The principle of detailed balance

Since the specific heats of the reactant and product are the same, AC° = 0, and 

On the other hand, the equilibrium constant in terms of conversion becomes 

In chemical kinetics, the reaction rates are proportional to concentrations or to some power of the concentrations. Phenomenological equations, however, require that the reaction velocities are proportional to the thermodynamic force or affinity. Affinity, in turn, is proportional to the logarithms of concentrations. Consider a monomolecular 

If the reaction proceeds in a closed system where there is no exchange of matter with the outside, we have 

The same chemical reaction system may be considered by the thermodynamic approach, in which the affinity 

We have seen that dynamic equilibrium requires that rate constants be related to the equilibrium constant in a single-step reaction. For sequential reactions the overall equilibrium constant is the product of the equilibrium constants for the individual reactions, and, in an -step process, the extension of (4.11) is 

A more interesting situation arises in systems of which the following is a well-known example  

It would appear possible that equilibrium could be established by means of the cycle A B C — A even if the forward and reverse steps in 

Newtonian mechanics and quantum mechanics have in common the property of time-reversal invariance. The Newtonian equations of motion for a system of N interacting particles are 

Changing t —t does not affect the accelerations on the other hand the velocities Xi do change sign. As a consequence, for every solution of (4.18) there is an identical one in which all particles have their velocities reversed.For classical point particles the dynamics of a colliding pair 

Suppose a small time interval Tc exists such that, during Tc, t/n-i changes without strongly affecting (j/n-al Vn J + Tc) in Eq. (E.IO). In the limit Te —+ 0 we may then expand the left. side of Eq. (E.IO) in a Taylor series according to 

Inserting this expression into Eq. (E.IO), we find after rearranging terms 

At this point it is convenient to introduce the transition probability per time interval Tc via 

There are a few important aspects of the state of chemical equilibrium, and the state of thermodynamic equilibrium in general, that must be noted. The principle of detailed balance is one of them. 

We have noted earlier that, for a given reaction, the state of equilibrium depends only on the stoichiometry of the reaction, not its actual mechanism. For example, in the reaction X + Y 2Z considered above, if the forward and reverse reaction rates were given by 

However, the equilibrium relation a /ax Y = K T) was not obtained using any assumption regarding the kinetic mechanism of the reaction. It remains valid even if there is a complex set of intermediate reactions that result in the overall 

In the state of equilibrium, every elementary transformation is balanced by its exact opposite or reverse. 

The principle of detailed balance implies a /axa = K(T), regardless of the mechanism this can be seen through the following example. Assume the reaction really consists of two steps  

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In accordance with the principle of detailed balance the set (3) with regard to (2) after some mathematics can be rewritten as ... 

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

These are applications of the principle of detailed balancing, which can be stated ... 

To make further progress specific forms for the rate constants are required. In the steady state, the principle of detailed balance gives ... 

One of the most significant recent insights in surface chemical dynamics is the idea that the principle of detailed balance may be used to infer the properties of a dissociative adsorption reaction from measurements on an associative desorption reaction.51,52 This means, for example, that the observation of vibrationally-excited desorption products is an indicator that the dissociative adsorption reaction must be vibrationally activated, or vice versa the observation of vibrationally-cold desorption products indicates little vibrational promotion of dissociative adsorption. In this spirit, it is... 

The principle of detailed balancing provides an automatic check on the self-consistency of postulated reaction mechanisms when equilibrium can be approached from both sides. 

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. 

Rips and Silbey (1991) have reexamined the thermalization of photoelectrons (of a few eV in energy) with a master equation approach for the time rate of energy loss. Their method is quite general, and it includes both direct (energy loss) and inverse (energy gain) collisions according to the principle of detailed balance. As in the Frohlich-Platzman method, they first calculate the time rate... 

Capture and dissociation rates are related by the principle of detailed balance. Returning to our example of H+ + A- 5 AH, let us define rAH as the lifetime of an AH complex with respect to breakup into H+ and A-. 

Thanks to the principle of detailed balance, an equivalent descriptor is the lifetime r0+ for carrier emission via the inverse reaction, i.e., for the... 

The principle of detailed balance which is also valid for the quantities fVqq- enables the diagonalization of the nonsymmetric matrix Wqq< with nonnegative elements ... 

The requisite value for k x was only approximately defined by the fitting procedures, and because of uncertainty in the standard potential for the Br2/Br redox couple it was likewise deemed unsuitable to use the value of kx and the principle of detailed balancing to derive the value of k x. Further reason to be doubtful of the derived value of k x was a major disagreement between it and the value predicted by the cross relationship of Marcus theory. 

