Principle of detailed balancing

The principle of microscopic reversibility at equilibrium, which comes from classical mechanics, states that the reaction paths in the forward direction and in the reverse direction are identical, apart from the direction, which corresponds to an inversion of time in mechanics equations. This principle therefore forbids the following reaction sequence  

According to this mechanism, an equilibrium state would be observed if the following conditions were obeyed (the superscripts and refer to the forward and reverse reactions, respectively)  

The principle of detailed balancing states that this rate r is equal to zero when the overall equilibrium is reached, that is to say that the forward and reverse elementary reactions are themselves at equilibrium the global equilibrium is therefore a consequence of the detailed equilibria. 

The kinetic principle of detailed balancing dictates the relationships between the kinetic parameters and the thermodynamic properties of an elementary reaction. 

The identification of the k /k ratio given by equation (73) on the one hand, and equations (74) and (75) on the other, lead to the following relationships  

The principle of detailed balancing states that when a system is at equilibrium, the rate in the forward direction equals the rate in the reverse direction for each individual step in the process as well as for the overall reaction. 

This can be of use in simple systems because it makes it possible to express one of the rate constants in terms of the others and the overall 

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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 ... 

The plot of CE = Pout/Ps (from Eqs (5.10.33) and (5.10.37)) versus Ag for AM 1.2 is shown in Fig. 5.65 (curve 1). It has a maximum of 47 per cent at 1100 nm. Thermodynamic considerations, however, show that there are additional energy losses following from the fact that the system is in a thermal equilibrium with the surroundings and also with the radiation of a black body at the same temperature. This causes partial re-emission of the absorbed radiation (principle of detailed balance). If we take into account the equilibrium conditions and also the unavoidable entropy production, the maximum CE drops to 33 per cent at 840 nm (curve 2, Fig. 5.65). 

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. 

The principle we have applied here is called microscopic reversibility or principle of detailed balancing. It shows that there is a link between kinetic rate constants and thermodynamic equilibrium constants. Obviously, equilibrium is not characterized by the cessation of processes at equilibrium the rates of forward and reverse microscopic processes are equal for every elementary reaction step. The microscopic reversibility (which is routinely used in homogeneous solution kinetics) applies also to heterogeneous reactions (adsorption, desorption dissolution, precipitation). 

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... 

To determine n, we utilize the condition of electron equilibrium on the surface. In the absence of illumination, this is of the form (principle of detailed balance)... 

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 ). 

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