Question

In a first-order reaction, 50 minutes time is taken for the completion of 93.75 of a reaction. What is the half life of the reaction?
A. 25 min
B. 12.5 min
C. 20 min
D. 10 min

Answer 1

For solving this question, first we need to understand the concept of a first order reaction. A reaction that proceeds at a rate which depends linearly on only one reactant concentration is known as a first order reaction. For example, the hydrolysis of aspirin and the reaction of t-butyl bromide with water give t-butanol.

Complete step by step answer:
In a half life period the formula for a first order reaction is:
k= $\dfrac{{2.303}}{t}\log \dfrac{{{{[A]}_0}}}{{[A]}}$
or the value of ‘t’ can be described as:
t= $\dfrac{{2.303}}{k}\log \dfrac{{{{[A]}_0}}}{{[A]}}$
As it is a first order reaction so t = ${t_{1/2}}$
So, [A] = $\dfrac{1}{2}{[A]_0}$
${t_{1/2}} = \dfrac{{2.303}}{k}\log \dfrac{{{{[A]}_0}}}{{\dfrac{1}{2}{{[A]}_0}}} = \dfrac{{2.303}}{k}\log 2$
${t_{1/2}} = \dfrac{{0.693}}{k}$
To calculate the half-life of a reaction we need to know that half-life of a chemical reaction can be defined as the time taken for the concentration of a given reactant to reach 50% of its initial concentration. Step by step we will calculate the values.
Now, let’s start solving the question:
$A \to B$
$\text{a 0}$
$\text{(a − 0.9375 a) (0.9375 a)}$
K $= \dfrac{{2.303}}{t}\log (\dfrac{a}{{a - x}})$
K $= \dfrac{{2.303}}{{50}}\log (\dfrac{a}{{(1 - 0.9375)}})$
K = 0.0554
Now we substitute the value of k in half-life equation we get,
$\Rightarrow {t_{1/2}} = \dfrac{{0.693}}{k} = \dfrac{{0.693}}{{0.0554}}$
${t_{1/2}} = 12.5{\text{mins}}$
So, the half life of the reaction is 12.5 mins.

So, the correct answer is Option C.

Note: After solving this question, we need to know whether a first order reaction completes or not. But it doesn’t complete because according to the integrated rate equation of first order reaction the complete amount of reactant gets converted into products only when the condition t=infinity is fulfilled. Thus, this condition is not possible. As the first order reaction is 20% which is completed in 20 minutes.