Select the correct relation of half life for a zero order reaction.

$t_{0.5}=\frac{[R]_0}{2k}$

The correct answer is Option (1) → $t_{0.5}=\frac{[R]_0}{2k}$

Core Concept

For a zero order reaction:

$Rate = k$

Integrated form:

$[R] = [R]_0 - kt$

At half-life:

$[R] = \frac{[R]_0}{2}$

Stepwise Derivation

$\frac{[R]_0}{2} = [R]_0 - k t_{0.5}$

$k t_{0.5} = [R]_0 - \frac{[R]_0}{2}$

$k t_{0.5} = \frac{[R]_0}{2}$

$t_{0.5} = \frac{[R]_0}{(2k)}$

Explanation of Each Option

Option 1

This is correct because in zero order reactions, half-life depends directly on initial concentration. Unlike first order reactions, it is not constant and varies with [R]o. This relation is derived directly from the integrated rate law.

Option 2

This is incorrect because it reverses the relationship between concentration and rate constant. In zero order reactions, increasing initial concentration increases half-life, not decreases it as this expression suggests.

Option 3

This is incorrect because it ignores the factor of 1/2 that comes from the definition of half-life. Half of the initial concentration is consumed, so the denominator must include 2k.

Option 4

This is incorrect because it omits the rate constant entirely. The half-life must depend on both initial concentration and rate constant for zero order reactions.