For the reaction \(\mathrm{A}+\mathrm{B} \longrightarrow 2 \mathrm{C},\) which proceeds by a single-step bimolecular elementary process, (a) \(t_{1 / 2}=0.693 / k ;\) (b) rate of appearance of C= - rate of disappearance of \(\mathrm{A} ;\) (c) rate of reaction = \(k[\mathrm{A}][\mathrm{B}] ;\) (d) \(\ln [A]_{t}=-k t+\ln [A]_{0}.\)

Reaction kinetics is the branch of physical chemistry that studies the rates of chemical processes. It examines how various factors such as concentration, temperature, and catalysts affect the speed of a chemical reaction. Understanding kinetics helps predict how a reaction proceeds over time and the time it takes for reactants to transform into products.

In the context of a bimolecular elementary process, reaction kinetics focuses on how two molecules come together to react. Bimolecular reactions are one subset of elementary reactions, indicating that the reaction occurs in one step and involves the simultaneous collision of two reactant molecules.

The rate of a reaction measures the speed at which reactants are converted into products over time. It is typically expressed in terms of concentration change per unit time, such as moles per liter per second. Factors like temperature, the presence of catalysts, and the concentrations of reactants can influence this rate.

For a bimolecular reaction such as \(\mathrm{A} + \mathrm{B} \longrightarrow 2\mathrm{C}\), the rate is proportional to the product of the concentrations of the reactants, \(k[\mathrm{A}][\mathrm{B}]\), where \(k\) is the rate constant. The rate of appearance of product \(\mathrm{C}\) is directly related to the rate of disappearance of reactants \(\mathrm{A}\) and \(\mathrm{B}\).

The half-life of a reaction, often represented by \(t_{1/2}\), is the time required for half the quantity of a reactant to be consumed or for the concentration of the reactant to fall to half its initial value. For first-order reactions, \(t_{1/2}\) is a constant that is independent of the initial concentration and can be calculated using the formula \(t_{1/2} = 0.693 / k\).

However, for second-order reactions, the half-life depends on the initial concentration of the reactants, and the formula for the first-order reactions does not apply. In a half-life context, this highlights the importance of knowing the reaction order when conducting kinetic analysis.

First-order reactions are characterized by a rate that is directly proportional to the concentration of one reactant. The integrated rate law for a first-order reaction is given by \(\ln[A]_t = -kt + \ln[A]_0\), where \(\ln[A]_t\) is the natural logarithm of the concentration of reactant \(A\) at time \(t\), \(\ln[A]_0\) is the natural logarithm of the initial concentration of \(A\), and \(k\) is the first-order rate constant. In these reactions, the half-life is constant and does not depend on the initial concentration of reactants.

Second-order reactions are defined by a rate proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. The rate expression for a second-order reaction with two reactants \(A\) and \(B\) would be \(rate = k[A][B]\).

Unlike first-order reactions, the half-life of a second-order reaction depends on the initial concentrations of the reactants, and therefore, it changes over time. The integrated rate law and half-life expressions for second-order reactions differ significantly from those of first-order reactions, reflecting the dependency on initial reactant concentrations.