Understanding Non-Equilibrium Thermodynamics - Springer 2008 Episode 4 ppt

78 3 Coupled Transport Phenomena The maximum efficiency is then η max = T h − T c T h 2+ZT − 2 √ 1+ZT ZT . (3.28) If the ratio x is significantly different from the optimum value (3.27), the material is not efficiently converting heat energy into electric energy. Once x is optimized, the optimal current may be found, for a given value of the temperature gradient. Note that for ZT  1, one finds back Carnot’s but materials currently used in thermoelectric devices have relatively low values of ZT, between 0.4 and 1.5, so that in reality, the maximum Carnot value is far from being attained. In general, the temperature difference between the two sides of the generator will be so high that the assumption of homogeneity used in (3.25)–(3.28) is not tenable. In this case, one may consider the thermoelectric device as formed by a series of small quasi-homogeneous elements, at different average temperatures. For instance, for two elements thermally in series, the com- bined efficiency is (see Problem 3.9), η = P el,1 + P el,2 ˙ Q 1 =1− (1 − η 1 )(1 − η 2 ). (3.29a) In the continuum limit, in which the generator is constituted of many layers at different temperatures, (3.29a) must be replaced by (see Problem 3.9), η =1− exp  −  T h T c η r (x, T ) T dT  . (3.29b) Using this expression, one finds for the total efficiency of the device η =1− ε c T c + x −1 c ε h T h + x −1 h . (3.30) In this configuration, it may happen that the optimum current determined by x opt in one segment (for instance, the hot side) is significantly different from the optimum value x opt in another segment (for instance, the cold side); in this case, there will be no suitable current for which both parts of the generator are operating with optimal efficiency. This is a challenge in materials sciences, as x opt is temperature dependent through the transport coefficients of the material (namely Z and ε) and it would be highly desirable to find materials with suitable temperature dependence of these coefficients to optimize the generation. In terms of the coefficient Z and assuming constant transport coefficients ε, λ,andr, the maximum efficiency of the power generation is given by (see Problem 3.7) η max = T h − T c T h (1 + ZT av ) 1/2 − 1 (1 + ZT av ) 1/2 +(T c /T h ) , (3.31) 3.3 Thermodiffusion: Coupling of Heat and Mass Transport 79 where T av is the average temperature T av = 1 2 (T h + T c ). When T c ≈ T h ,the ratio T c /T h in the denominator of (3.31) is close to 1 and the efficiency (3.28) is recovered. Analogously, the coefficient of performance for refrigeration (i.e. the ratio between the heat extracted per unit time and the electric power consumed by the corresponding engine) is η = ˙ Q P el = T c T h − T c (1 + ZT av ) 1/2 − (T h /T c ) 1+(1+ZT av ) 1/2 . (3.32) In many practical situations, two parallel generators are used, one of n-type semiconductors (current carried by electrons, with ε<0) and another with p-type semiconductors (current brought by holes, with ε>0). The global efficiency may be derived from (3.23) or (3.31) by using an average for both generators, namely η np = η p Q p + η n Q n Q p + Q n , (3.33) where Q p and Q n are the amounts of heat supplied to the hot side of the p and n elements per unit time. 3.3 Thermodiffusion: Coupling of Heat and Mass Transport By thermodiffusion is meant the coupling between heat and matter transport in binary or multi-component mixtures. In the case of a binary mixture, a natural choice of the state variables is ρ 1 , ρ 2 (individual mass densities), v (barycentric velocity), and u (internal energy), but a more convenient choice is ρ(= ρ 1 + ρ 2 ), c 1 =(ρ 1 /ρ), v ,andu. In the classical theory of irreversible processes, one is more interested by the behaviour of the barycentric velocity than by the individual velocities of the components and this is the reason why only the velocity v, and not the individual velocities v 1 and v 2 ,figuresamong the space of variables. In absence of chemical reactions, viscosity, external body forces and energy sources, the corresponding evolution equations are given by dρ dt = −ρ∇·v , (3.34) ρ dc 1 dt = −∇ · J 1 , (3.35) ρ dv dt = −∇p, (3.36) ρ du dt = −∇ · q −p∇·v . (3.37) 80 3 Coupled Transport Phenomena Moreover, we assume that the system is in mechanical equilibrium with zero barycentric velocity and zero acceleration, which is the case for mixtures confined in closed vessels. It follows then from (3.34) and (3.36) that the total mass density and the pressure remain uniform throughout the system and that the last term in (3.37) vanishes. Substituting the balance equations of mass fraction (3.35) and internal energy (3.37) in the Gibbs’ equation ds dt = T −1 du dt − T −1 (¯µ 1 − ¯µ 2 ) dc 1 dt , (3.38) where use has been made of dc 2 = −dc 1 , results in the following balance of entropy ρ ds dt = −∇·{T −1 [q −(¯µ 1 −¯µ 2 )J 1 ]}+q ·∇T −1 −J 1 ·∇[T −1 (¯µ 1 −¯µ 2 )], (3.39) with the entropy flux given by J s = T −1 [q − (¯µ 1 − ¯µ 2 )J 1 ], (3.40) and the rate of entropy production by σ s = q ·∇T −1 − J 1 ·∇[T −1 (¯µ 1 − ¯µ 2 )]. (3.41) With the help of the Gibbs–Duhem’s relation c 1 (∇¯µ 1 ) T,p + c 2 (∇¯µ 2 ) T,p =0, (3.42) where subscripts T and p indicate that differentiation is taken at constant T and p, and the classical result of equilibrium thermodynamics T ∇(T −1 ¯µ k )=−h k T −1 (∇T )+(∇¯µ k ) T , (3.43) with h k the partial specific enthalpy of substance k(k =1, 2), we are able to eliminate the chemical potential from (3.41) of σ s , which finally reads as σ s = −q  · ∇T T 2 − µ 11 Tc 2 J 1 ·∇c 1 . (3.44) The new heat flux q  is defined by q  = q −(h 1 −h 2 )J 1 , and it is equal to the difference between the total flux of heat and the transfer of heat due to diffusion while the quantity µ 11 stands for µ 11 =(∂ ¯µ 1 /∂c 1 ) T,p . The derivation of (3.44) exhibits clearly the property that the entropy production is a bilinear expression in the thermodynamic fluxes q  and J 1 and forces taking the form of gradients of intensive variables, easily accessible to direct measurements. It is obvious that when the mixture reaches thermodynamic equilibrium, the heat and mass flows as well as the temperature and mass fraction gradients 3.3 Thermodiffusion: Coupling of Heat and Mass Transport 81 vanish. Expression (3.44) of σ s suggests writing the following phenomenological relations between fluxes and forces: q  = −L qq ∇T T 2 − L q1 µ 11 Tc 2 ∇c 1 , (3.45) J 1 = −L 1q ∇T T 2 − L 11 µ 11 Tc 2 ∇c 1 , (3.46) with the Onsager’s reciprocal relation L q1 = L 1q and the following inequalities resulting from the positiveness of entropy production: L qq > 0,L 11 > 0,L qq L 11 − L q1 L 1q > 0. (3.47) After introducing the following identifications: L qq T 2 = λ (heat conductivity), L 11 µ 11 ρc 2 T = D (diffusion coefficient), L q1 ρc 1 c 2 T 2 = D F (Dufour coefficient), L 1q ρc 1 c 2 T 2 = D T (thermal diffusion coefficient), the phenomenological laws take the form q  = −λ∇T −ρT µ 11 c 1 D F ∇c 1 , (3.48) J 1 = −ρc 1 c 2 D T ∇T −ρD∇c 1 . (3.49) Inequalities (3.47) imply in particular that λ>0andD>0 while from Onsager’s relation is inferred that D F = D T . (3.50) This last result is a confirmation of an earlier result established by Stefan at the end of the nineteenth century. Starting from the law of conservation of momentum, Stefan was indeed able to demonstrate the above equality at