The diffusion of an isolated pair

On the right-hand side are some delta functions. These simply show that the species A and B where both formed at time t — 0, A at rA° and B at rB°. It is a source term and unless t — 0, rA = rA°, and rB = rB°, it is zero. Before either pair is formed (t 0), n = 0, but just after t — 0, the probability that A and B exist, /drA/drBn, is 1. Note also that /drA/drB5(rA — rA°)5(rB — rB°) = 1. Now eqn. (197) is of little use as it is, because both A and B could both be in London, on the moon or anywhere else The exact location is of little interest. The relative positions of A and B is of considerable interest, because when they are separated by a distance rB — rA — R, they can react. The relative position of B from A is r = rB — 

Now it is worth recalling that in the theory of gas phase collisions between two molecules, the motion of both species A and B can be separated into their mutual approach along the intermolecular axis and the motion of the centre of mass of the pair of molecules [475]. After collision, though the relative velocity of A and B has changed, that of the centre of mass has not. The centre of mass X is determined by weighting the positions of A and B by the fractional mass of A and B, and X = (mArA + mBrB)/(mA + mB). The relative position of B about A is r = 

A similar procedure may be applied when considering the relative 

The Laplacians V2 refer to the relative co-ordinate r, and Vx2 to the coordinate X. In eqn. (198), the species A and B are formed with relative separation r° and centre of diffusion coefficient coordinate X°. 

The two terms on the left-hand side are, respectively, functions of (X, t) and (r, t) only. The term on the right-hand side is zero unless r = r°, X = X° and t = 0. If these conditions are met, it has a value 5(f). Consequently, the equations can be factored into 

Let US consider an isolated pair of reactants A and B whose positions in space are r and re at time t. For simplicity, these reactants will be taken as uncharged and there is no potential of mean force consideration to be included either. Such factors may always be included later. The probability that species A is in the small volume cIta at r and species B is in the small volume drg at rg at time t is n rp, r, t) dr drs- Both species can diffuse independently of one another. They can even diffuse through each other, in principle, though in practice this is improbable because A and B will react before they b in to pass through each other. This can always be prevented by including a hard core repulsive term in the interaction evergy, U, which tums-on just inside the reaction distance, R. The probability density rs, t) can change with time due to diffusion of either A or B or both species. The rate of loss of A from is and, similarly, the loss of B from tg is D V n. In these terms, the subscript A and B refer to the co-ordinates of A or B, respectively, and the diffusion coefficients of A and B are, respectively, I a The dif- 

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