Step 1: Understand the Problem

This is a first-order reaction: A → products. The concentration of A changes from 0.1 M to 0.025 M in 40 minutes. We are asked to find the rate of the reaction when [A] = 0.01 M.

For a first-order reaction, the rate law is: $\mathrm{Rate}=k\left[A\right]$

Therefore, to find the rate, we first need to find the rate constant, $k$.

Step 2: Find the Rate Constant (k)

The integrated rate law for a first-order reaction is:

$k=\frac{2.303}{t}\log \frac{[A{]}_0}{\left[A\right]}$

Where:

• $[A{]}_0=0.1M$ (initial concentration)
• $\left[A\right]=0.025M$ (concentration at time t)
• $t=40\min$

Plugging in the values:

$k=\frac{2.303}{40}\log \frac{0.1}{0.025}$

$k=\frac{2.303}{40}\log \left(4\right)$

We know that $\log \left(4\right)=\log \left(2^2\right)=2\times 0.3010=0.6020$

$k=\frac{2.303}{40}\times 0.6020$

$k=\frac{1.3865}{40}$

$k=0.0346625{\min }^{-1}$

So, $k\approx 0.03466{\min }^{-1}$

Step 3: Calculate the Rate of Reaction

Now we use the rate law: $\mathrm{Rate}=k\left[A\right]$

We need the rate when $\left[A\right]=0.01M$.

$\mathrm{Rate}=\left(0.03466{\min }^{-1}\right)\times \left(0.01M\right)$

$\mathrm{Rate}=0.0003466M/\min$

Expressing this in scientific notation:

$\mathrm{Rate}=3.466\times {10}^{-4}M/\min$

This rounds to $3.47\times {10}^{-4}M/\min$.

Final Answer

The rate of reaction when the concentration of A is 0.01 M is $3.47\times {10}^{-4}M/\min$.

This corresponds to the second option.

Related Topics & Formulae

First-Order Reactions

A first-order reaction is one whose rate depends on the concentration of a single reactant raised to the first power.

Differential Rate Law: $\mathrm{Rate}=-\frac{d\left[A\right]}{dt}=k\left[A\right]$

Integrated Rate Law: $\ln \left[A\right]=-kt+\ln [A{]}_0$ or $\log \left[A\right]=-\frac{k}{2.303}t+\log [A{]}_0$

Half-life ($t_{1/2}$): The time taken for the concentration of the reactant to reduce to half its initial value. It is constant for first-order reactions and is given by: $t_{1/2}=\frac{0.693}{k}$

Key Theory

• The rate constant (k) for a first-order reaction is independent of the initial concentration of the reactant.
• A plot of $\ln \left[A\right]$ versus time (t) gives a straight line with a slope of $-k$.
• The unit of the rate constant for a first-order reaction is ${\mathrm{time}}^{-1}$ (e.g., s⁻¹, min⁻¹).