Q.47b. What is the age in years of a bo... [FREE SOLUTION]

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Chapter 42: Q.47b. (page 1237)

What is the age in years of a bone in which the $^{14} C^{12} C$ ratio is measured to be $1.65 脳 10^{鈭� 13}$ ?

Short Answer

Expert verified

The age of bone is 17,058.783 years.

Step by step solution

Step.1.

Ratio of $^{14} C$ to $^{12} C$ in living organisms is,

$\frac{N_{0} (^{14} C)}{N_{0} (^{12} C)} = 1.3 脳 10^{鈭� 12}$

Ratio of $^{14} C$ to $^{12} C$ in bone is,

$\frac{N (^{14} C)}{N (^{12} C)} = 1.65 脳 10^{鈭� 13}$

As $N (^{12} C) = N_{0} (^{12} C)$ ,

$\begin{array}{l} \frac{N (^{14} C)}{N_{0} (^{14} C)} = \frac{1.65 脳 10^{鈭� 13}}{1.3 脳 10^{鈭� 12}} \\ = 1.27 脳 10^{鈭� 1} \\ = 0.127 \end{array}$

Therefore,

$N (^{14} C) = 0.127 N_{0} (^{14} C)$

Step.2.

Decay equation in terms of half-life is,

$\begin{array}{l} N = N_{0} {(\frac{1}{2})}^{\frac{t}{t_{1}}} \\ {(\frac{1}{2})}^{\frac{t}{t_{1}}} = \frac{N}{N_{0}} \end{array}$

Here, $N$ is the present number of $^{14} C$ atoms, $N_{0}$ represent the initial number of atoms $^{14} C$ atoms and $t_{\frac{1}{2}}$ represents the half-life of $^{14} C$ .

Substitute 5730 years for $t_{\frac{1}{2}}$ and 0.127 for $\frac{N}{N_{0}}$ ,

${(\frac{1}{2})}^{\frac{1}{5730 years}} = 0.127$

$(0.5) \frac{t}{5730 years} = 0.127$

Take natural log on both sides,

$\begin{array}{l} \frac{t}{5730 years} \ln (0.5) = \ln (0.127) \\ t = \frac{\ln (0.127)}{\ln (0.5)} (5730 years) \\ = (2.9771) (5730 years) \\ = 17, 058.783 years \end{array}$

Step.3.

Hence, the age of bone is 17,058.783 years.

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91影视

What is the age in years of a bone in which the $^{14} C^{12} C$ ratio is measured to be $1.65 脳 10^{鈭� 13}$ ?

Short Answer

Step by step solution

Step.1.

Step.2.

Step.3.

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91影视

What is the age in years of a bone in which the 14C12Cratio is measured to be 1.65脳10鈭�13?

Short Answer

Step by step solution

Step.1.

Step.2.

Step.3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

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What is the age in years of a bone in which the $^{14} C^{12} C$ ratio is measured to be $1.65 脳 10^{鈭� 13}$ ?