Chapter 24: Q24.47 P (page 1104)

A rock that contains ${3.1×10}^{-15}$ mol of $^{232} Th$ ( $t_{1/2} {=1.4×10}^{10} yr$ ) has ${9.5×10}^{4}$ ﬁssion tracks, each track representing the ﬁssion of one atom of $^{232} Th$ . How old is the rock?

Short Answer

Expert verified

The age of the rock is $3.78 \times 10^{6}$ years

Step by step solution

Radioactive decay

Radioactive decay follows the first order kinetics. The expression for rate constant is shown below:

$k = \frac{2.303}{t} \ln \frac{[N_{t}]}{[N_{0}]}$

Calculation

- A rock contains_{$N_{t} {=3.1×10}^{-15} {mol}^{232} Th$}

- Half-life of_{$^{232} {Thist}_{1/2} {=1.4×10}^{10} yr$}

- The number of decayed $^{232} {Thatomsis9.5×10}^{4}$ atoms

- The number of $232$ Th atoms in a rock is

_{$\begin{matrix} N_{t} {=3.1×10}^{-15} mol× \frac{{6.022×10}^{23} atoms}{1.0mol} \\ {=1.86682×10}^{9} atoms \end{matrix}$}

First, let us calculate the initial amount of $^{232} Th (N_{0})$

_{$\begin{matrix} N_{0} {=N}_{t} {+9.5×10}^{4} atoms \\ {=1.86682×10}^{9} {atoms+9.5×10}^{4} atoms \\ {=1.866915×10}^{9} atoms \end{matrix}$}

Now, let us calculate the decay constant

$\begin{matrix} t_{1/2} = \frac{ln(2)}{k} \\ k= \frac{ln(2)}{t_{1/2}} \\ = \frac{0.693}{{1.4×10}^{10} yr} \\ {=4.95×10}^{-11} {yr}^{-1} \end{matrix}$

Therefore, the age of the rock is (t)

_{$\begin{matrix} ln (\frac{N_{t}}{N_{0}}) =-kt \\ t= \frac{-ln (\frac{N_{N}}{N_{0}})}{k} \\ = \frac{-ln}{{4.95×10}^{-11} {yr}^{-1}} \\ {=3.78×10}^{6} yr \end{matrix}$}

Hence,the age of the rock is_{$3.78 \times 10^{6}$} years.

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A rock that contains ${3.1×10}^{-15}$ mol of $^{232} Th$ ( $t_{1/2} {=1.4×10}^{10} yr$ ) has ${9.5×10}^{4}$ ﬁssion tracks, each track representing the ﬁssion of one atom of $^{232} Th$ . How old is the rock?

Short Answer

Step by step solution

Radioactive decay

Calculation

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A rock that contains3.1×10-15 mol of232Th ( t1/2=1.4×1010yr) has 9.5×104ﬁssion tracks, each track representing the ﬁssion of one atom of 232Th. How old is the rock?

Short Answer

Step by step solution

Radioactive decay

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Chemistry Textbooks

Kinetics

Ionic and Molecular Compounds

Chemical Analysis

Physical Chemistry

Nuclear Chemistry

Chemistry Branches

Study anywhere. Anytime. Across all devices.

A rock that contains ${3.1×10}^{-15}$ mol of $^{232} Th$ ( $t_{1/2} {=1.4×10}^{10} yr$ ) has ${9.5×10}^{4}$ ﬁssion tracks, each track representing the ﬁssion of one atom of $^{232} Th$ . How old is the rock?