Question

For a first order reaction ${\text{A }} \to {\text{ B}}$, the reaction rate at reactant concentration of 0.01 M is found to be ${\text{2}}{\text{.0 }} \times 1{0^{ - 5}}{\text{ mol }}{{\text{L}}^{ - 1}}{{\text{s}}^{ - 1}}$. The half period of the reaction is
A. 220 s
B. 30 s
C. 300 s
D. 347 s

Answer 1

For a first order reaction, rate of reaction is directly proportional to concentration of one reactant. Once we find the value of reaction constant $({\text{k)}}$, we can determine the half-life period of a first order reaction.

Complete step by step answer:
We have been given the first order reaction ${\text{A}} \to {\text{B}}$
Concentration of the reactant is given as 0.01M and the rate of reaction is given as $2 \times {10^{ - 5}}{\text{ mol }}{{\text{L}}^{ - 1}}{\text{ }}{{\text{s}}^{ - 1}}$.

We know that, for a first order reaction, ${\text{Rate = }}\dfrac{{ - {\text{d[A]}}}}{{{\text{dt}}}}{\text{ }} = {\text{ k }} \times {\text{ [A]}}$

When we substitute the given values in this equation, we get
$2 \times {10^{ - 5}}{\text{ mol }}{{\text{L}}^{ - 1}}{\text{ }}{{\text{s}}^{ - 1}}{\text{ = k }} \times {\text{ 0}}{\text{.01 M}}$

Therefore, ${\text{k = }}\dfrac{{2 \times {{10}^{ - 5}}{\text{mol }}{{\text{L}}^{ - 1}}{\text{ }}{{\text{s}}^{ - 1}}}}{{{\text{0}}{\text{.01 M}}}}$
${\text{ = 2 }} \times {\text{ 1}}{{\text{0}}^{ - 3}}{\text{ }}{{\text{s}}^{ - 1}}$

So, we have the value of reaction constant $({\text{k)}}$as ${\text{2 }} \times {\text{ 1}}{{\text{0}}^{ - 3}}{\text{ }}{{\text{s}}^{ - 1}}$

We know that for a first order reaction, half life period is given as –
${{\text{t}}_{{}^{1}/{}_{2}}}\text{ = }\dfrac{0.693}{\text{k}}\text{ = }\dfrac{0.693}{\text{2 }\times \text{ 1}{{\text{0}}^{-3}}}$

{{\text{t}}_{{}^{1}/{}_{2}}}\text{ }=\text{ 346}\text{.5}$$$$\approx 347\text{ s}
Hence, the half period of the given first order reaction ${\text{A }} \to {\text{ B}}$ is approximately 347 s.

So, option D is correct.

Additional information:
The speed at which reactants are converted into products is called the Rate of reaction. It gives us information about the time taken for a reaction to reach completion. Rate of reaction varies with concentration of the reactant, temperature, pressure and order of reaction.
The power dependence of the rate of reaction on concentration of all reactants is called the order of that reaction. Reactions where rate of reaction depends only on concentration of one reactant are called as first order reactions. In such reactions, multiple reactants may be present but rate is affected only by one reactant.
The time taken for the concentration of reactant to reach 50% of its initial concentration is called the Half-life period.

Note: Equations of rate of reaction and half life period are different for reactions of different order. If order of reaction is not given in the question check the units of constant (k) because units of constant (k) are different for different order reactions.