Question

For a reaction, \(\mathrm{A} \longrightarrow \mathrm{B}\), rate equation is \(\mathrm{r}=\mathrm{k}[\mathrm{A}]^{\circ}\). If initial concentration of reactant is 'a'mol dm \({ }^{-3}\) find half life time of reaction.

• A. \(\frac{\mathrm{a}}{4\mathrm{k}}\)
• B. \(\frac{\mathrm{a}}{\mathrm{k}}\)
• C. \(\frac{5\mathrm{a}}{2\mathrm{k}}\)
• D. \(\frac{\mathrm{a}}{2\mathrm{k}}\)

Hint:

For a zero order reaction the rate is constant, so half of the initial concentration is used up at a fixed rate.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The rate equation \(r=k[\text{A}]^{0}\) shows a zero order reaction, so the rate is constant and equal to \(k\).

Step 2: Integrated rate law
\(k=\frac{[\text{A}]_0-[\text{A}]}{t}\).

Step 3: Calculation
At half life, \([\text{A}]=\frac{a}{2}\), so \(k=\frac{a-\frac{a}{2}}{t_{1/2}}=\frac{a}{2\,t_{1/2}}\).
Therefore \(t_{1/2}=\frac{a}{2k}\).

Step 4: Compare the options
This matches option (D).

Final Answer:
The half life of the reaction is \(\frac{a}{2k}\). \[ \boxed{\text{(D) }\dfrac{a}{2k}} \]