least in the case of binary mixtures. By setting the gradient of the mass fraction equal to zero, (3.48) is identical to Fourier’s equation so that λ can be identified with the heat conductivity coefficient. Similarly, relation (3.49) reduces to Fick’s law of diffusion when temperature is uniform and therefore D represents the diffusion coefficient; in general, the phenomenological coefficients in (3.48) and (3.49) are not constant. In multi-component mixtures, the mass flux of substance i is a linear function of not only ∇c i but also of all the other mass fractions gradients ∇c j (j = i). Such “diffusion drag” forces have been invoked to interpret some biophysical effects and play a role in the processes of separation of isotopes (see Box 3.3). 82 3 Coupled Transport Phenomena Box 3.3 The Soret’s Effect and Isotope Separation Thermal diffusion is exploited to separate materials of different molecular mass. If a fluid system is composed of two kinds of molecules of different molecular weight, and if it is submitted to a temperature gradient, the lighter molecules will accumulate near the hot wall and the heavier ones near the cold wall. This property was used in the 1940s for the separation of 235 U and 238 U isotopes in solutions of uranium hexafluoride, in the Manhattan project, leading to the first atomic bomb. Usually this process is carried out in tall and narrow vertical columns, where convection effects reinforce the separation induced by thermal diffusion: the light molecules near the hot walls have an ascending motion, whereas the heavy molecules near the cold wall sunk towards the lowest regions. This process accumulates the lightest isotope in the highest regions, from where it may be extracted. This process is simple but its consumption of energy is high, and therefore it has been substituted by other methods. However, it is still being used in heavy water enrichment, or in other processes of separation of light atoms with the purpose of, for instance, to generate carbide layers on steel, alloys or cements, thus hardening the surface and making it more resistant to wear and corrosion. Soret’s effect plays also a role in the structure of flames and in polymer characterization. More recently, high temperature gradients have been produced by means of laser beams rather than by heating uniformly the walls of the container. The Dufour’s effect, the reciprocal of thermal diffusion, has not so many industrial applications. It plays nevertheless a non-negligible role in some natural processes as heat transport in the high atmosphere and in the soil under isothermal conditions but under a gradient of moisture. The coefficient D T in (3.49) is typical of thermal diffusion, i.e. the flow of matter caused by a temperature difference; such an effect is referred to as the Soret’s effect in liquids with the quotient D T /D called the Soret coefficient. The reciprocal effect, i.e. the flow of heat caused by a gradient of concentration as evidenced by (3.48) is the Dufour’s effect. It should be observed that the cross-coefficients D T and D F are much smaller than the direct coefficients like the heat conductivity λ and the diffusion coefficient D.The latter turns out to be of the order of 10 −8 m 2 s −1 in liquids and 10 −5 m 2 s −1 in gases while the coefficient of thermal diffusion D T varies between 10 −12 and 10 −14 m 2 s −1 K −1 in liquids and from 10 −8 to 10 −12 m 2 s −1 K −1 in gases. The Soret’s effect is mainly observed in oceanography while the Dufour’s effect, which is negligible in liquids, has been detected in the high atmosphere. The smallness of the coupling effects is the reason why they are hard to be observed and measured with accurateness. Defining