By use of well-established standard potentials, the reported values for K and kg, and the principle of detailed balancing, one can calculate that the reverse of reaction (10) has a rate constant (k g) of 2x103M-1s-1. Normal ligand substitution reactions at Fe2+ are much faster than this, which raises questions regarding the nature of the transition state for this reaction. 

In the example discussed above, the transition X- X sta s simply for a single spin flip at a randomly chosen lattice site, and W(X - ) = 1 if 5< 0 while W(X- X ) = Q — 3 lkgT) for >0 should be interpreted as transition probability per unit time. Note that other choices for W would also be possible provided they satisfy the principle of detailed balance ... 

Rate parameters for some decycUzation processes are also presented in Table XI. From the principles of detailed balancing, rate parameters for the reverse reactions, i.e., cyclizations, can be calculated. 

The backward velocity is obtained by Plummer et al. (1978) by application of the principle of detailed balancing. Denoting as Ki, K2, Kc, and respectively, the equilibrium constants of the processes... 

The Principle of Microscopic Reversibility and its large-scale consequence, known as the Principle of Detailed Balancing enable investigators to understand the mechanism of the reverse reaction to the same level of accuracy as that achieved for the forward reaction. 

The pairwise rates are presumed to be the same in both the forward and backward directions. The fact that the lines in the spectrum are all equally intense places little restriction on the three pairwise rates. The principle of detailed balance shows that the symmetry of the individual processes (i.e., equal forward and reverse rates) is sufficient to ensure that all the lines have equal intensity at equilibrium. 

The principle of detailed balance (and the micro reversibility) [48] requires that 2D exchange spectrum A Tm) is always symmetric. The matrix A(0) represents a 2D exchange spectrum recorded at = 0. It is a diagonal... 

Aq is the spectral peak volume of a single proton and n is the number of protons at the spin site i. Obviously, when different spin sites have different populations, i.e., rii rij, neither the product matrix A(rm) A(0) nor the exchange matrix L is symmetric, eq. (11). This also follows from the principle of detailed balance [28, 48],... 

The value of k2 is determined according to the principle of detailed balancing. For thermal energies less than or equal to 300°K, the last term in Eq. (48) may be neglected for the initial decay, provided a large excess of ground state atoms is not present (Table I). Thus the measured decay coefficients and data derived from these relate to k ... 

The Principle of Detailed Balance and Classification of Trapping States 2... 

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. 

The thermally activated emission rates are proportional to a Boltzmann factor, and by use of the principle of detailed balance can be related to the capture cross section (a ) ... 

Under equilibrium conditions the currents and i , and also icp and i, are equal to each other by the absolute value, in accordance with the principle of detailed balancing (see, for example, Landau and Lifshitz, 1977). These equilibrium values (i ) = (i )° = i° and (i )° = (i )° = i represent, by definition, exchange currents of an electrode reaction passing through the valence band (i°) and through the conduction band (i ). 

Theoretical models of correlation functions and line shapes have been proposed which satisfy the principle of detailed balance [35, 36, 41, 232]. These profiles, along with a number of extensions that were later added [69, 295, 47, 48], describe the known profiles well. Especially the BC and K0 functions, Eqs. 5.105 and 5.108, model multipole- and overlap-induced lines of the rototranslational bands closely. These three-parameter functions are simple analytical expressions that are readily computable, even on computers of small capacity (pocket calculators) the parameters can be computed from the lowest three spectral moments, see Chapter 5. 

Detailed balance. Classical line shapes are symmetric so that all classical, odd spectral moments M of the spectral function vanish. The odd moments of actual measurements are, however, non-vanishing because measured spectral density profiles satisfy the principle of detailed balance, Eq. 5.73. This problem of classical relationships may be largely alleviated by symmetrizing the measured profile prior to determining the moments, using the inverse Egelstaff procedure (P-4) discussed on p. 254 this generates a close approximation to the classical profile from the measurement and use of classical formulae is then justified. 

Here one arrives at expressions which are equivalent to those in the initial macrocanonical approach but with the expression for the total energy replaced by the potential energy. One can now apply the principle of detailed balancing to these expressions. Thus, if the probability of the potential energy changing from (J, to U, is given by P j and the converse probability by /, then these expressions will be related by the equation... 

In a system of connected reversible reactions at equilibrium, each reversible reaction is individually at equilibrium. This is the principle of microscopic reversibility or its corollary, the principle of detailed balance. 

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