First Order Kinetics for JEE

Chemical kinetics is the branch of physical chemistry that helps us to understand the rates of chemical reactions. The reaction can be classified as fast, moderate, or slow by studying the rate associated with it. Kinetics also allows us to analyze the effect of temperature, and catalyst on the rate of reaction and rate constant. It gives us information about mechanisms of reaction and allows us to assign specific rate constants to individual mechanistic steps. 

Order of Reaction 

Order of reaction refers to the relationship between the rate of reaction and the concentration of reactants taking part in a chemical reaction for example, in a first-order reaction the rate of reaction is directly proportional to the concentration of one reactant involved in the chemical reaction.

The order of reaction does not always depend on the stoichiometric coefficients in the balanced chemical equation. The sum of the power of concentration of reactants involved in reaction in the rate law expression is called the order of that chemical reaction.

The reaction can be zero, first or second order, and so on. The order of reaction can be an integer or fraction or zero. Methods to find an order of reaction are:-

Method of Initial Rates

The power-law equation is first converted to its natural logarithm form. It's calculated as $\ln r=\ln k+x \cdot \ln [A]+y \cdot \ln [B]+\ldots$

The partial order for each reactant is now estimated by running the reaction with different concentrations of the reactant in the issue while keeping the concentrations of the other reactants constant.

If the partial order of A is known, the rate equation's power-law expression becomes \ln$ \mathrm{r}=\mathrm{x} \cdot \ln [\mathrm{A}]+\mathrm{C}$, where C is a constant. The partial order, given by x, is plotted as a function of $\ln [A]$, and the corresponding slope is '$\ln r$' as a function of $\ln [A]$.

Integral Method

This method is commonly used to verify the order of reaction derived from the initial rates method. The integral form of the rate law is compared to the measured reactant concentrations.

For example, if the value of $\ln [A]$ corresponds to a linear function of time (integrated rate equation of a first-order reaction:

$\ln [A]-\ln \left[A_{0}\right]=-k t$, the rate law for a first-order reaction is verified.

Differential Method

This is the simplest method for determining the reaction order.

First, write the reaction's rate expression ($\mathrm{r}=\mathrm{k}[\mathrm{A}]^{\mathrm{x}}[\mathrm{B}]^{\mathrm{y}} \ldots$).

The final value of the reaction order is the sum of the exponents x+y+...

First Order Reaction 

A first-order reaction is a chemical reaction in which the rate of reaction depends only on the concentration of one of the reactants.  The order of the reaction is 1.

For a reaction $A\rightarrow B$

  The rate of reaction can be defined as 

                                         $R=k[A]^t$

Where A is the concentration of reactant  

This is an example of first-order reaction which gives the relationship of rate of reaction and reactant for first-order reaction.

Unit of rate constant for any order is given by 

(mol L-1)1-ns-1

For first-order reaction n=1. The unit of rate constant for the first-order reaction will be s-1.

Examples 

First-order reaction examples are:-

$\begin{align} &\mathrm{SO}_{2} \mathrm{Cl}_{2} \rightarrow \mathrm{SO}_{2}+\mathrm{Cl}_{2} \\ &2 \mathrm{H}_{2} \mathrm{O}_{2}(1) \rightarrow 2 \mathrm{H}_{2}  \mathrm{O}(1)+\mathrm{O}_{2}(\mathrm{~g}) \\ &\mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{~s}) \rightarrow \mathrm{NO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \end{align}$

All nuclear reactions are first-order reactions. The number of radioactive atoms decaying per unit time in radioactive decay is proportional to the total number of radioactive atoms present at the time. Because the rate of decay is related to the first power of the number of radioactive atoms present, radioactive decay is first-order kinetics.

Rate Law for First-Order Reaction 

For a reaction $A\rightarrow B$ 

The differential rate law for a first-order reaction is given as-

$\text { Rate }=-\dfrac{d a}{d t}=k[A]$

‘A’ is the concentration of reactant 

‘K’ is the rate constant 

The integral rate law is given by:

$\begin{align} &-\dfrac{\mathrm{da}}{\mathrm{dt}}=\mathrm{k}[\mathrm{A}] \\ &-\dfrac{\mathrm{da}}{\mathrm{A}}=k d t \\ &\int_{\left[\mathrm{A}_{\mathrm{e}}\right]}^{[\mathrm{A}]} \dfrac{d A}{A}=-\int_{t}^{t} k d t \end{align}$

First-order reaction formula 

$\ln [A]-\ln \left[A_{0}\right]=-k t$

It can also be written as:

$\mathrm{A}=\mathrm{A}_{0} \ell^{-k t}$

Half Life of First Order Reaction 

The half life time of reaction is defined as the time at which the concentration of reactant is half the initial concentration.

$\begin{align} &\mathrm{At}~\mathrm{t}=\dfrac{\mathrm{t}_{1}}{2} \mathrm{~A}=\dfrac{\mathrm{A}_{0}}{2} \\ &\dfrac{[\mathrm{A}]_{0}}{2}=[A]_{0} \ell^{-k t\dfrac{1}{2}}  \end{align}$

Hence half life time for first-order reaction is given by 

                                        $t\dfrac{1}{2}=\dfrac{0.693}{k}$

Where k is the rate constant 

Graphs 

The first-order reaction graph between The concentration v/s time is given as:-

Graph for a first-order reaction

The graph for $\ln (A)$ v/s t for a first-order reaction is a straight line with slope -k  The graph is a straight line with a negative slope which shows that the concentration of reactant decreases as the reaction proceeds. 

Graph shows the variation of logarithm of A wrt time

Summary 

The kinetics of first-order reaction is the reaction in which the rate of reaction is directly proportional to the concentration of reaction. The order of the reaction is 1 hence called first-order reaction. Units of first-order reaction rate constant are given by s-1. The first order reaction equation is $\ln [A]-\ln \left[A_{0}\right]=-k t$. The graph for a first-order reaction between ln $\ln (A)$, and time is a straight line with a negative slope.