Chapter 14, Problem 1KSP

Interpretation Introduction

Interpretation:

The half-life of the isotope is to be calculated.

Concept introduction:

The amount of time required for the concentration of a radioactive decay to reduce to half of its original valueis known as half-life $\left(t_{1/2}\right)$.

The expression for the first order reaction is as follows:

$A_t=A_0{e}^{-kt}$

Here, $t$ is the time, $k$ is rate constant, $e$ is the base of the natural logarithm and $A_0$ is the initial concentration, and $A_t$ is the concentration at time $t$.

Answer to Problem 1KSP

Solution: Option (d)

Explanation of Solution

Reason for the correct option:

Let the initial concentration $\left(A_0\right)$ be 100 and the concentration at time $t$ $\left(A_t\right)$ be 10

Calculate the rate constant from the expression of first order reaction as follows:

$A_t=A_0{e}^{-kt}$

Rearrange the above equation for rate constant as follows:

$k=-\ln \left(\frac{A_t}{A_0}\right)\left(\frac{1}{t}\right)$

Substitute 10 for $A_t$, 100 for $A_0$, and $218h$ for $t$ in the above equation,

Thus,

$\begin{array}{c}k=-\ln \left(\frac{10}{100}\right)\left(\frac{1}{218\text{ h}}\right)\\ =0.01055\text{ h}\end{array}$

Calculate the half-life of the isotope as follows:

$t_{1/2}=\frac{0.693}{k}$

Substitute $0.01055\text{ h}$ for $k$ in the above equation,

Thus,

$\begin{array}{c}{t}_{1/2}=\frac{0.693}{0.01055\text{ h}}\\ =65.6\text{ h}\end{array}$

So, the half-life of the isotope is $65.6\text{ h}$

Hence, option (d) is correct.

Reason for the incorrect option:

Option (a) is incorrect because the calculated value of half-life of the isotope does not match with the given value in option (a).

Option (b) is incorrect because the calculated value of half-life of the isotope does not match with the given value in option (b).

Option (c) is incorrect because the calculated value of half-life of the isotope does not match with the given value in option (c).

Option (e) is incorrect because the calculated value of half-life of the isotope does not match with the given value in option (e).

Hence, options (a), (b), (c), and (e) are incorrect.