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Appendix a Rate of Entropy Production and Kinetic Implications

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Appendix a Rate of Entropy Production and Kinetic Implications Appendix A Rate of Entropy Production and Kinetic Implications According to the second law of thermodynamics, the entropy of an isolated system can never decrease on a macroscopic scale; it must remain constant when equilib- rium is achieved or increase due to spontaneous processes within the system. The ramifications of entropy production constitute the subject of irreversible thermody- namics. A number of interesting kinetic relations, which are beyond the domain of classical thermodynamics, can be derived by considering the rate of entropy pro- duction in an isolated system. The rate and mechanism of evolution of a system is often of major or of even greater interest in many problems than its equilibrium state, especially in the study of natural processes. The purpose of this Appendix is to develop some of the kinetic relations relating to chemical reaction, diffusion and heat conduction from consideration of the entropy production due to spontaneous processes in an isolated system. A.1 Rate of Entropy Production: Conjugate Flux and Force in Irreversible Processes In order to formally treat an irreversible process within the framework of thermo- dynamics, it is important to identify the force that is conjugate to the observed flux. To this end, let us start with the question: what is the force that drives heat flux by conduction (or heat diffusion) along a temperature gradient? The intuitive answer is: temperature gradient. But the answer is not entirely correct. To find out the exact force that drives diffusive heat flux, let us consider an isolated composite system that is divided into two subsystems, I and II, by a rigid diathermal wall, the subsystems being maintained at different but uniform temperatures of TI and TII, respectively. It was shown in Sect. 2.8 that the entropy change of the composite system is given by Eq. (2.8.2) 1 1 dS = ␦qI − TI TII 443 444 Appendix A where ␦qI is the heat absorbed by the subsystem I. (It was also shown in Sect. 2.8 that because of the second law, ␦qI > 0ifTII > TI, and vice versa). Thus, the rate of entropy production is given by dS ␦qI 1 1 = − (A.1.1) dt dt TI TII If ⌬x is the length of the composite system and A is the cross sectional area of the wall separating the two subsystems (Fig. 2.9), then A(⌬x) is the total volume of the system. Thus the rate of entropy production per unit volume, commonly denoted by ␴, is given by 1 dS ␴ = (A.1.2) A(⌬x) dt so that ␦qI 1/TI − 1/TII ␴ = (AI.1.3) Adt ⌬x Considering infinitesimal change, the quantity within the second parentheses can be written as d(1/T)/dX. The term within the first parentheses represents the heat flux, JQ, (i.e. rate of heat flow per unit area) across the wall separating the subsystems I and II. Thus, d(1/T) ␴ = J (A.1.4) Q dx The parenthetical derivative term in the above equation represents the force driving the heat flux,JQ. Note that the force is the spatial gradient of the inverse temper- ature instead of that of temperature itself. Noting that d(1/T) = – dT/T2,thelast equation can be written as J dT ␴ =− Q (A.1.5) T2 dx Equation (A.1.4) shows a general property of the rate of entropy production in a system due to an irreversible process or several irreversible processes. For each pro- cess, k, the rate of entropy production per unit volume, ␴k, is given by the product of afluxterm,Jk, associated with the process k (or a rate per unit volume for chemical reaction, as shown below), and the force, ␹k, driving the flux. That is ␴ = (Either flux or rate per unit volume)(conjugate force) (A.1.6) = Jkχk A.1 Rate of Entropy Production: Conjugate Flux and Force Irreversible Processes 445 Thus, consideration of the entropy production during a process leads to the proper identification of the conjugate force and flux terms for the process. The rate of total entropy production per unit volume in a closed system due to several irreversible processes is given by ␴ = ␴k = Jkχk (A.1.7) k k Let us now try to identify the appropriate forces that drive diffusion and chemical reaction by considering the rate of entropy production in a system due to these processes. For the first problem, let us again consider an isolated composite system. as above, but make the wall separating the two subsystems porous to the diffusion I II of the component i, and denote dni and dni as the change in the number of moles of i in