Question

For the liquid phase reaction A \(\to\) P, in a series of experiments in a batch reactor, the half-life \(t_{1/2}\) was found to be inversely proportional to the square root of the initial concentration of A. The order of the reaction is

• A. 3/2
• B. 1
• C. +1/2
• D. -1/2

Hint:

A quick guide to half-life dependency on initial concentration:
Zero-order ($n=0$): $t_{1/2} \propto C_{A0}$ (directly proportional).
First-order ($n=1$): $t_{1/2}$ is independent of $C_{A0}$.
Second-order ($n=2$): $t_{1/2} \propto 1/C_{A0}$ (inversely proportional).
Fractional-order ($n=1.5$): $t_{1/2} \propto 1/\sqrt{C_{A0}}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to determine the order ($n$) of a chemical reaction given the relationship between the reaction's half-life ($t_{1/2}$) and the initial concentration ($C_{A0}$) of reactant A in a batch reactor.

Step 2: Key Formula or Approach:
For an $n$-th order reaction where the rate is $-r_A = k C_A^n$, the half-life $t_{1/2}$ is given by the general proportional relation:
\[ t_{1/2} \propto \frac{1}{C_{A0}^{n-1}} = C_{A0}^{1-n} \] This formula is valid for all orders of reaction except $n = 1$ (for which half-life is independent of initial concentration).
By equating the given proportional relationship to the general formula, we can solve for $n$.

Step 3: Detailed Explanation:
Let us formulate the problem based on the given information:
We are told that the half-life is inversely proportional to the square root of the initial concentration:
\[ t_{1/2} \propto \frac{1}{\sqrt{C_{A0}}} \] We can rewrite the square root using an exponent: \[ \sqrt{C_{A0}} = C_{A0}^{1/2} \] So the relationship becomes: \[ t_{1/2} \propto \frac{1}{C_{A0}^{1/2}} = C_{A0}^{-1/2} \] Now, compare this with the general order-dependence relation: \[ t_{1/2} \propto C_{A0}^{1-n} \] Equating the exponents of $C_{A0}$ from both expressions: \[ 1 - n = -1/2 \] Rearranging the equation to solve for the reaction order $n$: \[ n = 1 + \frac{1}{2} = \frac{3}{2} \] Thus, the reaction is of order $3/2$ (or $1.5$).

Step 4: Final Answer
Therefore, the order of the reaction is $3/2$, which corresponds to option (A).