere, A = 4Π r² ; and M = 4Π r3ρ /3 from which Substituting into the rate equation and integrating gives Similar expressions can be derived for pellet/wire and for cube, etc. Rates can also be expressed in terms of fraction converted (X), 𝑥= 𝑀𝑜 −𝑀 allowing for the radius of sphere (r) to be expressed in terms of X, which can 𝑀𝑜 then be substituted into the expression for surface area, A, and mass of solid, M, and finally into the rate equation: − 𝑑𝑀 𝑑𝑡 = 𝑘𝐴𝐶 For example, in the case of a sphere, 𝑀= 4 3 𝜋𝑟 3 𝜌 Therefore, Substituting the above expression into the expressions for mass(M) and area (A) gives: 4 𝑀 = 3 𝜌𝑟𝑜3 (1 − 𝑥) 𝐴 = 4𝜋𝜌𝑟𝑜2 (1 − 𝑥)2/3 and Finally, substitution into the differential rate expression: − 𝑑𝑀 𝑑𝑡 = − 𝑑𝑀 𝑑𝑟 ∙ 𝑑𝑟 𝑑𝑡 = 𝑘𝐴𝐶 from which Simplifying and integrating gives: 𝑟𝑜 (1 − 𝑥)1/3 = 𝑘′𝑡 12 𝑑𝑟 4𝜋𝜌𝑟 2 𝑑𝑡 = 4𝜋𝑟 2 𝑘𝐶 6.1.5 Boundary Layer (Nernst boundary layer) This is a stagnant film of fluid covering a solid in contact with fluid through which reactants have to diffuse to the reaction interface before reaction can take place. The existence of this boundary layer is due to two hydrodynamic factors: (i) the adhesion of the fluid to the surface (in laminar flow adhering fluid has zero velocity) (ii) the viscosity of the fluid. The interaction between solid and fluid follows the following steps: 6.1.6 1. diffusion of reacting molecules from the bulk of fluid to the interface, 2. adsorption at the interface 3. reaction at the interface, 4. desorption of reaction products from the interface, 5. diffusion of reaction products from the interface into the bulk of fluid Effect of fluid velocity For diffusion-controlled reactions increases in rate of fluid flow (eg. by agitation) around the solid particle in solid-liquid reactions causes an increase in the rate of reaction. This is because the thickness of the boundary layer decreases with increase in the flow rate of liquid (eg by stirring). 𝛿= 2𝑟𝑜 𝑆ℎ ; 𝑆ℎ = 𝑘𝑚 𝑑𝑝 𝐷 where δ is the thickness of the boundary layer around a particle, ro is the radius of the spherical particle, km is the mass transfer coefficient dp is the particle diameter D diffusion coefficient Sh is the Sherwood number (compares the relative magnitudes of the inertial forces to the viscous forces and is a measure of rate of mass transfer across the boundary layer) 𝐷 𝑟 = 𝛿 𝐴𝐶 therefore, as δ decreases the rate increases. However, this increase is not infinite because air pockets begin to form in the liquid at high agitation and interfere with mass transport. Other reasons for agitation include: - the suspension of solid particles in the liquid to ensure adequate solid-liquid contact. - to ensure the homogenous distribution of fluid reactant in the fluid. 13 In solid-gas reactions the rate of reaction depends on the intensity of the flow of gas. The rate of chemical reaction-controlled reactions are not affected by stirring. 6.1.7 Effect of temperature Diffusion-controlled reactions are only slightly affected by temperature. According to Stokes-Einstein equation, 𝐷= 𝑅𝑇 𝑁 1 ∙ 3𝜋𝑑𝜂, i.e. D~T where D is diffusion constant, m²/s R is the gas constant, J/mol.K N is the Avogadro number, 1/mol d is the diameter of the diffusing molecule, m η is dynamic viscosity, Pa.s For chemical-controlled reactions temperature strongly affects the rate through the rate constant, k, which has an exponential dependence on temperature. 𝐸 𝑘 = 𝐴𝑜 exp [𝑅𝑇𝐴 ] If the value of EA in the Arrhenius equation is 4 - 13 kJ/mol, then the process is diffusion controlled 21 - 33 kJ/mol, intermediate or mixed controlled > 42 kJ/mol, chemical reaction controlled For solid-solid reactions the diffusion coefficient has an exponential dependence on temperature: 𝐸 𝐷 = 𝐷𝑜 exp [𝑅𝑇𝐴 ] E is high, 840 - 1680 kJ/mol The mechanism of the reaction may change from chemically controlled at low temperature to diffusion controlled at high temperatures, because as the temperature increases chemical reaction rate increases, becoming faster than diffusion rate. The change in mechanism may also be due to the formation of a nonporous reaction product on the surface of the solid reactant. 