Hydrolysis of an ester in an aqueous solution was studied by titrating the liberated carboxylic acid against sodium hydroxide solution. The concentrations of the ester at different time intervals are given below:

Time (in minute)	$0$	$30$	$60$	$90$
Ester concentration ($\mathrm{mol} {\mathrm{L}}^{-1}$)	$0.85$	$0.80$	$0.754$	$0.71$

Show that, the reaction follows first order kinetics.

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 $\mathrm{k}={\mathrm{Ae}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}$

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