Thermodynamically consistent rate laws

The Monod equation differs from the Michaelis-Menten equation in that it includes as a factor biomass concentration [X], which can change with time. A microbe as it catalyzes a redox reaction harvests some of the energy liberated, which it uses to grow and reproduce, increasing [X], At the same time, some microbes in the population decay or are lost to predation. The time rate of change in biomass 

In considering the energetics of a microbially catalyzed reaction, it is important to recall that progress of the redox reaction (e.g., Reaction 18.7) is coupled to synthesis of ATP within the cell, so the overall reaction is the redox reaction combined with ATP synthesis. The free energy liberated by the overall reaction is the energy liberated by the redox reaction, less that consumed to make ATP. The overall reaction s equilibrium point is where this difference vanishes at this point, 

To account for reverse as well as forward reaction, the Monod (and dual Monod) equation can be modified by appending to it a thermodynamic potential factor, as shown by Jin and Bethke (2005), in which case the equation predicts the net rate of reaction. The thermodynamic factor Ft, which can vary from zero to one, is given 

The free energy change of the redox reaction is given by, 

Appending Ft to the Monod equation, we write a thermodynamically consistent rate law, 

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We see that at equilibrium (-r = 0) the rate law for the reversible reaction is indeed thermodynamically consistent ... 

In each case, make sure that the rate laws at high temperatures are thermodynamically consistent at equilibrium (cf. Appendix C). 

We need to check to see if the rate law given by Equation (3-14) is thermodynamically consistent at equilibrium. Using Equation (3-10) and substituting the appropriate species concentration and exponents, thermodynamics tells us that... 

More importantly, as written, the partial derivative implies that a single rate constant is to be varied, holding all the others constant, and indeed this is the way it is implemented in many sensitivity analysis routines. The index y runs out to 2Areactions because each reaction has two rate constants, one for the forward direction, and one for the reverse. However, in order for the model to remain consistent with the laws of thermodynamics, the rate constant for the reverse of reaction y must vary simultaneously with the forward reaction, since the two rate constants must maintain a detailed-balance ratio related to the AGreaction, Eq. (2). This can be assured by specifying that the partial derivative is taken only for the forward reaction, while holding the thermochemistry fixed. Note that this also cuts the number of partial derivatives to be computed in half. These sensitivities should then be supplemented with sensitivities to the individual species thermochemistry as in Eq. (25) overall the number of partial derivatives to be computed per model prediction M, is... 

Postulate a mechanism and show that it is consistent with the behavior described above. Derive the rate law and determine the activation energy of as many of the individual steps in the mechanism as possible. The following thermodynamic information may prove useful ... 

At equilibrium, the rate law must reduce to an equation consistent wth thermodynamic equilibrium. 

There are several ways to check for thermodynamic consistency. If both the forward and reverse rate equations are power-law expressions, a value of N can be calculated from Eqn. (2-18) for each of the reactants and products. The values of v,- for this calculation can come from any balanced stoichiometric equation for the reaction in question, e.g., Eqn. (2-A) for this example. If all of the calculated values of are the same, the rate equations are thermodynamically consistent. 

The advantage of the procedure outlined in the preceding paragraph is that it can be used for rate equations that are not power-law expressions. We will use this approach to analyze the thermodynamic consistency of the proposed rate equations for phosgene synthesis. 

Chapter 2 is an overview of rate equations. At this point in the text, the subject of reaction kinetics is approached primarily from an empirical standpoint, with emphasis on power-law rate equations, the Arrhenius relationship, and reversible reactions (thermodynamic consistency). However, there is some discussion of collision theory and transition-state theory, to put the empiricism into a more fundamental context. The intent of this chapter is to provide enough information about rate equations to allow the student to understand the derivations of the design equations for ideal reactors, and to solve some problems in reactor design and analysis. A more fundamental treatment of reaction kinetics is deferred until Chapter 5. The discussion of thermodynamic consistency... 

It should be clear that the most likely or physical rate of first entropy production is neither minimal nor maximal these would correspond to values of the heat flux of oc. The conventional first entropy does not provide any variational principle for heat flow, or for nonequilibrium dynamics more generally. This is consistent with the introductory remarks about the second law of equilibrium thermodynamics, Eq. (1), namely, that this law and the first entropy that in invokes are independent of time. In the literature one finds claims for both extreme theorems some claim that the rate of entropy production is... 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

The dependence of nucleation rate on solubility is also consistent with Ostwald s (1897) law of stages regarding the preferential crystallization of metastable crystalline phases. It states that When leaving an unstable state, a system does not seek out the most stable state, rather the nearest metastable state which can be reached with minimal loss of free energy. This indicates that a metastable (more soluble) crystalline phase will generally crystallize before a more thermodynamically stable (less soluble) crystalline phase, because it will have a higher nucleation rate. 

Consider a thermal machine consisting of an insulated piston-cylinder assembly attached to a container as shown in Kg. 1.2. Initially, the matter in the cylinder is separated from that of the container by a partition. The partition is ruptured and, following an infinitesimal process, the mass A mf within the cylinder is slowly pushed by the piston into the container. Assume the container to be a control volume. During this process, the heat received and the shaft work done by the control volume, respectively, are Agcv and AWcv> subscript cv denoting the control volume. We wish to find the rate of the first law of thermodynamics for this control volume. 

Transition strengths can be given in terms of Einstein rate coefficients. For a pair of states j > and k > it is shown in elementary texts that these are related in a simple way. If one assumes that, for any pair of microstates i and j, the rate from i to j is equal to the rate from j to i one has the principle of detailed balance). Then, the relation between the coefficients is consistent with thermodynamics (Planck s black-body radiation law). 

Careful observation teaches us that, left undisturbed, every material system tends to evolve to a unique equilibrium state that is consistent with any imposed constraints. The rates of such evolutions cannot be determined from thermodynamics, but thermodynamics does provide quantitative criteria both for identifying the directions of such evolutions and for identifying equilibrium once it is reached. Those criteria are obtained by combining the first and second laws. 

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