6.1.8 Effect of reagent concentration For solid-liquid reactions the mechanism of the reaction may change from diffusion to chemical reaction control with increasing reagent concentration. This may be due to low diffusion rate at low concentration, which increases over chemical reaction rate as concentration increases. Sometimes a change in initial concentration may even result in change of control. 14 6.1.9 Electrochemical nature of some heterogeneous reactions The very nature of the formation of bonds makes some materials conductors, whilst others are semiconductors or insulators. The formation of minerals happens from complex systems containing various elements and depends on the conditions present at the time of mineral formation. Some of the elements or components are very similar in structure (eg. Similar ionic radii, or isomorphous), as a result of which there are the substitutions of elements from one to the other. These processes lead to disparities within the crystal lattice as a result of which certain portions acts as electro donors and others as electron acceptors. During heterogeneous reactions charges may be transferred (electrochemical in nature), eg. in the dissolution of iron in dilute HCl, the halfreactions are: At the anode: Fe = Fe2+ + 2e- (Fe is oxidized) At the cathode: 2H+ + 2e- = H2 (H+ is reduced to H2 gas) Overall reaction: Fe + 2H+ = Fe2+ + H2 Other examples are the leaching of gold in a cyanide medium in the presence of oxygen gas, the reaction between metal and gas, etc.: 4Au + 8CN- + O2 + 2H2O = 4Au[CN]2- + 4OH2Au + 4CN- + O2 + 2H2O = 2Au[CN]2- + H2O2 + 2OH4Fe + 3O2 = 2Fe2O3 6.1.10 Effect of the ratio of the reacting phases Again in solid-liquid reactions, when the mass of solid (M) and the concentration of reagent (C) remains the same, but the volume of reagent and thus the solid/liquid ratio (pulp density) varies, then the rate of reaction also varies. If the volume of reactant is small, the concentration decreases during reaction leading to a decrease in rate with time. However, when the volume of reagent is large then the concentration remains approximately constant and therefore rate is high throughout. 6.1.11 Nucleation Nucleation plays an important role in heterogeneous processes such as thermal decomposition in solids, crystallization, precipitation of solids from liquids, gas evolution in liquids, deposition of solids from gases and roasting. Three steps are involved in such reactions: 1) nucleation stage or induction period, which depends on imperfections in the crystal lattice of the decomposing solid, vibrations of atoms at the lattice points breaking of bonds and formation of new ones, 2) formation and growth of the reaction interface during the period of acceleration. Once the nuclei are formed the reaction accelerates, 15 3) propagation of the reaction interface, thus the initiated reaction accelerates until the reaction interface reaches a maximum, thereafter, the reaction slows down as the interface decreases. In the case of liquids and gases nucleation takes place at the walls of the container or on foreign particles present in the liquid or gas. When nucleation and rate of growth of nuclei are slowest (ie. rate determining), then the rate equation can be derived assuming equal probability of nucleation at each active site, thus the rate of reaction is proportional to the mass of reacting substance at time t. 6.1.12 Autocatalytic reactions This type of reaction is possible in solid-liquid reactions also when the liquid product reacts