A reaction is first order in \(A\) and second order in \(B\) : (i) Write differential rate equation. (ii) How is the rate affected when the concentration of \(B\) is tripled? (iii) How is the rate affected when the concentration of both \(A\) and \(B\) is doubled?

Understanding the differential rate equation is crucial for analyzing the kinetics of a chemical reaction. Essentially, it relates the rate of change in concentration of reactants over time. The equation is tailored to the reaction order with respect to each reactant.

In our exercise, given a reaction first order in reactant A and second order in reactant B, the differential rate equation is formulated as: \[ \text{Rate} = k[A]^1[B]^2 \].

This equation reveals that the rate is directly proportional to the concentration of A and the square of the concentration of B. The powers of the concentrations ([A] and [B]) are indicative of the reaction order, which in turn defines the relationship between the concentration of reactants and the rate of the reaction. The presence of the rate constant 'k' in the equation denotes that this proportionality is unique for every reaction under specific conditions.

The rate of a chemical reaction is heavily influenced by the concentrations of its reactants. According to the rate law, modifying the concentration of one or more reactants will change the reaction rate.

For a reaction like the one in our exercise, where the order with respect to B is second, tripling the concentration of B would exponentially affect the rate. To visualize, if [B] is tripled: \[ \text{Rate}' = k[A][3B]^2 \], this simplifies to \[ \text{Rate}' = k[A](9[B]^2) \]. Consequently, the rate becomes nine times faster, demonstrating a squared dependency on the concentration of B, due to its second-order nature.

This concept is an indispensable part of chemical kinetics, as it allows chemists to control and optimize reaction rates for industrial and laboratory processes.

The rate constant 'k' in the rate equation is a crucial factor that determines the speed of a chemical reaction at a given temperature. While reactant concentrations can vary, the rate constant remains fixed under constant temperature and pressure, characteristic of the specific reaction.

In our given problem: \[ \text{Rate} = k[A]^1[B]^2 \], 'k' is unaffected by changes in concentration levels but is sensitive to temperature changes. The magnitude of 'k' reflects the rate at which reactants transform into products, and different reactions have different rate constants. Knowing 'k' and the reaction order allows chemists to predict how changes in concentration or conditions will affect the reaction rate.