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.

The half-life period of a reaction

decrease with an increase in temperature

The correct answer is option 2. decrease with an increase in temperature.

Let us explain why the half-life period of a reaction typically decreases with an increase in temperature.

The rate constant \( k \) of a reaction is temperature-dependent, described by the Arrhenius equation:
\(k = A e^{-\frac{E_a}{RT}}\)

where:

\( k \) is the rate constant,

\( A \) is the pre-exponential factor (frequency factor),

\( E_a \) is the activation energy,

\( R \) is the gas constant,

\( T \) is the temperature in Kelvin.

As temperature \( T \) increases, the term \( e^{-\frac{E_a}{RT}} \) increases because the exponent \(-\frac{E_a}{RT}\) becomes less negative (or smaller in magnitude), leading to an increase in \( k \).

Half-Life and Reaction Order

The half-life (\( t_{1/2} \)) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. The relationship between half-life and rate constant varies with the order of the reaction.

For first-order reactions, the half-life is given by:
\( t_{1/2} = \frac{\ln 2}{k} \)

Here, \( t_{1/2} \) is inversely proportional to the rate constant \( k \). As \( k \) increases with increasing temperature, \( t_{1/2} \) decreases.

For second-order reactions, the half-life is given by:
\(t_{1/2} = \frac{1}{k [A]_0} \)

Again, \( t_{1/2} \) is inversely proportional to the rate constant \( k \). As \( k \) increases with increasing temperature, \( t_{1/2} \) decreases.

For zero-order reactions, the half-life is given by:
\(t_{1/2} = \frac{[A]_0}{2k} \)

Here too, \( t_{1/2} \) is inversely proportional to the rate constant \( k \). As \( k \) increases with increasing temperature, \( t_{1/2} \) decreases.

In summary, for reactions of any order, the half-life \( t_{1/2} \) is inversely proportional to the rate constant \( k \). Since the rate constant \( k \) increases with temperature (according to the Arrhenius equation), the half-life \( t_{1/2} \) decreases as temperature increases.

Therefore, the half-life period of a reaction decreases with an increase in temperature because the rate constant increases with temperature, leading to faster reaction rates and shorter times required to reach half of the initial concentration.

Thus, the correct answer is option 2. decreases with an increase in temperature