Q24.47 P A rock that contains3.1脳10-15聽... [FREE SOLUTION]

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Chapter 24: Q24.47 P (page 1104)

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

Short Answer

Expert verified

The age of the rock is $3.78 脳 10^{6} 鈥�$ 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_{$N_{t} {=3.1脳10}^{-15} {mol}^{232} Th$}

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

- The number of decayed $^{232} {罢丑补迟辞尘蝉颈蝉9.5脳10}^{4}$ atoms

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

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

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

_{$\begin{matrix} N_{0} {=N}_{t} {+9.5脳10}^{4} atoms \\ {=1.86682脳10}^{9} {补迟辞尘蝉+9.5脳10}^{4} atoms \\ {=1.866915脳10}^{9} atoms \end{matrix}$}

Now, let us calculate the decay constant

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

Therefore, the age of the rock is (t)

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

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

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91影视

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

Short Answer

Step by step solution

Radioactive decay

Calculation

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Most popular questions from this chapter

Recommended explanations on Chemistry Textbooks

Nuclear Chemistry

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91影视

A rock that contains3.1脳10-15 mol of232Th ( t1/2=1.4脳1010yr) has 9.5脳104铿乻sion tracks, each track representing the 铿乻sion 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

Nuclear Chemistry

Kinetics

Inorganic Chemistry

Making Measurements

Chemical Reactions

The Earths Atmosphere

Study anywhere. Anytime. Across all devices.

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