Step 1: Understanding the Question:
We need to find the rate constant ($k$) of a first-order reaction from the slope of the integrated rate law graph. Step 2: Key Formula or Approach:
For a first-order reaction:
\[ k = \frac{2.303}{t} \log \frac{[\text{A}]_0}{[\text{A}]_t} \]
Rearranging into $y = mx + c$ form:
\[ \log[\text{A}]_t = -\frac{k}{2.303}t + \log[\text{A}]_0 \]
The slope ($m$) of $\log[\text{A}]$ vs $t$ is $-\frac{k}{2.303}$. Step 3: Detailed Explanation:
Given: Slope $= -2.5 \times 10^{-3} \text{ s}^{-1}$
Calculation:
\[ -\frac{k}{2.303} = -2.5 \times 10^{-3} \]
\[ k = 2.303 \times 2.5 \times 10^{-3} \]
\[ k = 5.7575 \times 10^{-3} \text{ s}^{-1} \] Step 4: Final Answer:
The rate constant is $5.757 \times 10^{-3} \text{ s}^{-1}$.