Question

the element thorium has a half life of 24 days and undergoes beta decay. calculate the time it would take for 10 grams to decay to 1.25 grams

Answer 1

`72 \; \text{days}`

It takes a half life
`t_(1/2) = 24 \; \text{days}` for the mass of thorium in a sample of to decrease by
`1/2`, two half lives
`2 \; t_(1/2) = 48 \; \text{days}` to decrease by yet another
`1/2`- which leaves thorium atoms of mass
`1/4` that of the initial mass in the sample. Another half life
`t_(1/2) = 24 \; \text{days}` would bring this ratio down to
`1/2 * 1/4 = 1/8`. It happens that

`1.25/ 10 \\= 1 /8`

Meaning that it would take precisely

`3 \; t_(1/2) = 72 \; \text{days}`

for the mass of thorium in the
`10 \; \text{gram}`-sample to reach
`1.25 \; \text{g}`.

Note, that it might take a calculator to find the time required in case no integer powers of
`1/2 = 0.5` (
`3` in this case) matches the desired ratio or percentage. An exponential decay formula (which naturally works for this question as well) is given below:

`t = t_(1/2) \cdot log_2((m_0/m))`