a stationary state by the absence of matter flow (J 1 = 0), it turns out from (3.49) that (∆T ) st = − D D T c 1 c 2 (∆c 1 ) st , (3.51) 3.4 Diffusion Through a Membrane 83 which indicates that a difference of concentration is able to generate a temperature difference, called the osmotic temperature. This is typically an irreversible effect because the corresponding entropy production is non-zero as directly seen from (3.41). This effect should not be confused with the osmotic pressure, which expresses that a difference of concentration between two reservoirs kept at the same uniform temperature but separated by a semi-permeable membrane gives raise to a pressure drop, called the osmotic pressure. The latter is a pure equilibrium effect resulting from the property that, at equilibrium, the chemical potential ¯µ(T, p, c 1 ) takes the same value in both reservoirs so that (∆¯µ) T =∆p/ρ 1 + µ 11 ∆c 1 =0,and ∆p = −ρ 1 µ 11 ∆c 1 , (3.52) with ρ 1 the specific mass of the species crossing the membrane; it is directly checked that in the present situation, the entropy production (3.41) is indeed equal to zero. The phenomena studied in this section are readily generalized to multi- component electrically charged systems, like electrolytes. 3.4 Diffusion Through a Membrane The importance of transport of matter through membranes in the life of cells and tissues justify that we spend some time to discuss the problem. In biological membranes, one distinguishes generally between two modes of transport: purely passive transport due to a pressure gradient or a mass concentration gradient and active transport involving ionic species, electrical currents, and chemical reactions. Here we focus on some aspects of passive transport. Our objective is to present a simplified analysis by using a minimum number of notions and parameters; in that respect, thermal effects will be ignored but even so, the subject keeps an undeniable utility. We consider the simple arrangement formed by two compartments I and II separated by a homogeneous membrane of thickness ∆l, say of the order of 100 µm. Each compartment is filled with a binary solution consisting of a solvent 1 and a solute 2 (see Fig. 3.3). The membrane is assumed to divide the system in two discontinuous sub- systems that are considered as homogeneous. 3.4.1 Entropy Production In absence of thermal gradients, it is inferred from (3.41) that the rate of dissipation, measured per unit volume of the membrane, will take the form 84 3 Coupled Transport Phenomena Fig. 3.3 System under study consisting in a membrane separating a binary solution Tσ s = −J 1 ·∇¯µ 1 − J 2 ·∇¯µ 2 . (3.53) After integration over the thickness ∆l, the rate of dissipation per unit surface, denoted as Φ, can be written as Φ = −J 1 ∆¯µ 1 − J 2 ∆¯µ 2 , (3.54) where ∆¯µ i designates the difference of chemical potential of species i across the membrane, J 1 and J 2 are the flows of solvent and solute, respectively. Instead of working with J 1 and J 2 , it is more convenient to introduce the total volume flow J V across the membrane and the relative velocity J D of the solute with respect to the solvent, defined, respectively, by J V =¯v 1 J 1 +¯v 2 J 2 , (3.55) J D = v 2 − v 1 , (3.56) the quantities ¯v 1 and ¯v 2 stand for the partial specific volumes of the solvent and the solute, v 1 and v 2 are their respective velocities given by v 1 =¯v 1 J 1 and v 2 =¯v 2 J 2 . With the above choice of variables, (3.54) reads as (Katchalsky and Curran 1965; Caplan and Essig 1983) Φ = −J V ∆p − J D ∆π, (3.57) where ∆p = p I −p II and ∆π = c 2 (∆¯µ 2 ) p is the osmotic pressure; the quantity (∆¯µ 2 ) p is that part of the chemical potential depending only on the concentration and defined from ∆¯µ 2 = V 2 ∆p +(∆¯µ 2 ) p , c 2 is the number of moles of the solute per unit volume. For ideal solutions, one has (∆¯µ 2 ) p = RT∆c 2 . 