Step 1: Understanding the Question:
We are given the total time ($x$) for a specific concentration change in a first-order reaction and need to find the half-life ($t_{1/2}$). Step 2: Key Formula or Approach:
For first-order reactions, the concentration decreases by half in every half-life period. Step 3: Detailed Explanation:
Initial concentration $[A]_0 = 0.4$ M
Final concentration $[A]_t = 0.1$ M
Let's trace the decay:
$0.4\text{ M} \xrightarrow{1^{st}\text{ half-life}} 0.2\text{ M}$
$0.2\text{ M} \xrightarrow{2^{nd}\text{ half-life}} 0.1\text{ M}$
Total number of half-lives = $2$.
Total time taken = $x$ hours.
Therefore, $2 \times t_{1/2} = x$
$t_{1/2} = \frac{x}{2}$ Step 4: Final Answer:
The half-life of the reaction is $\frac{x}{2}$ hours.