the subsystems I and II, respectively. Since the total number of moles of a component (ni) in the overall composite system is conserved, we obviously have I II dni = –dni . For simplicity, we assume that the two subsystems are at the same temperature. The change of total entropy, S, of the composite isolated system due to a change in the number of moles of the component i in the subsystem I is given by dS dSI + dSII dSI dSII = = − (A.1.8) dnI dnI dnI dnII i ni i i i where SI and SII are the entropies of the subsystems I and II, respectively. Now, since the volumes of2 the subsystems are constant, we have, from Eq. (8.1.4) (i.e. dU = TdS – PdV + ␮idni) dSI dUI ␮I = − i . I I (A 1 9a) dni Tdni T and dSII dUII ␮II = − i dnII TdnII T i i (A.1.9b) dUII ␮II =− − i I Tdni T Substituting the last two relations in Eq. (A.1.8), and noting that d(UI +UII)=0,we obtain dS ␮II − ␮I = i I . I (A 1 10) dni T Using chain rule, Eq. (A.1.2) can be written as 446 Appendix A 1 dS dnI ␴ = i (A.1.11) ⌬ I A x dni dt Substituting Eq. (A.1.10) into the last expression and rearranging terms, we obtain ␮II − ␮I dnI ␴ = i i i (A.1.12) T⌬x Adt Thus, we can write the following expression for the rate of entropy production due to chemical diffusion: Ѩ(␮i/T) ␴ =− J , (A.1.13) Ѩx d i where Jd,i, which stands for the parenthetical term in Eq. (A.1.12), is the diffusive flux of the component i (rate of diffusive transfer of the component i per unit surface area). From the above equation, we see that, in the absence of an external force, the 1 driving force for diffusion of i is the negative gradient of ␮i/T. As we shall see later, recognition of the appropriate driving force leads to the formal understanding of how diffusion within a solution is affected by its thermodynamic mixing property. Let us now consider a system undergoing irreversible chemical reaction under constant P-T condition within an isolated system. From Eq. (10.2.5) dSint = A d␰ T where Sint is the internal entropy production due to chemical reaction, and A is the affinity of the reaction, as defined Eq. (10.2.6), and equals – ⌬rG. Thus, dS dS d␰ A d␰ = = (A.1.14) dt d␰ dt T dt Consequently, the rate of entropy production per unit volume due to chemical reaction is given by 1 dS A d␰ ␴ = = (A.1.15) V dt T Vdt where V is the volume of the system. The parenthetical quantity is the reaction rate per unit volume, which we would denote by Ri for the i th reaction. Thus, the term A/T (or – ⌬rG/T) represents the driving force for a chemical reaction. 1If there is an external force acting on the diffusing species i, then the driving force becomes – grad (␮i/T) + Fi,whereFi is the force per unit mass acting on i. For example, if there is an electrical field Ed, then there is a driving force zEd, where z is the charge of the diffusing species. A.2 Relationship Between Flux and Force 447 Table A.1 Summary of fluxes and conjugate forces for some irreversible processes Process Flux Conjugate force Heat Conduction JQ grad (1/T) (or heat diffusion) (Joules/s-cm2) ∗ Chemical Diffusion Jd,i –[grad(␮i/T) – Fi] (mol/s-cm2) Chemical reaction Rate of reaction A/T or – ⌬rG/T per unit volume ∗ Fi: External force per unit mass acting on i The fluxes and conjugate forces for heat conduction (or heat diffusion), chemical diffusion and chemical reaction are summarized in Table A.1. Using Eq. (A.1.7) and the conjugate flux and force terms tabulated above, the total rate of entropy production in a system within which both heat and chemical diffusion and irreversible reactions are taking place, is given by JQ dT d(␮i/T) Ai ␴ =− − J , − F + R (A.1.16) T T2 dx d i dx i i T i i A.2 Relationship Between Flux and Force AfluxJk depends on the conjugate force, χ k. If there are other forces in the system, then Jk depends also on those other forces. In general, Jk could become a compli- cated function of the forces. In the simplest approximation, one can write that J1 = L11χ1 + L12χ2 + L13 + χ3 + ......... or n J1 = L1jχj (A.2.1) j=1 where it is assumed that there are n independent forces in the system. The L terms represent phenomenological coefficients. Because of the linear nature of the above equation, it is called an expression of flux in the domain of linear irreversible thermodynamics. As remarked by Lasaga (1998), the linear approximation works quite well for heat and chemical diffusion processes, but may not work well for all problems of fluid flow of geological interest and fails for chemical reactions far from equilibrium for which one needs to introduce higher order terms.
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