Reversibility, microscopic

This principle is concerned with an analysis of the individual pathways that chemical reactions take when the reactant is transformed to the product, and when the product is transformed back to the reactant. It states that the pathway for conversion of the product back to the reactant is the exact microscopic reverse of the forward pathway. The same intermediates and transition states are achieved in either direction. An example of how useful this notion becomes is given in the next Connections highlight. 

Reaction coordinate diagrams that show two paths interconverting reactant to product. 

Applying the Principle of Microscopic Reversibility to Phosphate Ester Chemistry 

Kluger, R., Covitz, F., Dennis, E., Williams, D., and Westheimer, F. H. pH-Product and pH-Rale Profile for the Hydrolysis of Methyl Ethylene-phosphate. Rate limiting Pseudorotation. /. Am. Chem. Soc., 91,6066-6072 (1%9). 

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. 

The trajectory of a classical particle may be found by integrating Newton s equation of motion 

As t varies from to to t, r is seen from Eq. (B.2) to vary from t to to, so the substitution is equivalent to a time reversal. Since the equations of motion are identical, the system will follow exactly the same trajectory the only difference will be that in one 

The Schrodinger equation for a quantum system also has time-reversal symmetry. The solution to the time-dependent Schrodinger equation 

Then it is remembered that the wave function itself does not have a physical meaning. So let us, for example, determine the transition probability from say state k(to)) at time to to state m(t )) at time t. Then we have 

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It might be thought that since chemisorption equilibrium was discussed in Section XVIII-3 and chemisorption rates in Section XVIII-4B, the matter of desorption rates is determined by the principle of microscopic reversibility (or, detailed balancing) and, indeed, this principle is used (see Ref. 127 for... 

The condition that the process a(t) is a stationary process is equivalent to the requirement tiiat all the distribution fimctions for a t) are invariant under time translations. This has as a consequence that W a, t) is independent of t and that 1 2(0, t 2, 2) depeirds on t = 2 -1. An even stationary process [4] has the additional requirement that its distribution fimctions are invariant under time reflection. For 1 2, this implies fV2(a 02> t) = 2 2 1 caWcd microscopic reversibility. It means that the quantities are even... 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

Light J C, Ross J and Shuler K E 1969 Rate coefficients, reaction cross sections and microscopic reversibility Kinetic Processes in Gases and Piasmas ed A R Hochstim (New York Academic) pp 281-320... 

A consideration of the transition probabilities allows us to prove that microscopic reversibility holds, and that canonical ensemble averages are generated. This approach has greatly extended the range of simulations that can be perfonned. An early example was the preferential sampling of molecules near solutes [77], but more recently, as we shall see, polymer simulations have been greatly accelerated by tiiis method. 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

The observation that in the activated complex the reaction centre has lost its hydrophobic character, can have important consequences. The retro Diels-Alder reaction, for instance, will also benefit from the breakdown of the hydrophobic hydration shell during the activation process. The initial state of this reaction has a nonpolar character. Due to the principle of microscopic reversibility, the activated complex of the retro Diels-Alder reaction is identical to that of the bimoleciilar Diels-Alder reaction which means this complex has a negligible nonpolar character near the reaction centre. O nsequently, also in the activation process of the retro Diels-Alder reaction a significant breakdown of hydrophobic hydration takes placed Note that for this process the volume of activation is small, which implies that the number of water molecules involved in hydration of the reacting system does not change significantly in the activation process. 

In any equilibrium process the sequence of intermediates and transition states encountered as reactants proceed to products m one direction must also be encountered and m precisely the reverse order m the opposite direction This is called the principle of microscopic reversibility Just as the reaction... 

Microscopic reversibility (Section 6 10) The pnnciple that the intermediates and transition states in the forward and back ward stages of a reversible reaction are identical but are en countered in the reverse order... 

When the addition and elimination reactions are mechanically reversible, they proceed by identical mechanistic paths but in opposite directions. In these circumstances, mechanistic conclusions about the addition reaction are applicable to the elimination reaction and vice versa. The principle of microscopic reversibility states that the mechanism (pathway) traversed in a reversible reaction is the same in the reverse as in the forward direction. Thus, if an addition-elimination system proceeds by a reversible mechanism, the intermediates and transition states involved in the addition process are the same as... 

The iodide-induced reduction is essentially the reverse of a halogenation. Application of the principle of microscopic reversibility would suggest that the reaction would proceed through a bridged intermediate as shown below. ... 

The principle of microscopic reversibility requires that the reverse process, ring closure of a butadiene to a cyclobutene, must also be a coiuotatory process. Usually, this is thermodynamically unfavorable, but a case in which the ring closure is energetically favorable is conversion of tra s,cis-2,4-cyclooctadiene (1) to bicyclo[4.2.0]oct-7-ene (2). The ring closure is favorable in this case because of the strain associated with the trans double bond. The ring closure occurs by a coiuotatory process. 

Detailed balance is a chemical application of the more general principle of microscopic reversibility, which has its basis in the mathematical conclusion that the equations of motion are symmetric under time reversal. Thus, any particle trajectory in the time period t = 0 to / = ti undergoes a reversal in the time period t = —ti to t = 0, and the particle retraces its trajectoiy. In the field of chemical kinetics, this principle is sometimes stated in these equivalent forms ... 

A catalyst is a substance that increases the rate of a reaction without affecting the position of equilibrium. It follows that the rate in the reverse direction must be increased by the same factor as that in the forward direction. This is a consequence of the principle of microscopic reversibility (Section 3.3), which applies at equilibrium, and rates are often studied far from equilibrium. 

Notice that the condition of detailed balance is eciuivaleiit to microscopic reversibility from equation 7.96, we see that if a given PGA in a stationary state satisfies detailed balanc.e, then a motion-picture of the sy.stem will appear the same whether the film is run forwards or backwards. 

Lack of termination in a polymerization process has another important consequence. Propagation is represented by the reaction Pn+M -> Pn+1 and the principle of microscopic reversibility demands that the reverse reaction should also proceed, i.e., Pn+1 -> Pn+M. Since there is no termination, the system must eventually attain an equilibrium state in which the equilibrium concentration of the monomer is given by the equation Pn- -M Pn+1 Hence the equilibrium constant, and all other thermodynamic functions characterizing the system monomer-polymer, are determined by simple measurements of the equilibrium concentration of monomer at various temperatures. 

The rate equations determine the rate of change of the probability of a particular configuration, a, within an ensemble of growing crystals. They must include the rate constants for adding or subtracting units, which are assumed to obey microscopic reversibility. The net flux between configurations a and a which occur with probability P(a) and P(a ) respectively, and differ by one unit is ... 

By the principle of microscopic reversibility, it follows that protodeiodination must in all steps be the reverse of iodination, and since this latter reaction is partly rate-determining in loss of a proton (see pp. 94-97, 136) it follows that attachment of a proton should be rate-determining in the reverse reaction this was found to be the case, the first-order rate coefficients for reaction in H20 and 97.5 % D20 being 76.6 and 13.1 x 10"6 respectively, so that kH20jkD20 = 5.8. 

N-methyl-N,2,4-trinitroaniline, nitration of, 12 microscopic reversibility, and protodeiodina-tion, 356... 

For every positive term there is a corresponding negative one, a manifestation of the principle of microscopic reversibility, discussed further in Section 7.8. 

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