Question

Element X decays radioactively with a half life of 5 minutes. If there are 340 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 4 grams?

Answer 1

Answer: 25.6

Explanation:

Y= a(.5)^t/h

Answer 2

32.0 min

Explanation:

The equation that describes the decay of a radioactive isotope is:

`m(t)=m_0 e^(-\lambda t)` (1)

where:

`m_0` is the mass of the element at time t = 0

`m(t)` is the mass of the element left at time t

`\lambda` is the decay constant

The decay constant is related to the half-life of the element by

`\lambda=(ln 2)/(t_(1/2))`

where

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

For element X, we have

`t_(1/2)=5 min`

So the decay constant is

`\lambda=(ln 2)/(5)=0.139 min^(-1)`

We also know that for element X:

`m_0 = 340 g` is the initial mass

`m(t)=4 g` is the final mass

So, from eq(1) we can now find the time:

`t=-(ln((m(t))/(m_0)))/(\lambda)=-(ln(4/340))/(0.139)=32.0 min`