Reversible relation between equilibrium

These equations show that whereas the kinetic coefficients of an individual reaction can assume any value, the coefficients of its forward and reverse process are always interrelated. The relation between the standard equilibrium potential EP and the rate constants and is analogous to the well-known physicochemical relation between equilibrium constant K and the rate constants of the forward and reverse process. 

The reverse reaction is determined by the nature of R. The generating [Pt(PR3)2] moiety is a rather stable intermediate. A comparison of the results obtained with the NMR data (22) of substituted phenylacetylenes led to the conclusion that there is a relation between equilibrium constant, first-order rate constants, and chemical shifts of the acetylenic proton they all depend on substituent effects on the electron density in the triple bond. 

This chapter will be concerned mainly with the relation between the equilibrium constants of acid-base reactions and their forward and reverse rates. Relations between equilibrium constants and structure have already been considered in Chapter 6, so that the present discussion also implies relations between rates and structure. Moreover, there are many cases in which rates are easier to measure (though more difficult to interpret) than equilibria and can be compared directly with structures. We shall first consider the general basis and experimental evidence for this type of relation, followed by its molecular interpretation, with special reference to exceptional cases. We have seen in the two preceding chapters that the rates of proton-transfer reactions can be measured either directly, or indirectly through the study of acid-base catalysis, and in the following discussion information from both sources will be used indifferently. 

As stated above, under certain conditions, the kinetics of chemical reactions is defined by equations involving only total concentrations and macroscopic rate constants. In the state of thermodynamic equilibrium, the overall reaction rates are zero, and this permits to establish a relation between equilibrium concentrations and macroscopic rate constants. On the other hand the equilibrium concentrations are related via the thermodynamic equilibrium constants K which themselves do not depend on the reaction mechanism and are expressed through partition functions of reacting molecules (see Chapter I). This establishes the relation between equilibrium constants and the rate constants for forward and reverse reactions. Though this relation indisputable for equilibrium reactions, i.e. for reactions for which the perturbation of the Boltzmann distribution is small at any stage of the reaction, for non-equilibrium reactions the above relation is not strictly justifiable. 

Relation between Equilibrium and Rate of Forward and Reverse Reaction... 

Although derived for a reversible process, this equation relates properties only and is valid for any change between equilibrium states in a closed system. It may equally well be written... 

That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

At equilibrium, the rates of the forward and reverse reactions are equal. Because the rates depend on rate constants and concentrations, we ought to be able to find a relation between rate constants for elementary reactions and the equilibrium constant for the overall reaction. 

We derived the relation between the equilibrium constant and the rate constant for a single-step reaction. However, suppose that a reaction has a complex mechanism in which the elementary reactions have rate constants ku k2, and the reverse elementary reactions have rate constants kf, k2, . .Then, by an argument similar to that for the single-step reaction, the overall equilibrium constant is related to the rate constants as follows ... 

Since the reverse of the reaction Nl is the ionisation of the ester, the equilibrium position for any one system depends critically on the nature, especially the polarity, of the solvent, which determines the AHS terms. The accumulation of the necessary thermochemical data is essential to a rationalisation of the relation between cationic and pseudocationic polymerisations but the prevalence of the former at low temperatures and of the latter at high temperatures is surely related to the fact that the dielectric constant, and with it solvation energies, increases as the temperature of a polar solvent is reduced, so that decreasing temperature favours ionisation. 

Most discussions, such as those cited above, of monolayer films are presented within the context of equilibrium thermodynamics. The applications of the two-dimensional gas law, ttA = kT, the phase rule, and relations between surface tension and surface pressure to free energy all assume reversibility. Perhaps it seems odd to... 

The principle of detailed balance is a result of the microscopic reversibility of electron kinetics. A prerequisite for the establishment of thermal equihbrium requires that the forward and reverse rates are identical. For isothermal reactions, the equilibrium constant remains unchanged. The principle of detailed balance is of fundamental importance to estabhsh helpful relations between reaction and equilibrium constants because both are at the initial thermal equilibrium in addition, at the new equihbrium after the relaxation of the perturbation, the net forward and reverse reaction rates are zero. 

Assume that typical time scales of both direct and reverse reactions in (3.3.2) are much shorter than any other time scale in the system. Then the reaction (3.3.2) yields the Langmuir s local equilibrium relation between Ci and 0 of the form... 

Once more the known equilibrium distribution leads to a relation between the rates of a transition and its reverse. Relation (3.3) is called the law of mass action . 

