Question

# The half life period of a first order reaction is 10 minutes. The time required for the concentration of the reactant to change from 0.08 M to 0.02 M is:a.) 10 minb.) 20 minc.) 30 mind.) 40 min

Hint: The first order reaction is the one in which the rate depends only on the one reactant concentration.
We have for the first order reaction
$\ln \frac{{\left[ {{A_0}} \right]}}{{\left[ {{A_t}} \right]}} = kt$
Where $\left[ {{A_0}} \right]$ is the initial concentration
$\left[ {{A_t}} \right]$ is the final concentration
k is the rate constant and
t is the time taken for the reaction
Thus, by putting the values of certain variables, we can find the time taken for the reaction.

We know that for first order reaction, the half life time period can be given by –
${t_{\frac{1}{2}}} = \frac{{0.693}}{k}$
Where k is the rate constant.
In the question, a half life period is given to us which is 10 min.
So, by filling the value of half life time, we can get value of rate constant (k) as-
$k = \frac{{0.693}}{{{t_{\frac{1}{2}}}}}$
k = $\frac{{0.693}}{{10}}$
k = 0.0693 ${\min ^{ - 1}}$
Further, for the first order reaction, we have the formula as-
$\ln \frac{{\left[ {{A_0}} \right]}}{{\left[ {{A_t}} \right]}} = kt$
Where $\left[ {{A_0}} \right]$ is the initial concentration
$\left[ {{A_t}} \right]$ is the final concentration
k is the rate constant and
t is the time taken fo$t = \frac{{\ln 4}}{{0.0693}}\min$r the reaction
As we have to find the time for the reaction. Thus, the equation can be modified as-
$\frac{{\ln \frac{{\left[ {{A_0}} \right]}}{{\left[ {{A_t}} \right]}}}}{k} = t$
So, from the question we have initial value of concentration = 0.08 M
The final value of the concentration = 0.02 M
And rate constant = 0.0693 ${\min ^{ - 1}}$
Thus, by putting the values; we have
$t = \frac{{\ln \frac{{\left[ {0.08M} \right]}}{{\left[ {0.02M} \right]}}}}{{0.0693}}\min$

$t = \frac{{1.38}}{{0.0693}}\min$
t = 19.91 min
So, the value is approximately 20 minutes.

Thus, option b.) is our correct answer.

Note: The half life time of a first order reaction is independent of the reactant concentration. Thus, for one type of reaction if we have reactant concentration different in two sets of reactions, still both will show the same half life time. The half life time of one type of first order reaction will however be different for another type.