COLLISION THEORY OF THE RATE CONSTANT CHEMISTRY 213

COLLISION THEORY OF THE RATE CONSTANT CHEMISTRY 213 The simplest collisional model for the rate constants in bimolecular elementary reactions, e.g., A + B → Products, is to assume that reaction occurs every time an A and B molecule collide. Thus the rate becomes: RATE = Z A,B

(1)

where Z A,B is the number of AB collisions occurring per unit time, per unit volume of the system. The kinetic theory of gases gives Z A,B = π (R A + R B )2 c AB n A n B ,

(2)

where R A and R B are the van der Waals radii of A and B, respectively, n A and n B are their number densities, and c AB is the mean relative speed of A with respect to B; i.e., 1/2

c AB

 8k B T  . ≡  π µ AB 

(3)

In Eq. (3), k B is Boltzmann’s constant and µ AB ≡ m A m B /(m A + m B ) is the AB reduced mass (m A,B denotes the mass of A or B). Equation (2) is obtained by considering how many B molecules will lie in the cylinder swept out by an A molecule in time dt. The radius of the cylinder is the maximum distance and A and B can be in order to collide, i.e., R A + R B . Thus the volume of the cylinder for molecules moving with relative speed v is V (v) ≡ π (R A + R B )2 vdt.

(4)

The number of B molecules in the cylinder with relative speed v to v + dv is V (v)n B f (v)dv, where µ AB  f (v) = 4π v  2π k B T  2

3/2

e

−

µ AB v 2

2k B T

(5)

is the Maxwell-Boltzmann speed distribution function governing the relative motion of B with respect to A. Finally, our expression for the rate is obtained by adding (integrating) the contributions for the different relative velocities, multiplying by the number of A molecules per unit volume and dividing out the factor of dt; i.e, RATE = n A n B π (R A + R B )

∞ 2

∫0 dv

Winter Term 2001-2002

f (v)v,

(6)

Collision Rate Constants

-2-

Chemistry CHEM 213W

which gives the expression presented above. As was mentioned in class, Eqs. (1) - (3) in general do not correctly describe the temperature dependence of rate constants since the activation energy does not appear. In order to correct for this, we will modify our initial hypothesis; specifically, now assume that only those molecules whose relative speed is greater than v min can react. In this case, Eq. (6) becomes: RATE = n A n B π (R A + R B )

∞ 2

∫

dv f (v)v,

(7)

v min

which upon evaluating the integral gives E   RATE = π (R A + R B )2 c AB e−E A /k B T 1 + A n A n B ,  k BT 

(8)

where E A ≡ µ AB v 2min /2 is the activation energy. This expression has the correct Arrhenius form except for the extra nonexponential factors containing temperature. However, compared with the exponential factor, these usually do not change very rapidly in temperature and can approximately be treated as constant. Thus we have derived an approximate expression for the bimolecular rate constant which has at least some of the qualitative features of those found in gas reactions.

Winter Term 2001-2002