Question

In a first order reaction the concentration of reactant is reduced to 1/8 of initial concentration in 75 minutes at 298 K. What is the half-life of reaction in minutes ?
A. 50 min
B.15 min
C. 30 min
D. 25 min

Answer 1

We can easily find the half life of a first order reaction by using the relation given by ${t_{\dfrac{1}{2}}} = \dfrac{{0.693}}{k}$
where k = rate constant and k can be evaluated by using
$k = \dfrac{{2.303}}{t}\ln \dfrac{{\left[ {{R_ \circ }} \right]}}{{\left[ R \right]}}$

Complete solution:
Half-life is a time interval in which the reactant's concentration reaches half of its initial value
first order reaction is a chemical reaction in which the rate of reaction is directly proportional to the concentration of reactant
$r \propto \left[ A \right]$
In the question we are given,
Final concentration of reactant = 1/8 of initial concentration
Time taken, $t = 75$ min
Temperature, $T = 298{\text{ K}}$
Let Initial concentration of reactant $= \left[ {{R_ \circ }} \right]$
So, final concentration of reactant, $\left[ R \right] = \dfrac{1}{8}\left[ {{R_ \circ }} \right]$
Formula used to evaluate the half-life of the reaction is given by,
${t_{\dfrac{1}{2}}} = \dfrac{{0.693}}{k}$ $...\left( 1 \right)$
Where ${t_{\dfrac{1}{2}}} =$ half life of reaction.
First of all, we should find the value of k
$k = \dfrac{{2.303}}{t}\ln \dfrac{{\left[ {{R_ \circ }} \right]}}{{\left[ R \right]}}$
On substituting the values, we get
$k = \dfrac{{2.303}}{{75}}\ln \dfrac{{\left[ {{R_ \circ }} \right]}}{{\dfrac{1}{8}\left[ {{R_ \circ }} \right]}}$
$k = 2.77 \times {10^{ - 2}}{\text{ mi}}{{\text{n}}^{ - 1}}$
On replacing the value of k in equation $...\left( 1 \right)$
${t_{\dfrac{1}{2}}} = \dfrac{{0.693}}{{2.77 \times {{10}^{ - 2}}}}{\text{ min}}$
${t_{\dfrac{1}{2}}} = 25{\text{ minutes}}$

Hence the correct option is (D).

Note: The time for half reaction for a first order reaction is independent of initial concentration of reactants. All radioactive decays are first order reaction.
All first order reaction must follow the form of rate law for all time instants,
$r \propto \left[ A \right]$
A reaction can be a zero order reaction, first order reaction, second order reaction or third order reaction.