(p.304) B Microscopic reversibility and detailed balance
(p.304) B Microscopic reversibility and detailed balance
Both Newton’s equation of motion for a classical system and Schrödinger’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.
B.1 Microscopic reversibility
B.1.1 Transition probability
The trajectory of a classical particle may be found by integrating Newton’s equation of motion
As t varies from t 0 to t 1, τ is seen from Eq. (B.2) to vary from t 1 to t 0, 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 (p.305) case it is followed forward in time from t 0 to t 1, and in the other case backward in time from t 1 to t 0. In other words, it is not possible from the equation of motion to decide whether it describes a forward or a reverse propagation in time. This is summarized by saying that the classical equation of motion has time-reversal symmetry. As a consequence, the probability for a forward scattering process must equal the probability for the reverse scattering process, where all velocities dr/dt and time have changed sign (see Eq. (B.3)).
The Schrödinger equation for a quantum system also has time-reversal symmetry.
The solution to the time-dependent Schrödinger equation
(p.306) 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(t 0)> at time t 0 to state |m(t 1)> at time t 1. Then we have
This transition probability is now compared to the transition probability for the reverse process, where we consider the transition from |m(t1)> to |k(t 0)>. We find
The transition probabilities per unit time and per scattering center in Eqs (B.11) and (B.12) are for a collision process, often written in a more explicit form to emphasize the particular transition considered:
From the fundamental relation in Eqs (B.12) or (B.13), we are now going to derive a relation between the differential cross-section for the forward and the reverse scattering process.
As defined previously in Eq. (2.7), differential cross-sections are defined as the flux of particles scattered into a range of solid angle dΩ around the solid angle Ω per unit initial flux and per scattering center. The state-to-state transition probabilities are already per unit scattering center and, since we do not scatter into one single final state but a range of final states as given by dΩ, we need to determine how many final states are consistent with that uncertainty in the final solid angle. It is then assumed that the transition probability to that small range of final states is the (p.307) same, irrespective of the state; then the product of the transition probability and the number of final states will be the total transition probability. The scattering into the solid angle dΩ around Ω with a relative final velocity vf is equivalent to scattering into a final state with momentum pf, pf + dpf. The number of final states in this range may be determined in the following way. We consider a free particle in one dimension where the momentum eigenfunction is given by ψp(x) = c(p) exp (ipx/ħ). In a (macroscopic) box of length L, periodic boundary conditions on the wave function ψp(x + L) = ψ p(x) imply that exp(ipL/ħ) = 1 or p/ħ = 2nπ/L, where n is an integer. That is, the spacing between momentum values is h/L, which implies that the density is L/h. This argument can be extended to three dimensions, and it can be shown that the density, ρ(p), is (L/h)3 and
If the density of particles in the incident beam of A(i) molecules with relative velocity vi with respect to the B molecules is na(i), then the magnitude of the incoming flux is
From the definition of the differential cross-section, we obtain1
The factor δ(Ef – Ei)dE f on the left-hand side of the equation has been introduced to emphasize energy conservation for the process indicated in the argument to σ R . The transition probability P on the right-hand side is determined directly from the equation of motion, so it is therefore not necessary to include the delta function on the right-hand side.
(p.308) The cross-section for the reverse reaction may be written analogously:
When we use Eq. (B.13) and cancel common factors, we find from Eqs (B.17) and (B.18) that
From the energy balance
The desired relation between cross-sections for the forward and the reverse reactions is thus found to be
This relation expresses the principle of microscopic reversibility for the differential cross-sections.
Alternatively, using Eq. (4.173) and the symmetry of the S-matrix element, using Eq. (B.12), we immediately obtain
Equation (B.23) is the desired relation (at the level of integrated cross-sections), and it is equivalent to Eq. (B.22), since p = ħk.
