Question

Watch dials are often painted with a compound containing tritium, which has a half-life of 12.3 years. When the tritium atom undergoes beta decay the emitted electron strikes a fluorescent pigment, which then glows.If your tritium-dial watch is giving off a certain amount of light, how many half-lives must pass until the watch gives off only one-eighth as much light?

Answer 1

Answer:

3 half-lives

Explanation:

The decay rate of the watch light intensity is given by:

$I_{t} = I_{0} \cdot e^{(-\lambda \cdot t)}$  

where $I_{t}$: is the intensity of light that changes with time, $I_{0}$: is the initial intensity of light, λ: is the decay constant and t: is the decay time

For light to be only one-eighth as much light, it means:

$\frac{I_{0}}{8} = I_{0} \cdot e^{(-\lambda \cdot t)}$

$Ln (\frac{1}{8}) = -\lambda \cdot t$

$t = - \frac{Ln (\frac{1}{8})}{\lambda}$ (1)

The half-life $t_{\frac{1}{2}}$ of the tritium decay is:

$t_{\frac{1}{2}} = \frac{Ln(2)}{\lambda}$        

$\lambda = \frac{Ln(2)}{t_{\frac{1}{2}}}$ (2)

By entering equation (2) into equation (1), we can find the time that must pass until the watch gives off only one-eighth as much light:

$t = - \frac{Ln (\frac{1}{8})}{Ln(2)} \cdot t_{\frac{1}{2}}}$

$t = 36.9y$

Which correspond to the next half-lives:

$t = \frac{1 half-lives}{12.3y} \cdot 36.9y = 3 half-lives$

So, until the watch gives off only one-eighth as much light must pass 3 half-lives.

I hope it helps you!