Chapter 19, Problem 19.48QP

(a)

Interpretation Introduction

Interpretation: The year of the first and second volcanic eruption and the year of blackened grains found from the site of ancient Jerico are given. The percent difference in the ${}^{14}\text{C}$ decay rate in the biological samples of the volcanic eruptions and the $\frac{{}^{14}\text{C}}{{}^{12}\text{C}}$ ratio in the blackened grains compared to grains harvested last year is to be calculated.

Concept introduction: The relation between the initial and the final concentration of the carbon decay is given by the formula,

$\frac{\text{Final}\text{concentration}\text{of}{}^{\text{14}}\text{C}}{\text{Initial}\text{concentration}\text{of}{}^{\text{14}}\text{C}}={\text{e}}^{\frac{\text{Final}\text{time}\times 0.693}{\text{Half}\text{life}\text{of}{}^{\text{14}}\text{C}}}$

To determine: The percent difference in the ${}^{14}\text{C}$ decay rate in the biological samples between the two volcanic eruptions.

Answer to Problem 19.48QP

Solution:

The percent difference in the ${}^{14}\text{C}$ decay rate in the biological samples between the two volcanic eruptions is $\underset{\_}{2.15\%}$.

Explanation of Solution

Given

The first volcanic eruption occurs at $1360\text{BCE}$.

The second volcanic eruption occurs at $1628\text{BCE}$.

The present year is $2018$. Therefore, for the conversion of $\text{BCE}$ into $\text{AD}$, the year $2017$ is added to the given $\text{BCE}$ year, since $2017$ is a completed year.

Therefore, the first volcanic eruption occurred,

$1360+2017=3377$ years ago

The second volcanic eruption occurred,

$1628+2017=3645$ years ago

The half life of ${}^{14}\text{C}$ is $5730$ years.

The relation between the initial and the final concentration of the carbon decay is given by the formula,

$\frac{\text{Final}\text{concentration}\text{of}{}^{\text{14}}\text{C}}{\text{Initial}\text{concentration}\text{of}{}^{\text{14}}\text{C}}={\text{e}}^{\frac{\text{Final}\text{time}\times 0.693}{\text{Half}\text{life}\text{of}{}^{\text{14}}\text{C}}}$

Substitute the value of final time of first volcanic eruption and half life of ${}^{14}\text{C}$ in the above equation.

$\begin{array}{c}\frac{\text{Final}\text{concentration}\text{of}{}^{\text{14}}\text{C}}{\text{Initial}\text{concentration}\text{of}{}^{\text{14}}\text{C}}={\text{e}}^{-\frac{3377\times 0.693}{\text{5730}}}\\ ={\text{e}}^{-0.4084}\\ =0.665\times 100\\ =66.5\%\end{array}$

Substitute the value of final time of second volcanic eruption and half life of ${}^{14}\text{C}$ in the above equation.

$\begin{array}{c}\frac{\text{Final}\text{concentration}\text{of}{}^{\text{14}}\text{C}}{\text{Initial}\text{concentration}\text{of}{}^{\text{14}}\text{C}}={\text{e}}^{-\frac{3645\times 0.693}{\text{5730}}}\\ ={\text{e}}^{-0.4408}\\ =0.6435\times 100\\ =64.35\%\end{array}$

Therefore, the percent difference in the ${}^{14}\text{C}$ decay rate in the biological samples between the two volcanic eruptions is,

$66.5\%-64.35\%=\underset{\_}{2.15\%}$

(b)

Interpretation Introduction

To determine: The $\frac{{}^{14}\text{C}}{{}^{12}\text{C}}$ ratio in the blackened grains compared to grains harvested last year is to be calculated.

Answer to Problem 19.48QP

Solution:

The $\frac{{}^{14}\text{C}}{{}^{12}\text{C}}$ ratio in the blackened grains compared to grains harvested last year is to be calculated is $\underset{\_}{0.6683}$.

Explanation of Solution

Given

The year of blackened grains found from the site of ancient Jerico is $1315\text{BCE}$.

The present year is $2018$. Therefore, for the conversion of $\text{BCE}$ into $\text{AD}$, the year $2017$ is added to the given $\text{BCE}$ year, since $2017$ is a completed year.

The molar mass of hydrogen gas is,

$2\times 1.0079\text{g}/\text{mol}=2.0158\text{g}/\text{mol}$

The year of blackened grains found from the site of ancient Jerico is,

$1315+2017=3332$ years ago

The relation between the initial and the final concentration of the carbon decay is given by the formula,

$\frac{\text{Final}\text{concentration}\text{of}{}^{\text{14}}\text{C}}{\text{Initial}\text{concentration}\text{of}{}^{\text{14}}\text{C}}={\text{e}}^{\frac{\text{Final}\text{time}\times 0.693}{\text{Half}\text{life}\text{of}{}^{\text{14}}\text{C}}}$

Substitute the value of final time and half life of ${}^{14}\text{C}$ in the above equation.

$\begin{array}{c}\frac{\text{Final}\text{concentration}\text{of}{}^{\text{14}}\text{C}}{\text{Initial}\text{concentration}\text{of}{}^{\text{14}}\text{C}}={\text{e}}^{-\frac{3332\times 0.693}{\text{5730}}}\\ ={\text{e}}^{-0.403}\\ =0.6683\end{array}$

Therefore, the $\frac{{}^{14}\text{C}}{{}^{12}\text{C}}$ ratio in the blackened grains compared to grains harvested last year is to be calculated is $\underset{\_}{0.6683}$.

Conclusion:

1. a. The percent difference in the ${}^{14}\text{C}$ decay rate in the biological samples between the two volcanic eruptions is $\underset{\_}{2.15\%}$.
2. b. The $\frac{{}^{14}\text{C}}{{}^{12}\text{C}}$ ratio in the blackened grains compared to grains harvested last year is to be calculated is $\underset{\_}{0.6683}$