VELOCITY O F X BIMOLECULAR REACTIOX I N A HOMOGEKEOUS SYSTEM B Y SAMUEL C. HORNIXG
The following derivation is an attempt to derive the equation for the velocity of a reaction from the simplest plsssible assumptions. The reaction A B =C D is accompanied by the free energy change -AF. This reaction will be caused by a collision between the molecules of X which are in a state x and the molecules of B which are in a state y. For the reverse reaction, the molecules of C must be in the state s’ and the molecules of D in the state y’. The free energies of the reacting states will then be F, F, and Fx, F,, and the free energy of formation of these states will be AFx AF, and AFxp AFyt. I t can easily be seen that, F, F, = Fxl F,! - AFX, - AFy, AFx AF, = - AF and
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The free energy in the state A B must be increased to that in the state x,y in order that the reaction x,y to C D may occur. Similarly, the state C D must change to x’,y’ for the reaction x’,y’ to h B. If the direction of the reaction x’,y’ to X B is reversed, then a t some point x”, y” the free energy will equal that in the state x,y. The reactions x‘,y‘ to XI’, y“ and x,y to x‘,y‘ or its equal x“,y“ to x‘,y‘ occur spontaneously. For this to be true, x,y; x’y’; and x“, y” must coincide. The rate of reaction may be expressed as dx,/dt = k‘l (A-x,) (B-x,). This will equal the number of molecules in state x,y which collide. The number of molecular collisions between these two unlike molecules’ is Z = zNxNyuxy24 2 n k T (lf% l f b ) /(Ma I&,). From the Boltzrnann distribution law : N, -AF, - = e RT
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Ka
S,
and
- = e
- AF, RT
Nb
The rate of reaction given by the two expressions can be equated.
By inspection, it will be seen that (A - x,) (B - x,) which represents the activity product of the reacting substances is equal to: S s S b 6x,’
Tolman: “Statistical llechanics,”
411s 242.
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SAMUEL C. HORNING
The activity of a substance depends, therefore, upon the size and weight of the reacting molecules. h similar equation will hold for the reverse reaction.
K2(C)(D)=
-AFx' -AFy' 2
N,K,e
RT
4
2
T K T (M,+Md)/(MLlo Rid)
The equilibrium constant is K
Since -AF,
- AF,
+ AF,, + AF,,
=
-
AF
-AFx-AFy
ills0
Or
IC, =
2 ~ ' Z i i Fe
+
RT
d- _12 Ki - AF, AF, tiT li T'
I
.t2
T
