Chapter 30, Problem 61GP

To determine

To Find: The age of the bottle of wine.

Answer to Problem 61GP

  $\mathrm{t}=40.9\text{ yrs}$

Explanation of Solution

Given info:

Half-life of tritium, $\mathrm{t}=12.33\text{ yr}$

Formula used:

Number of half-life can be given as:

  $\frac{\mathrm{N}}{{\mathrm{N}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}$

Here,

  ${\mathrm{N}}_0$ is the initial number of nuclei present

  $\mathrm{N}$ is the number remaining after time t

  $\mathrm{n}$ is the number of half-lives

Calculation:

Number of half-lives can be given as:

  $\begin{array}{l}\frac{\mathrm{N}}{{\mathrm{N}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ \frac{1}{10}={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ \mathrm{n}=-\frac{\ln \left(0.10\right)}{\ln \left(2\right)}\\ \mathrm{n}=3.32\end{array}$

So, the age of the bottle of the wine is given by:

  $\begin{array}{l}\mathrm{t}=\mathrm{n}{\mathrm{T}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ \mathrm{t}=\left(3.32\right)\left(12.33\text{ yr}\right)\\ \mathrm{t}=40.9\text{ yrs}\end{array}$

Conclusion:

Thus, the age of the bottle of wine is $\mathrm{t}=40.9\text{ yrs}$ .