If the system has an internal angular momentum (associated with rotational states of molecules) there will, in the absence of an external field, be degeneracies in the system that will be practical to display explicitly in the expression for microscopic reversibility in Eq. (B.22). For systems with angular momenta, time reversal of the quantum equations of motion reverses the signs of both the momenta and their projections on a given direction, just like in a classical system. To express this explicitly Eq. (B.13) is written as
The ‘starred’ quantum states differ from the ‘unstarred’ ones by the sign of the projection of their angular momenta, so if the projection of the angular momentum of (p.309) state i on the z-axis is m i then the projection of i* on the z-axis is -m i, etc. By an analysis similar to that used to derive Eq. (B.22), we obtain
Experimental and calculated cross-sections are usually averaged over the degenerate quantum states associated with the angular momenta when there is no external field. A more useful statement of microscopic reversibility can therefore be obtained by working with average cross-sections , which refer to transitions between sets of degenerate levels in the initial and final states. Let Ji denote the total angular momentum of state i and mi its projection on the z-axis. Then by summing Eq. (B.24) over all degenerate states of the reactants and products one obtains
Since the pi and pf are constant when summed over degenerate states, we may now use Eqs (B.17) and (B.26) to introduce a cross-section, σ R (s), where we have summed over all degenerate states:
So the principle of microscopic reversibility may be rewritten as
The σR(s) refers to the sum of cross-sections for the transition from the degenerate set of states to defined for unit flux from each initial state (see Eqs (B.15)–(B.17)). We may instead operate with an average cross-section based on an initial flux for the total set of degenerate states:
It is noted that this relation only holds in the absence of an external field, when the sets of states are degenerate. Other degeneracies may also occur in the internal molecular states. In this case, the g factors should include those degeneracies.
(p.310) B.2 Detailed balance
Detailed balance provides a relation between the macroscopic rate constants k f and k r 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.
On a macroscopic level in the absence of an external field, we cannot distinguish between sets of degenerate states in the reactants and products; so the most detailed relation between macroscopic rate constants and microscopic cross-sections will be one where we have summed over all degenerate states as in Eq. (B.27). The macroscopic rate constant for a particular transition between degenerate states is then given by
To relate the rate constants in Eqs (B.32) and (B.33), we substitute Eq. (B.28) and use the conservation of energy to relate the differentials and limits of integration. Conservation of energy requires
From Eq. (B.32), we then obtain the following relation between the forward and reverse rate constants:
It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell–Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium.
To obtain the statement of detailed balance for complete equilibrium, with both translational and internal degrees of freedom in thermal equilibrium, we must sum over the rate constants in Eqs (B.32) and (B.33), weighting each by its equilibrium Boltzmann distribution; that is (as in Eq. (2.18)),
Here kf is the rate constant for the forward reaction and k r for the reverse reaction. is the probability (mole fraction) of A in any of the set of states , which is given by statistical mechanics according to Eq. (A.4), and similarly for the other constituents. If we substitute these relations into Eqs (B.38) and (B.39), we obtain
Note that the degeneracies gi do not appear explicitly in the equations, because we use ‘barred’ quantities as indices in the sums. They imply a sum over degenerate states; had we used ‘unbarred’ indices, then the degeneracy factor gi should be included (p.312) explicitly. From Eqs (B.40) and (B.41) and using Eq. (B.37), we find the following relation between the rate constants:
In summary, we have seen that the application of microscopic reversibility for the forward and reverse cross-sections and the use of complete equilibrium distributions for the evaluation of the statistical rate constant lead to the usual results known from equilibrium statistical mechanics. If one knows the cross-section for a forward reaction, one can always determine the inverse cross-section through the principle of microscopic reversibility. Also, if one knows the cross-section for the forward reaction, and in addition one knows that the translational and internal distribution functions of reactants and products have reached equilibrium, one can calculate the rate constant. Detailed balance then permits the calculation of the reverse rate constant.
J.C. Light, J. Ross and K.E. Shuler, in Kinetic processes in gases and plasmas (Academic Press, 1969) Chapter 8