Energy Barrier

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In this theory, the rate constant is related to three factors Z, f and p by the equation

k = Zfp b ……(19)

where Z is the collision frequency

f is the fraction of collisions having energy greater than the threshold energy and p is the fraction of the collisions that occurs with the reactant molecules properly oriented.

Z, the collision frequency increases with temperature because Z is directly proportional to the root mean square (rms) as given by,

where, s is the molecular diameter

N/V is the number of molecules per cc.

From kinetic theory of gases, it is known that V_rms has a square-root dependency on temperature, that is,

Therefore, for a given molecular diameter, the collision frequency will also have a square-root dependency on temperature. The ratio of Z values at two temperature values, for e.g., at 298 K and 308 K will be

A ten-degree rise in temperature near the room temperature usually doubles the rate of a reaction, but the factor Z does not change much with temperature and therefore, does not explain doubling of rate of reaction.

The factor that contributes to the acceleration of a rate of a reaction is (f), the fraction of collisions having energy greater than the threshold energy.

Again, from kinetic theory of gases it is seen that f has an exponential dependence with temperature.

f = e^-Ea/RT …… (22)

The effect of this factor on k can be best explained with an example.

Examples 13:

The reaction of nitric oxide (NO) with chlorine (Cl₂) is believed to occur in a single step. The activation energy for the reaction is 8.5 x 10⁴ J / mol. At 298 K the fraction of collision with the energy of activation is 1.2 x 10^-15. This is an extremely small value. However, the frequency of collision Z is very large therefore, the product of Z and f is not small. For a 10^o rise in temperature the fraction of collisions with energy greater than the threshold increases to 3.8 x 10^-15, this increase is threefold. Therefore, the rate of the reaction or the rate constant increases threefold.

The effect of temperature on f can be explained from the energy distribution among molecules at different temperatures. From the energy curves shown below.

fig 6.12 - Distribution of kinetic energy of molecules in a gas

It is observed that the energy distribution curve is flatter at higher T and is shifted toward higher energy region. This is because, the kinetic energy of the molecules are higher at higher temperatures. Also, the fraction of molecules with energy greater than the threshold as indicated by the shaded portion is almost double.

It is very clear, that the rate of reaction increases with temperature, mainly due to increase in the number of collisions which occur with sufficient energy, i.e., with energy greater than the threshold.