Explain briefly the collision theory of bimolecular reactions.

1. Rate law and rate equation:

(i) Chemical Kinetics is the branch of chemistry which deals with the study of reaction rates and their mechanism.

(ii) Rate of Reaction: It is the rate of change of concentration of any of the reactant or product with time at any particular moment of time.

(iii) Average Rate: The rate of reaction measured over a long time interval is called average rate. It is given as $\mathrm{\Delta x}/\mathrm{\Delta t}$.

For the reaction $\mathrm{aA} + \mathrm{bB}\to \mathrm{cC} + \mathrm{dD}$

Average Rate $=-\frac{1}{\mathrm{a}}\frac{\mathrm{\Delta }\left[\mathrm{A}\right]}{\mathrm{\Delta }\mathrm{t}}=-\frac{1}{\mathrm{b}}\frac{\mathrm{\Delta }\left[\mathrm{B}\right]}{\mathrm{\Delta }\mathrm{t}}=\frac{1}{\mathrm{c}}\frac{\mathrm{\Delta }\left[\mathrm{C}\right]}{\mathrm{\Delta }\mathrm{t}}=\frac{1}{\mathrm{d}}\frac{\mathrm{\Delta }\left[\mathrm{D}\right]}{\mathrm{\Delta }\mathrm{t}}$

(iv) Instantaneous Rate: It is the rate of a reaction at a given instant of time i.e., $\frac{\mathrm{\Delta x}}{\mathrm{\Delta t}}$ (average rate) becomes $\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}$ when $\mathrm{\Delta t}$ approaches zero.

For the reaction $\mathrm{aA} + \mathrm{bB}\to \mathrm{cC} + \mathrm{dD}$

Instantaneous Rate $=-\frac{1}{\mathrm{a}}\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{d}\mathrm{t}}=-\frac{1}{\mathrm{b}}\frac{\mathrm{d}\left[\mathrm{B}\right]}{\mathrm{d}\mathrm{t}}=\frac{1}{\mathrm{c}}\frac{\mathrm{d}\left[\mathrm{C}\right]}{\mathrm{d}\mathrm{t}}=\frac{1}{\mathrm{d}}\frac{\mathrm{d}\left[\mathrm{D}\right]}{\mathrm{d}\mathrm{t}}$

(v) Rate Expression: The mathematical expression giving the rate of a reaction in terms of concentration of reactants at a given temperature.

(vi) Rate Constant (k): It is the rate of the reaction when the concentration of each of reacting species is unity.

(vii) Rate Law: Describes the reaction rate in terms of concentrations of reactants.

2. Order and Molecularity:

(i) Molecularity: The number of reacting species which collide simultaneously to bring about the chemical change.

(ii) Order of Reaction: The sum of the exponents of the concentration terms in the experimental rate law of reaction. It can be $0, 1, 2, 3$ or a fractional value.

(iii) Rate Law: For a general reaction, $\mathrm{aA} + \mathrm{bB}\to \text{products}$, $\text{Rate}=\mathrm{k}{\left[\mathrm{A}\right]}^{\mathrm{m}}{\left[\mathrm{B}\right]}^{\mathrm{n}}$ $\mathrm{m}$ and $\mathrm{n}$ are determined experimentally

Order w.r.t. $\mathrm{A}=\mathrm{m}$; Order w.r.t. $\mathrm{B}=\mathrm{n}$

Overall Order $=\mathrm{m}+\mathrm{n}$

Units of Rate $=\mathrm{mol} {\mathrm{L}}^{-1}{\mathrm{s}}^{-1}\text{ or }\mathrm{Atm} {\mathrm{s}}^{-1}$

Units of k: For reaction of nth order $\mathrm{k}={(\mathrm{mol} {\mathrm{L}}^{-1})}^{1-\mathrm{n}}{\mathrm{s}}^{-1}\text{ or }{\left(\mathrm{atm}\right)}^{1-\mathrm{n}}{\mathrm{s}}^{-1}$

3. Half-life Period of Reaction ( t1/2):

The time taken for the concentration of reactants to be reduced to half of their initial concentration.

(i) Integrated Rate Equation:

(ii) The differential rate equations which are integrated to give a relationship between rate constant and concentrations at different times.

(iii) Rate Determining Step is the slowest step in the reaction mechanism.

(iv) Rate law for a zero order reaction $=\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}=\mathrm{k}{\left[\mathrm{A}\right]}^0$

$\text{Units of }\mathrm{k}=\mathrm{mol} {\mathrm{L}}^{-1}{\text{s}}^{-1}$

The integrated rate law equation for a zero order reaction, $\mathrm{R} \to \mathrm{P}$ is

$\mathrm{k}=\frac{\left[{\mathrm{R}}_0\right]-\left[\mathrm{R}\right]}{\mathrm{t}}$

Half-life Period (t1/2) of a zero order reaction is ${\mathrm{t}}_{1/2}=\frac{\left[{\mathrm{R}}_0\right]}{2\mathrm{k}}$

(v) Rate law for a first order reaction $=\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}=\mathrm{k}{\left[\mathrm{A}\right]}^1$

Units of $\mathrm{k}={\mathrm{s}}^{-1}\text{ or }\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}$

(a) The integrated rate law equation for a first order reaction, $\mathrm{A} \to \mathrm{B}$ is

$\mathrm{k}=\frac{2.303}{\mathrm{t}}\mathrm{l}\mathrm{o}\mathrm{g}\frac{\left[{\mathrm{A}}_0\right]}{\left[\mathrm{A}\right]}$

(b) The plot of log [A] vs time gives a straight line whose slope=-k2.303

(c) Half-life Period of a 1st order reaction, ${\mathrm{t}}_{1/2}=\frac{0.693}{\mathrm{k}}$

4. Collision theory:

(i) Activation Energy (Ea): The additional energy required by reacting species over and above their average PE to enable them to cross the energy barrier between reactants and products.

(ii) Catalyst: A substance which enhances the rate of a reaction without itself undergoing chemical change.

(iii) Effective Collisions: The collisions responsible for changing the reactant molecules into product molecules.

(iv) Threshold Energy: The minimum energy that a reacting species must possess in order to undergo effective collisions.

(v) Collision Theory: A chemical reaction takes place due to collisions between reacting molecules. For a bimolecular reaction, $\text{Rate}={\mathrm{Z}}_{\mathrm{AB}}.{\mathrm{e}}^{-\mathrm{Ea}/\mathrm{RT}}$. Here Z is collision frequency and ${\mathrm{e}}^{-\mathrm{Ea}/\mathrm{RT}}$ is fraction of molecules with energy equal to or greater than activation energy.

5. Arrhenius Equation $\mathrm{k}={\mathrm{Ae}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}$

$\mathrm{l}\mathrm{o}\mathrm{g}\frac{{\mathrm{k}}_2}{{\mathrm{k}}_1}=\frac{{\mathrm{E}}_{\mathrm{a}}}{2.303\mathrm{ }\mathrm{R}}\left[\frac{{\mathrm{T}}_2-{\mathrm{T}}_1}{{\mathrm{T}}_1{\mathrm{T}}_2}\right]$