Chapter 21, Problem 21.80QA

Interpretation Introduction

To explain:

If the radioactive dating distinguishes between debris from the eruptions of Mt. Vesuvius that occurred in years 472 and 512.

Answer to Problem 21.80QA

Solution:

Yes; we can distinguish between the radioactive dating debris from the eruptions of Mt. Vesuvius that occurred in years 472 and 512.

Explanation of Solution

1) Concept:

The half-life of $C$ is taken from the Table 21.2. Using the half-life, we get the value of rate constant. Plugging the values of rate constant and given time in the first order radioactive decay equation, we can calculate the ratio $\frac{C}{C}$  for both years. Then the difference between both ratios and given uncertainty in determining the ratio is compared.

2) Formula:

i) $t_{\frac{1}{2}}=\frac{0.693}{k}$

ii)  $ln\frac{{\left[X\right]}_t}{{\left[X\right]}_0}=-kt$

where, ${\left[X\right]}_t$ is the final concentration, ${\left[X\right]}_0$ is initial concentration, $k$ is rate constant,  $t$ is time, and $t_{\frac{1}{2}}$ is half-life.

3) Given:

i) Uncertainty in determining the ratio $=0.1 \%$

ii) $t_1=472 years$

iii)  $t_2=512 years$

4) Calculations:

The half-life of $C$ is taken from the Table 21.2 as $5730 years$.

So, the rate constant is calculated as follows:

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

$5730 yr=\frac{0.693}{k}$

$k=\frac{0.693}{5730 yr}=1.209\times {10}^{-4} {yr}^{-1}$

Calculating $C/ C$ for $t_1=472 years$:

$ln\frac{{\left[X\right]}_t}{{\left[X\right]}_0}=-kt$

$\ln \left[\frac{C}{C}\right]=-1.209\times {10}^{-4}{yr}^{-1}\times 472 yr$

$\left[\frac{C}{C}\right]= e^{-1.209\times {10}^{-4} {yr}^{-1}\times 472 yr}=0.9445$

$\left[\frac{C}{C}\right]=94.5 \%$

Calculating $C/ C$ for $t_1=512 years$:

$ln\frac{{\left[X\right]}_t}{{\left[X\right]}_0}=-kt$

$\mathrm{l}\mathrm{n}\left[\frac{C}{C}\right]=-1.209\times {10}^{-4} {yr}^{-1}\times 512 ys$

$\left[\frac{C}{C}\right]= e^{-1.209\times {10}^{-4} {yr}^{-1}\times 512 yr}=0.9400$

$\left[\frac{C}{C}\right]=94.0 \%$

Therefore, the difference in $\frac{C}{C}$ for both time is $=94.5\%-94.0\%=0.5\%$

While the uncertainty in determining the ratio is $0.1 \%$, this is less than the difference in the values of ratio $\frac{C}{C}$. So, we can distinguish between the radioactive dating debris from the eruptions of Mt. Vesuvius that occurred in years 472 and 512.

Conclusion:

The difference between carbon-14: carbon 12 for both years is larger than the uncertainty in determining the ratio.