Question

Rate of the reaction $\text{A}+\text{B}\rightarrow$ product is $3.6\times10^{-2}\text{ mol dm}^{-3}\text{s}^{-1}$ and rate law is $r=k[\text{A}][\text{B}]^{2}$. What is rate constant of the reaction if $[\text{A}]=0.2$ M and $[\text{B}]=0.1$ M?

• A. $18\text{ mol}^{-2}\text{dm}^{6}\text{s}^{-1}$
• B. $10\text{ mol}^{-2}\text{dm}^{6}\text{s}^{-1}$
• C. $24\text{ mol}^{-2}\text{dm}^{6}\text{s}^{-1}$
• D. $4.8\text{ mol}^{-2}\text{dm}^{6}\text{s}^{-1}$

Hint:

General formula for the units of a rate constant $k$ for an $n^{th}$ order reaction:
Units = $(\text{mol L}^{-1})^{1-n} \text{s}^{-1}$
Here, order $n = 3$, so $(\text{mol L}^{-1})^{-2} \text{s}^{-1} = \text{mol}^{-2} \text{L}^2 \text{s}^{-1}$ (where L is $\text{dm}^3$, so $\text{L}^2 = \text{dm}^6$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the experimental rate of a reaction, the theoretical rate law expression, and the instantaneous concentrations of the reactants. We must calculate the numerical value and units of the rate constant ($k$).

Step 2: Detailed Explanation:
The given rate law is:
$r = k [\text{A}] [\text{B}]^2$
We are provided with the following values:
Rate ($r$) = $3.6 \times 10^{-2} \text{ mol dm}^{-3}\text{s}^{-1}$
Concentration of A ($[\text{A}]$) = $0.2 \text{ M} = 0.2 \text{ mol dm}^{-3}$
Concentration of B ($[\text{B}]$) = $0.1 \text{ M} = 0.1 \text{ mol dm}^{-3}$
Rearrange the rate law to solve for the specific rate constant ($k$):
$k = \frac{r}{[\text{A}] [\text{B}]^2}$
Substitute the numerical values:
$k = \frac{3.6 \times 10^{-2}}{(0.2) \times (0.1)^2}$
$k = \frac{0.036}{0.2 \times 0.01}$
$k = \frac{0.036}{0.002}$
Multiply numerator and denominator by 1000 to simplify:
$k = \frac{36}{2} = 18$
Now, let's verify the complex units for a third-order reaction (order $1 + 2 = 3$):
Unit of $k = \frac{\text{mol dm}^{-3}\text{s}^{-1}}{(\text{mol dm}^{-3})(\text{mol dm}^{-3})^2}$
Unit of $k = \frac{\text{mol dm}^{-3}\text{s}^{-1}}{(\text{mol}^3 \text{dm}^{-9})}$
Unit of $k = \text{mol}^{-2} \text{dm}^6 \text{s}^{-1}$.
The numerical value matches perfectly with the correct units.

Step 3: Final Answer:
The rate constant is $18\text{ mol}^{-2}\text{dm}^{6}\text{s}^{-1}$, matching option (a).