Question

1) A substance has a half life of 20 years. what percentage would be left after 40 years?

2)After 4 half lifes of uranium, 10 grams of the uranium remains. how much uranium did you start with?

Answer 1

1) The half-life is the time required for a substance to reduce to half its initial value. In formulas:

`(m(t))/(m_0) = ( (1)/(2) )^{t/t_(1/2)}` (1)
where
m(t) is the amount of substance left at time t
m0 is the initial mass

`t_(1/2)` is the half-life

In this problem, the half-life of the substance is 20 years:

`t_(1/2) = 20 y`
therefore, the fraction of sample left after t=40 years will be

`(m(t))/(m_0)=( (1)/(2))^ (40 y)/(20 y) = ( (1)/(2))^2 = (1)/(4)`

So, only 1/4 of the original sample will be left, which corresponds to 25%.

2) We can use again formula (1), by re-arranging it:

`m_0 = \frac{m(t)} {( (1)/(2) )^{ (t)/(t_(1/2) )}}`
If we use m(t)=10 g (mass of uranium left at time t), and
`t=4 t_(1/2)` (the time is equal to 4 half lifes), we get

`m_0 = (10 g)/( ((1)/(2))^4 ) =16 \cdot 10 g = 160 g`
So, the initial sample of uranium was 160 g.