3.4 Diffusion Through a Membrane 85 3.4.2 Phenomenological Relations By assuming linear relations between thermodynamic fluxes and forces, one has J V = L VV ∆p + L VD ∆π, (3.58) J D = L DV ∆p + L DD ∆π. (3.59) The advantage of (3.58) and (3.59) is that they are given in terms of parameters that are directly accessible to experiments. The corresponding Onsager’s relation is L VD = L DV , whose main merit is to reduce the number of parameters from four to three. To better apprehend the physical meaning of the phenomenological coefficients L VV , L DV , L DD ,andL VD , let us examine some particular experimental situations. First consider the case wherein the concentration of the solute is the same on both sides of the membrane such that ∆π = 0. If a pressure difference ∆p is applied, one will observe according to (3.58) a volume flow proportional to ∆p; the proportionality coefficient L VV is called the mechanical filtration coefficient of the membrane: it is defined as the volume flow produced by a unit pressure difference between the two faces of the membrane. A further look on relation (3.59) indicates that even in absence of a concentration difference (∆π = 0), there will be a diffusion flow J D = L DV ∆p caused by the pressure difference ∆p. This phenomenon is known in colloid chemistry under the name of ultrafiltration and the coefficient L DV is the ultrafiltration coefficient. An alternative possibility is to impose ∆p = 0 but different solute concentrations in compartments I and II. In virtue of (3.59), the osmotic difference ∆π will produce a flow of diffusion J D = L DD ∆π and L DD is identified as the permeability coefficient: it is the diffusional mobility induced by a unit osmotic pressure ∆π = 1. Another effect related to (3.58) is the occurrence of a volume flow J V = L VD ∆π caused by a difference of osmotic pressure at uniform hydrostatic pressure: the coupling coefficient L VD is referred to as the coefficient of osmotic flow. The above discussion has clearly shown the importance of the coupling coefficient L DV = L VD ; by ignoring it one should miss significant features about motions across membranes. The importance of this coefficient is still displayed by the osmotic pressure experiment illustrated by Fig. 3.4. The two phases I (solvent + solute) and II (solvent alone) are separated by a semi-permeable membrane only permeable to the solvent. The height of the solution in the capillary tube gives a measure of the final pressure difference obtained when the volume flow J V vanishes, indeed from (3.58) one obtains (∆p) J V =0 = − L VD L VV ∆π. (3.60) This result indicates that, contrary to what is sometimes claimed, ∆p is not a measure of the osmotic pressure, this is only true if L VD = −L VV . 86 3 Coupled Transport Phenomena Fig. 3.4 Osmotic pressure experiment This condition is met by so-called ideal semi-permeable membranes whose property is to forbid the transport of solute whatever the values of ∆p and ∆π. For membranes, which are permeable to the solute, it is experimentally found that r ≡− L VD L VV < 1. (3.61) This ratio that is called the reflection coefficient tends to zero for selective membranes, like porous gas filters, and has been proposed to act as a measure of the selectivity of the membrane. For r = 1 (ideal membranes), all the solute is reflected by the membrane, for r<1, some quantity of solute crosses the membrane, while for r = 0 the membrane is completely permeable to the solute. To clarify the notion of membrane selectivity, let us go back to the situation described by ∆π = 0. In virtue of (3.58) and (3.59), one has r = − L VD L VV = −  J D J V  ∆π=0 , (3.62) or, in terms of the velocities introduced in relation (3.56),  2v 2 v 1 + v 2  ∆π=0 =1− r. (3.63) For an ideal semi-permeable membrane (r = 1), one has v 2 =0andthe solute will not cross the membrane; for r =0(v 1 = v 2 ) the membrane is not selective and allows the passage of both the solute and the solvent; for negative values of r(v 2 >v 1 ), the velocity of the solute is greater than that of the solvent and this is known as negative anomalous osmosis, which is a characteristic of the transport of electrolytes across charged membranes. 3.5 Problems 87 Table 3. 1 Phenomenological coefficients for two biological membranes ωL VV Membrane Solute Solvent (10 −15 mol dyn −1 s −1 ) r (10 −11 cm 3 dyn −1 s −1 ) Human Methanol Water 122 – – blood cell Urea Water 17 0.62 0.92 Toad skin Acetamide Water 4 × 10 −3 0.89 0.4 Thiourea Water 5.7 × 10 −4 0.98 1.1 A final parameter of interest, both in synthetic and biological membranes, is the solute permeability coefficient ω = c 2 L VV (L VV L DD − L 2 VD ). (3.64) For ideal semi-permeable membranes for which L VD = −L VV = −L DD , one has ω = 0, and for non-selective membranes (r = 0), it is found that ω = c 2 L DD . The interest of irreversible thermodynamics is to show clearly that three parameters are sufficient to describe transport of matter across membrane and to provide the relationships between the various coefficients character- izing a semi-permeable membrane. In Table 3.1 are reported some values of these coefficients for two different biological membranes (Katchalsky and Curran 1965). 3.5 Problems 3.1. Entropy flux and entropy production. Determine (3.11) and (3.12) of the entropy flux and the entropy production in the problem of thermoelectricity. 3.2. Onsager’s reciprocal relations. In presence of a magnetic field, Onsager’s relations can be written as L(H)=L T (−H). Decomposing L in a symmetric and an antisymmetric part L = L s + L a , show that L s (H)=L s (−H)and L a (H)=−L a (−H). 3.3. Thermoelectric effects. The Peltier coefficient of a couple Cu–Ni is π Cu–Ni ≈−5.08 mV at 273 K, π Cu–Ni ≈−6.05 mV at 295 K, and π Cu–Ni ≈ −9.10 mV at 373 K. Evaluate (a) the heat exchanged in the junction by Peltier’s effect when an electric current of 10 −2 AflowsfromCutoNiat 295 K; (b) idem when the current flows from Ni to Cu. (c) The two junctions of a thermocouple made of Cu and Ni are kept at 305 and 285 K, respectively; by using the Thomson relation, determine the Seebeck coefficient and estimate the electromotive force developed by the thermocouple. [...]... per unit volume, namely ck ≡ Nk /V In this case, ˜ (4. 7b) may be expressed as 94 4 Chemical Reactions and Molecular Machines A = RT ln c2 (˜ν1 )eq (˜ν2 )eq · · · c1 cν1 cν2 · · · ˜1 ˜2 (4. 7c) To derive the expression of the rate of entropy production, we introduce the balance equations of mass (4. 2) and energy (4. 3) in Gibbs’ relation (4. 4a) and (4. 4b) Since in chemical reactions, the temperature is... c c (4. 1.6) Using (4. 1.6) to eliminate k−1 , k−2 , and k−3 in (4. 1 .4) and (4. 1.5), one obtains w1 − w3 = k1 (Ã )eq c δÃ c δ˜B c − (Ã )eq c (˜B )eq c c + k3 (˜C )eq w2 − w3 = k2 (˜B )eq c δ˜B c δ˜C c − (˜B )eq c (˜C )eq c c + k3 (˜C )eq δÃ c δ˜C c − , (Ã )eq c (˜C )eq c (4. 1.7) δÃ c δ˜C c − (Ã )eq c (˜C )eq c (4. 1.8) 4. 2 Coupled Chemical Reactions 99 In view of the results (4. 1.2) and (4. 1.3),... (Van den Broeck 2005) 4. 4 Chemical Reactions and Mass Transport: Molecular Machines The general developments of Sect 4. 3 will be applied to the study of molecular machines, mainly pumps across membranes and motors along filaments which are briefly introduced in Box 4. 