Entropy production due to chemical reactions

The formalism of the previous sections can now be used to relate entropy production to reaction rates more explicitly. In Chapter 4 we have seen from 

Our objective is to relate the affinity A to the reaction rates, so that the entropy production is written in terms of the reaction rates. Let us consider the reaction 

Since the forward and reverse rates must be equal at equilibrium, we have seen from (9.4.2) that 

We have already seen that the velocity of a reaction is simply the difference between the forward and reverse reaction rates and that the reaction rates can be expressed in terms of (equation 9.2.6)  

To obtain the velocity of reaction as a function of time, this differential equation has to be solved. An example is given below. 

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Finally, Example 8 is more complex in that it comprises entropy production due to chemical reaction in combination with heat transfer, and also diffusion the role of the latter appears as marginal. The example can also be regarded as an example of complex single-node balancing, a kind of thermodynamic analysis included. Concerning the entropy production (or loss of exergy), it turns out that the chemical portion, thus the term -(Gr / Tj ) Wr in (6.2.109) can represent an enormous item in the exergy balance it can be computed, but this is usually all that we can do in practice. In other terms, what kind of work has been actually lost, is a matter of theoretical speculations only. 

At thermodynamic equilibrium, the affinity A and the velocity dl /dt of each reaction is zero. We will consider explicit examples of entropy production due to chemical reactions in Chapter 9. 

Also, the entropy production due to chemical reactions in each phase should be separately positive.) Thus, the symmetry principle provides constraints for the coupling of, and the entropy production due to, irreversible processes. 

The first term is the entropy production due to chemical reactions, which can be dropped when considering only electrical circuit elements. For a resistor and a capacitor, (()> j — ( )2) in the second term may be identified as the voltage Vacross the element and dQ/dt as the electric current 7. If R is the resistance, according to Ohm s law, the voltage across the resistor = (cj) 1 — 4>2) = IF The entropy production is... 

Let us look at the time variation of the entropy production due to chemical reactions in an open system in the linear regime. As before, we assume homogeneity and unit volume. The entropy production is ... 

It is sometimes possible to neglect the entropy production term in the above equation because of its minor contribution. In such cases, the entropy production due to chemical change becomes simply proportional to the heat of reaction [1], so that... 

For the reaction S < P, the affinity is A = jits — Mp The local rate of entropy production due to chemical... 

Though Gibbs did not consider irreversible chemical reactions, equation (4.1.1) that he introduced included all that was needed for the consideration of irreversibility and entropy production in chemical processes. By making the important distinction between the entropy change S due to exchange of matter and energy with the exterior, and the irreversible increase of entropy djS due to chemical reactions [2, 3], De Bonder formulated the thermodynamics of irreversible chemical transformations. And we can now show he took the uncompensated heat of Clausius and gave it a clear expression for chemical reactions. 

CHEMICAL AFFINITY. The entropy production due to a chemical reaction has the form... 

Equation above shows the three contributions to the rate of entropy production due to heat flow, mass flow, and the chemical reaction, respectively, and excludes the viscous and electrical effects. As the membrane is assumed to be an isotropic medium, there will be no coupling between the vectorial heat and mass flows and scalar chemical reaction, according to the Curie-Prigogine principle. Under these conditions, entropy production equation identifies the conjugate forces and flows, and linear relations for coupled heat and mass flows become... 

The generalization to r distinct chemical reactions is simple every such reaction is specified by its own degree of advancement dXk, and the different chemical reactions are specified by VikAik = 0 ( = 1,2,..., r), for which the affinities are given by = — J ik ikdi- Note that the subscript k is missing from /l, (why ). The total rate of entropy production due to all the chemical reactions that run concurrently is then given by... 

The physical meaning of the terms (or group of terms) in the entropy equation is not always obvious. However, the term on the LHS denotes the rate of accumulation of entropy within the control volume per unit volume. On the RHS the entropy flow terms included in show that for open systems the entropy flow consists of two parts one is the reduced heat flow the other is connected with the diffusion flows of matter jc, Secondly, the entropy production terms included in totai demonstrates that the entropy production contains four different contributions. (The third term on the RHS vanishes by use of the continuity equation, but retained for the purpose of indicating possible contributions from the interfacial mass transfer in multiphase flows, discussed later). The first term in totai arises from heat fluxes as conduction and radiation, the third from diffusion, the fourth is connected to the gradients of the velocity field, giving rise to viscous flow, and the fifth is due to chemical reactions. 

Thus the total entropy change consists of the part supplied by the environment and of the entropy production with in the system due to chemical reaction. 

As in the case of entropy production due to heat conduction, the entropy production due to a chemical reaction is a product of a thermodynamic force A/T, and a thermodynamic f[ow d /dt. The flow in this case is the conversion of... 

As an example of entropy production due to an irreversible chemical reaction, consider the simple reaction... 

We emphasize that this is the entropy production due to the decay of a small fluctuation. The entropy production associated with the steady state reaction as such is discussed below in a separate example. The chemical potential fluctuations A/x,- (i = A, X, B) are... 

The tight and loose transition-state hypothesis is in contrast with the assumption that there is extensive cancellation of contributions due to chemical change in the entropic component of the EM (p. 81). Indeed, the uniform behaviours displayed by 0AS-data for reactions widely differing in nature (Figs 5, 23, and 24) clearly shows that no matter how loose a transition state or product is, the entropy contribution from such looseness will be cancelled out extensively by virtue of the operator 0. 

For nonequilibrium systems far from global equilibrium, the second law does not impose the sign of entropy variation due to the terms djS and d S, as illustrated in Figure 12.2. Therefore, there is no universal Lyapunov function. For a multicomponent fluid system with n components, entropy production in terms of conjugate forces Xu flows Jj, and / number of chemical reactions is... 

Then we can easily see that the left-hand side of Eq. 90 is the thermodynamical definition of the local entropy production as given by Eq. 17, although, in the present case, it includes tlie second summation corresponding to the production due to the viscous flow, but does not include the term for the chemical reactions. Thus the condition Eq. 79 implies, that the phenomenological entropy production does agree with the statistical expression of the entropy production, which is given as the brace symbol. This was first pointed out by the author. ... 

That is, the entropy production in the volume consists of three terms, each of which is due to an irreversible process. The first term is the heat conduction term, the second is the mass diffusion term, and the third is the chemical reaction term. The above equation is known as the entropy production equation. 

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