Question

Consider the following reaction
\[A + B \rightarrow C\]
The time taken for A to become 1/4th of its initial concentration is twice the time taken to become 1/2 of the same. Also, when the change of concentration of B is plotted against time, the resulting graph gives a straight line with a negative slope and a positive intercept on the concentration axis.
The overall order of the reaction is ____.

Answer 1

Correct Answer: 1

The overall reaction order for \(A + B \rightarrow C\) is determined by analyzing its kinetic behavior. Initially, the observation that the time for reactant \(A\) to reach \( \frac{1}{4} \) of its initial concentration is double the time required to reach \( \frac{1}{2} \) of its initial concentration signifies first-order kinetics for \(A\). This is consistent with the integrated rate law for a first-order reaction, \( [A] = [A]_0 e^{-kt} \), where the time taken to reach a specific fraction of the initial concentration is \( t = \frac{\ln(\text{fraction})}{k} \). Specifically, the half-life (\(t_1\)) is \( t_1 = \frac{\ln(2)}{k} \), and the time to reach \( \frac{1}{4} \) concentration (\(t_2\)) is \( t_2 = \frac{\ln(4)}{k} \). The given condition \(t_2 = 2t_1\) is mathematically verified as \( \frac{\ln(4)}{k} = 2 \times \frac{\ln(2)}{k} \), simplifying to \( 2\ln(2) = 2\ln(2) \). This confirms the first-order dependence of the reaction rate on \(A\). Subsequently, a plot of the change in concentration of \(B\) over time yields a straight line with a negative slope. This linearity indicates zero-order kinetics with respect to \(B\), as described by the equation \( [B] = [B]_0 - kt \), which depicts a constant rate of decrease in concentration. Therefore, the overall reaction order is the sum of the individual orders: Order with respect to \(A\) is 1, and order with respect to \(B\) is 0. The total order of the reaction is \( 1 + 0 = 1 \). This result aligns with the provided range of (1,1), consistent with overall first-order kinetics.