The mathematical expression of the rate of reaction on concentration terms of reactants is known as rate expression or rate equation or rate law.

For reaction \(A + B \rightarrow Products\), the rate equation is

\[rate ­\propto [A] [B]\]

\[rate = K [A] [B]\]

K is known as specific rate constant or rate per unit concentration of the reactants.

Units of rate constant are \((mole)^{1−n} (litre)^{n−1} s^{−1}\).

Rate law for any reaction cannot be predicted by looking at the balanced chemical reaction, that is, theoretically but must be determined experimentally.

Several chemical reactions take place in a sequence

of steps and the overall rate of reaction is governed

by the slowest step.

In certain cases, the slowest or rate-determining step may involve the formation of an unstable intermediate

from the reactant molecules. The total number of reactant molecules taking part in the slowest step may involve the formation of an unstable intermediate. The total number of reactant molecules taking part in the slowest step or limiting step in the formation of intermediate species is known as the molecularity of the reaction.

If a is the initial concentration then time required to decompose half of the substance for nth order is inversely proportional to

a¹⁻ⁿ

The correct answer is option 3. a¹⁻ⁿ.

The time required to decompose half of the substance for n^th order is inversely proportional to \(a^{n-1}\).

The half-life of a reaction is the time it takes for half of the reactants to be converted into products. The half-life period of an nth order reaction can be expressed as follows:

\[t_{1/2} =\frac{(n - 1)}{ka^{ n - 1} } \]

where:

\(t_{1/2}\) is the half-life period of the reaction

\(k\) is the rate constant

\(a\) is the initial concentration of the reactants

\(n\) is the order of the reaction

As the initial concentration of the reactants increases, the half-life period of the reaction increases. Therefore, the half-life period of a n^th order reaction is inversely proportional to \(a^{n-1}\).

Here is an explanation:

The rate of reaction is the change in the concentration of a reactant or product per unit time. It can be expressed as follows:

\[rate = k[A]^n\]

where:

\(rate\) is the rate of reaction

\(k\) is the rate constant

\([A]\) is the concentration of reactant \(A\)

\(n\) is the order of the reaction

The order of the reaction is the number of reactant molecules that must collide in order for the reaction to occur. The higher the order of the reaction, the slower the rate of reaction.

As the initial concentration of the reactants increases, the number of reactant molecules increases. This means that more of the reactant molecules have the opportunity to collide and react. Therefore, the rate of reaction increases with increase in initial concentration of the reactants.

The half-life period of a reaction is the time it takes for half of the reactant molecules to be converted into products. It can be expressed as follows:

\[t_{1/2} =\frac{(n - 1)}{ka^{ n - 1} } \]

where:

\(t_{1/2}\) is the half-life period of the reaction

\(k\) is the rate constant

\(a\) is the initial concentration of the reactants

\(n\) is the order of the reaction

As the rate of reaction increases, the half-life period of the reaction decreases. Therefore, the half-life period of a n^th order reaction is inversely proportional to \(a^{n-1}\).