2 In classical non-equilibrium thermodynamics, 4. 4 Chemical Reactions and Mass Transport: Molecular Machines 103 Box 4. 2 More About Biological... such a molecular motor is presented The expression for the dissipated energy is then T σ s = Aw + vF, (4. 43) v being the velocity of the machine along the filament and F the viscous resistance The phenomenological laws are, in the linear approximation, w = L11 A + L12 F, (4. 44) v = L21 A + L22 F, (4. 45) 4. 4 Chemical Reactions and Mass Transport: Molecular Machines 105 with the coefficient L22 inversely proportional... )eq A1 + c = RT w2 − w3 = (4. 1.9) k3 (˜C )eq A3 c RT 1 [k2 (˜B )eq + k3 (˜C )eq ]A2 , (4. 1.10) c c RT where use has been made of A3 = −(A1 + A2 ) Note that (4. 1.9) and (4. 1.10) have been derived exclusively by reference to chemical kinetics and the principle of detailed balance A look on the coefficients of A2 in (4. 1.9) and A1 in (4. 1.10), which referring to (4. 25a) and (4. 25b), can be identified as... L12 ∆µ, J = L21 A + L22 ∆µ, (4. 41) (4. 42) with L12 = L21 according to Onsager’s reciprocal relations The coefficient L21 in (4. 42) is responsible for the coupling between chemical reaction and mass transport For instance, the experimental values of the phenomenological coefficients for the transport across the membrane of a frog bladder are the following: L22 = 1 04, L12 = 5 .40 , and L11 = 0.37 in units of... ideas of such isothermal free-energy transfer can be well interpreted in the framework of linear non-equilibrium thermodynamics Consider two coupled reactions, for which the dissipated energy is T σ s = w1 A1 + w2 A2 (4. 32) The corresponding phenomenological equations describing the kinetics of the reactions are w1 = L11 A1 + L12 A2 , w2 = L21 A1 + L22 A2 (4. 33) (4. 34) The above results remain applicable... coefficient w2 /w1 and the efficiency η of the conversion defined by w2 A2 (4. 36) η≡− w1 A1 Introducing the quantities x ≡ A2 /A1 , Z ≡ (L22 /L11 )1/2 , and using (4. 33) and (4. 34) , it is easily checked that η=− Zx(q + Zx) , 1 + qZx (4. 37) which is maximum, namely, dη/dx = 0, at (1 − q 2 )1/2 − 1 , qZ (4. 38) q2 [1 + (1 − q 2 )1/2 ]2 (4. 39) xmax = thus leading to ηmax = This result is worth to be underlined:... these symmetry relations for a triangular reaction scheme is presented 4. 2.2 Cyclical Chemical Reactions and Onsager’s Reciprocal Relations A demonstration of the reciprocal relations (4. 24) , based on the cycle of chemical reactions A B C A depicted in Fig 4. 1, was proposed by Onsager in his celebrated papers of 1931 In Sect 4. 4, we will use a modified form of this scheme as a model for chemically driven... cCl2 − k− c2 = k+ cH2 cCl2 1 − ˜HCl 1 c2 , (4. 13) K cH2 cCl2 ˜ ˜ 4. 1 One Single Chemical Reaction 95 where k+ and k− are, respectively, the rate constants corresponding to the forcHCl c c ward and backward reactions, and K = k+ /k− = (˜2 )eq /[(˜H2 )eq (˜Cl2 )eq ] In terms of the affinity, and in virtue of (4. 7c), relation (4. 13) can be expressed as ˜ ˜ (4. 14) w = k+ cH2 cCl2 [1 − exp(−A/RT )] For reactions . (4. 2) and energy (4. 3) in Gibbs’ relation (4. 4a) and (4. 4b). Since in chemical reactions, the temperature is generally assumed to remain uniform, one obtains ρ ds dt = −∇· q T + A T dξ dt . (4. 8) By. k + /k − =(˜c 2 HCl ) eq /[(˜c H 2 ) eq (˜c Cl 2 ) eq ]. In terms of the affinity, and in virtue of (4. 7c), relation (4. 13) can be expressed as w = k + ˜c H 2 ˜c Cl 2 [1 − exp(−A/RT )] . (4. 14) For reactions near equilibrium, the affinity A is. relations, one has w i =  j L ij A j T , (4. 23) 4. 2 Coupled Chemical Reactions 97 with, in virtue of Onsager’s reciprocal relations, L ij = L ji . (4. 24) In Sect. 4. 2.2, a particular derivation of these