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

The watch is 40.9 years old.

Step-by-step 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 = (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^{-(ln(2))/(t_(1/2))t}`

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

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

`ln((I_((t)))/(I_(0))) = -(ln(2))/(t_(1/2))t`

`t = -(ln((I_((t)))/(I_(0))))/((ln(2))/(t_(1/2))) = -(ln((1)/(10)))/((ln(2))/(12.3)) = 40.9 y`

Therefore, the watch is 40.9 years old.

I hope it helps you!