For a 1st order change \(R \rightarrow P\), the concentration of Reactant R changes from \(0.1\) M to \(0.025\) M in \(40\) minutes. The rate of reaction when the concentration of R is \(0.01\) M is ____

The concentration of the reactant changes from \(0.1\text{ M}\) to \(0.025\text{ M}\), which is \(\dfrac{1}{4}\) of the initial concentration.

This corresponds to \(2\) half-lives.

\(2 t_{1/2} = 40\text{ min} \Rightarrow t_{1/2} = 20\text{ min}\)

The rate constant \(k\) for a first-order reaction is:

\(k = \dfrac{0.693}{t_{1/2}} = \dfrac{0.693}{20} = 0.03465\text{ min}^{-1}\)

The rate of reaction when \([R] = 0.01\text{ M}\) is:

\(\text{Rate} = k[R]\)

\(\text{Rate} = 0.03465 \times 0.01 = 3.465 \times 10^{-4}\text{ M min}^{-1}\)

\(\text{Rate} \approx 3.47 \times 10^{-4}\text{ M min}^{-1}\)

Answer: \(3.47 \times 10^{-4}\text{ M min}^{-1}\)