Question

The half life period of a first order reaction is $35\min$. What is the fraction of the reactant that remains after $75\min$?
A. $4.415$
B. $0.226$
C. $5.263$
D. $0.155$

Answer 1

Half-life of a reaction is the time taken by the reactant to decrease its concentration to half of its initial concentration. It is used in chemistry and other areas for the prediction of concentration of a particular substance over time.

Complete answer:
For first order reaction the relation between half-life of a reaction and the rate constant is given as
$k = \dfrac{{0.693}}{{{t_{1/2}}}}$
where $k$ is the rate constant and ${t_{1/2}}$ is the half-life of the reaction. Putting the given value of the half- life in the above equation, we get the value of rate constant
$\Rightarrow$ $k = \dfrac{{0.693}}{{35}} = 0.0198{\min ^{ - 1}}$
Rate of a reaction is the change in concentration of a reactant or change in the concentration of product per unit time. Expression for calculating rate of a reaction according to rate law is given as
$\Rightarrow$ $k = \dfrac{{2.303}}{t}\log \dfrac{a}{{(a - x)}}$
where t is the final time, a is the initial concentration of reactant and x is the final concentration of the reactant. Putting given values of final time and obtained value of rate constant in above reaction we get
$\Rightarrow$ $0.0198 = \dfrac{{2.303}}{{75}}\log \dfrac{a}{{(a - x)}}$
On further simplifying the equation
$\Rightarrow$ $\log \dfrac{a}{{(a - x)}} = 0.6449$
After solving the above equation, we get the fraction of the reactant
$\Rightarrow$ $\dfrac{a}{{(a - x)}} = 4.415$
Therefore, the right answer is option (A).

Note:
Remember the unit of time and rate constant if time is given in minutes, then calculate rate constant per minute or if time is given in seconds, then calculate the rate constant per second. Half-life of a first order reaction does not depend on the concentration of the reactants or concentration of the products.