In heterogeneous reactions we frequently find relations between rate of reaction and concentration quite different from those which the law of mass action would indicate to be valid for a homogeneous system. It is a little difficult, at first sight, to see how, by equating the rates of the forward and reverse reactions, we are still to arrive at the correct equilibrium relations. The general problem is very complex, but one simple example may be given to illustrate the manner in which conflict with the second law of thermodynamics is avoided. 

Given the Arrhenius equation, k = Ae E RT, and the relation between the equilibrium constant and the forward and reverse rate constants, Kc = kf/kr, explain why Kc for an exothermic reaction decreases with increasing temperature. 

If two steps in the sequence constitute a reversible reaction and equilibrium is set up and maintained throughout reaction, then formulation of the equilibrium constant, in terms of the ks for the forward and back reactions and the concentrations of the species involved, gives another relation between the rate constants. 

Chemical reactivity depends on the differences in free energy between pairs of closely related systems. In the case of a reversible reaction, the equilibrium constant (K) is determined by the difference ( AF) in free energy between the reactants and the products ... 

Reversible processes are those processes that take place under conditions of equilibrium that is, the forces operating within the system are balanced. Therefore, the thermodynamics associated with reversible processes are closely related to equilibrium conditions. In this chapter we investigate those conditions that must be satisfied when a system is in equilibrium. In particular, we are interested in the relations that must exist between the various thermodynamic functions for both phase and chemical equilibrium. We are also interested in the conditions that must be satisfied when a system is stable. 

Bejan and Tondeur [9] make a number of other observations in their paper. One is that the relation between j and x is not necessarily linear. Another observation is that a similar analysis can show that the force x should be equipartitioned in time, which is another way of saying that the steady state is optimal. Prigogine gave an earlier proof of this principle [11]. The steady state is common in nature and often the favored state in industrial operation. It can be considered to be the "stable state" of nonequilibrium thermodynamics, comparable to the equilibrium state of reversible thermodynamics (see Figure 4.2). Of course, the latter is characterized by Sgen = 0, whereas the former is characterized by a minimum value , larger than zero. 

Both Newton s equation of motion for a classical system and Schrodinger s equation for a quantum system are unchanged by time reversal, i.e., when the sign of the time is changed. Due to this symmetry under time reversal, the transition probability for a forward and the reverse reaction is the same, and consequently a definite relationship exists between the cross-sections for forward and reverse reactions. This relationship, based on the reversibility of the equations of motion, is known as the principle of microscopic reversibility, sometimes also referred to as the reciprocity theorem. The statistical relationship between rate constants for forward and reverse reactions at equilibrium is known as the principle of detailed balance, and we will show that this principle is a consequence of microscopic reversibility. These relations are very useful for obtaining information about reverse reactions once the forward rate constants or cross-sections are known. Let us begin with a discussion of microscopic reversibility. 

Detailed balance provides a relation between the macroscopic rate constants kf and kr for the forward and reverse reactions, respectively. On a macroscopic level, the relation is derived by equating the rates of the forward and reverse reactions at equilibrium. Here it will be shown that the principle of detailed balance can be readily obtained as a direct consequence of the microscopic reversibility of the fundamental equations of motion. 

The decrease of the decomposition voltage is at first sight difficult to comprehend and it cannot be explained by Nemst s equation, according to which the equilibrium deposition potentials of hydrogen and oxygen increase with increased pressure. The equation (X—8) expresses the relation between the reversible water decomposition potential and the gas pressure at 25 °C ... 

The Equilibrium of Atoms and Electrons.—From the cases we have taken up, wTe see that the kinetics of collisions forms a complicated and involved subject, just as the kinetics of chemical reactions does. Since this is so, it is fortunate that in cases of thermal equilibrium, we can get results by thermodynamics which are independent of the precise mechanism, and depend only on ionization potentials and similarly easily measured quantities. And as we have stated, thermodynamics, in the form of the principle of microscopic reversibility, allows us to get some information about the relation between the probability of a direct process... 

Since this relation involves properties only, it must be satisfied for ch in state of any closed system of uniform T and P, without restriction conditions of mechanical and thermal reversibility assumed in its derivation/ inequality applies to every incremental change of the system between equilibrium states, and it dictates the direction of change that leads to equilibrium. The equality holds for changes between equilibrium states (reve processes). Thus Eq. (6.1) is just a special case of Eq. (13.51). 

This equation describes the relation between two parameters characterizing the reversible polymerization ceiling temperature and equilibrium monomer concentration. For any temperature there is a certain value of [ML [Eq. (48)]. If the starting concentration of monomer is lower than this value, i.e., [M0] < [ML, the polymerization will not proceed. If [M]0 is higher than [ML, polymerization will proceed until concentration of monomer reaches [ML (i-e., consumption of monomer will be equal to [M]0 - [ML). 

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