Half Life Period Determination from Initial Concentration Graph

From the given graph of half-life ($t_{1/2}$) versus initial concentration ($[\text{A}]_0$), the plot is a straight line passing through the origin.

This indicates that the half-life $t_{1/2}$ is directly proportional to the initial concentration $[\text{A}]_0$, which is characteristic of a zero-order reaction: $t_{1/2} \propto [\text{A}]_0$

Thus, the ratio of half-life to initial concentration is constant: $\frac{t_{1/2}}{[\text{A}]_0} = \text{constant}$

Using the coordinates from the graph:

• For $[\text{A}]_{0, 1} = 4 \times 10^{-3} \text{ mol L}^{-1}$, $t_{1/2, 1} = 240 \text{ s}$
• For $[\text{A}]_{0, 2} = 1.5 \times 10^{-3} \text{ mol L}^{-1}$, $t_{1/2, 2} = x \text{ s}$

Setting up the proportion: $\frac{240 \text{ s}}{4 \times 10^{-3} \text{ mol L}^{-1}} = \frac{x \text{ s}}{1.5 \times 10^{-3} \text{ mol L}^{-1}}$

Solving for $x$: $x = \frac{240 \times 1.5 \times 10^{-3}}{4 \times 10^{-3}}$

$x = 60 \times 1.5 = 90$

The value of $x$ is $90$.