further with the solid reactant, eg. in the dissolution of copper in dilute sulphuric acid in the presence of oxygen or in aqueous ammonia: Cu + 2H+ + 0.5O2 = Cu2+ + H2O Cu + Cu2+ = 2Cu+ (which is rapidly oxidized by O2 as follows) 2Cu+ + 2H+ + 0.5O2 = 2Cu2+ + H2O (leading to increasing rate of Cu dissolution) 6.1.13 Nature of solid reaction products If solid product is formed on reacting solid, then the nature of the product (porous or nonporous) may determine the kinetics of reaction. In the case of a porous product, the fluid reactant diffuses easily to the reaction interface and therefore the rate is not affected by the coating. Resistance in this case may be due only to diffusion through the boundary layer. However, if a non-porous product layer is formed then the rate of reaction may be controlled by the diffusion through this layer. 6.2 Heterogeneous Reaction Rate Heterogeneous reactions can generally be represented by the following reactions: 1. Fluid-solid reactions: (i) A(fluid) + bB(solid) = cR (fluid) + dD(solid) (ii) A(fluid) + bB(solid) = cR (fluid) (iii) either of the above but involving charge transfer 2. Solid-solid reactions (eg. Ceramics, cement, ferrous metallurgy): (i) A(solid) + bB(solid) = cR 16 (ii) A(solid)+ bB(solid) = cR (solid) + dD(solid) Some of these reactions may take place through gaseous intermediates, eg. in the production of pig iron in the blast furnace, 3Fe2O3 + CO(g) = 2Fe3O4(s) + CO2(g) 2Fe3O4(s)+ CO(g) = 6FeO + CO2(g) CO2(g) + C = 2CO(g) In the case of solid-fluid reactions, the overall process may involve the following steps: 1. external mass transfer between the bulk fluid and the external surface of the solid. 2. diffusion of reactants and products within the pores of the solid. 3.* adsorption of reactants and products on the reaction surface. 4.* migration of reactants and products on the reaction surface. 5. chemical reaction between the reactants from the fluid phase and components in the solid. 6.* nucleation. Even though the steps indicated with asterisks do occur in heterogeneous reactions they shall not be considered in our discussions. Discussions will be limited to external mass transfer, pore diffusion and chemical reaction, since these are more common with reactions in extractive metallurgy. In addition to these steps other processes may also have significant effect on the overall rate. For example, - reaction heat transfer to and from the surrounding area where the reaction is taking place. - changes in the structure of the solid during reaction. The above steps show that diffusion is an important process in heterogeneous reactions. Diffusion is a result of differences in concentration and is in the direction of low concentration. It is described by Fick's Laws. Fick's First Law states that the rate of diffusion of substance A per unit cross section of a solid is proportional to the concentration gradient in the direction of diffusion (in other words the flux of a given particle A through a unit area perpendicular to the concentration gradient is related to the concentration gradient as): JA= - D∇CA 𝐽𝐴 = −𝐷 or written in the direction of x, where JA is the diffusion flux (∂nA/∂t), moles/m².s D is the diffusivity (diffusion coefficient), m²/s CA is concentration, moles/m3 ∂CA/∂x is the concentration gradient, moles/m3. m 17 𝜕𝐶𝐴 𝜕𝑥 The negative sign emphasizes that diffusion occurs in the direction of drop in concentration. The First Law applies when concentration gradient remains unchanged with the passage of time so that the rate of diffusion is constant. For unsteady-state diffusion, that is if the concentration gradient changes with time as a result of diffusion itself, then the Second Law applies: 𝐽𝐴 = 𝐷∇2 𝐶𝐴 or written in the x direction 𝜕𝑛𝐴 𝜕𝑡 −𝐷 𝜕2 𝐶𝐴 𝜕𝑥 2 =0 Generally, the overall rate of a heterogeneous reaction