Question

A certain watch’s luminous glow is due to zinc sulfide paint that is energized by beta particles given off by tritium, the radioactive hydrogen isotope 3 H, which has a half-life of 12.3 years. This glow has about 1/10 of its initial brightness. How many years old is the watch? g

Answer 1

Answer:

The watch is 40.9 years old.

Explanation:

To know how many years old is the watch we need to use the following equation:

$I_{(t)} = I_{0}e^{-\lambda t}$   (1)

Where:

$I_{(t)}$: is the brightness in a time t = (1/10)I₀

$I_{0}$: is the initial brightness

λ: is the decay constant of tritium

The decay constant is given by:

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

Where:

$t_{1/2}$: is the half-life of tritium = 12.3 years

By entering equation (2) into (1)  we have:

$I_{(t)} = I_{0}e^{-\lambda t} = I_{0}e^{-\frac{ln(2)}{t_{1/2}}t}$

$\frac{I_{(t)}}{I_{0}} = e^{-\frac{ln(2)}{t_{1/2}}t}$

By solving the above equation for "t" we have:

$ln(\frac{I_{(t)}}{I_{0}}) = -\frac{ln(2)}{t_{1/2}}t$

$t = -\frac{ln(\frac{I_{(t)}}{I_{0}})}{\frac{ln(2)}{t_{1/2}}} = -\frac{ln(\frac{1}{10})}{\frac{ln(2)}{12.3}} = 40.9 y$

Therefore, the watch is 40.9 years old.

 

I hope it helps you!