is determined by the slowest of the steps (rate controlling step). the reaction conditions can affect which of the steps becomes rate-controlling. Sometimes several steps may have more or less equal effects on determining the overall rate. An understanding of how each step relates to the other is therefore important in the determination of the overall rate of the process. In comparing and/or combining rates, we must define the rates in the same manner, for example if the rate of mass transfer, nA, (a physical transport) is defined per unit surface area, then the chemical reaction rate, r A, at the same surface must be defined per unit area of that surface for any comparison of both rates to be done, ie. rate of mass transfer, 1 𝑑𝑁𝐴 𝑛𝐴 = 𝑆 and rate of reaction, 𝑑𝑡 1 𝑑𝑁𝐴 𝑟𝐴 = 𝑆 where S is the area of reaction interface 𝑑𝑡 Consider the reaction between A (fluid reactant) and B (in the solid) giving a product R (fluid). (ii) A(fluid) + bB(solid) = cR (fluid) For the reaction to take place A must diffuse to the surface through a stagnant film, external mass transfer, and after reaction the product must also diffuse through the stagnant film into the bulk of fluid. Rate of mass transfer of A per unit surface area of the solid is 𝑛𝐴 = 1 𝑑𝑁𝐴 ∆𝐶 𝐷 (𝐶 − 𝐶𝑠 ) = 𝑘𝑚 (𝐶𝑏 − 𝐶𝑠 ) = −𝐷 = 𝑆 𝑑𝑡 ∆𝑥 ∆𝑥 𝑏 km is the mass transfer coefficient, Cb, Cs is concentration of A in bulk and surface respectively. The estimation of km depends on the system and on the mode of solid-fluid contacting and in terms of numbers. For a single particle in a large fluid volume (eg. dilute solution), Sh = 2.0 + 0.6 Re1/2 . Sc1/3 where Sh is Sherwood number, 𝑆ℎ = 𝑘𝑚 𝑑𝑝 𝐷 18 𝑅𝑒 = Re is Reynolds number, 𝑑𝑝 𝜌𝑢 𝜇 𝜇 𝑆𝑐 = 𝜌𝐷 Sc is Schmidt number, km : mass transfer coefficient, m/s D : diffusion coefficient, m2/s d : characteristic length (diameter of the particle), m u : average velocity. m/s ρ : density, kg/m3 μ : viscosity, kg/m.s At steady-state when external mass transfer rate is equal to reaction rate at the surface and assuming first-order reaction: nA = rA 𝐶𝑠 = −𝑘𝑚 (𝐶𝑏 − 𝐶𝑠 ) = −𝑘𝑠 𝐶𝑠 and 𝑘𝑚 𝑘𝑚 +𝑘𝑠 𝐶𝑏 𝑛𝐴 = 𝑟𝐴 = − (ks is the reaction rate constant) (Cs is difficult to measure and therefore expressed as a function of C b) 1 1 1 + 𝑘𝑚 𝑘𝑠 𝐶𝑏 = −𝑘𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝐶𝑏 Asymptotic cases: 1) if ks <<km , then Cs ≈ Cb , external mass transfer offers no resistance and reaction rate is chemical reaction control and r o = r A = - k s Cb 2) if ks >>km , then Cs → Cs* (equilibrium), chemical reaction is fast and overall process is mass transfer control and ro = rA = - km(Cb - Cs*) If solid reactant is porous, the diffusion of fluid species through the pores of the solid becomes important since a greater portion of the solid becomes available for reaction. It is also especially important when the product of a nonporous solid is porous allowing rapid diffusion through the porous product. However, pore diffusion is very complex because of complications as a result of the existence of tortuosity in the solid depending on the pore structure. In addition, when the pores are smaller than the mean free path of the various species, then the laws of unimolar molecular diffusion will no longer hold and the so-called Knudsen diffusion (ie. rate of diffusion is governed by the collision of molecules with the walls of the pore) becomes important. If the pores are large, diffusion from the fluid bulk into the pores is fast and fluid concentration in the pores would be the same as in the bulk of fluid and therefore mass transfer in such porous media is described by Fick's 1st Law: r A = - D ∇ CA For porous solids the diffusion of fluid reactants within the pores of the solid creates an additional regime where the overall reaction is strongly influenced by the pore diffusion but not controlled by it. 19