Question

Radium-221 has a half-life of 30 seconds. How long will it take for 94% of a sample to decay?

Answer 1

`t=121.749s`

Explanation:

Now, the equation for radiation of a radioactive substance is given by the equation
`N=N_oe^{-lambdat`
where `N` implies the weight of the original radioactive mass remaining, `N_o` being the original weight of the radioactive mass before the start of this experiment, `lamda` being the degradation constant, and `t` being time taken.

Now, apparently for the entire radioactive mass of Radium-221 taken to degrade such that only half of the original mass remains, 30 seconds pass by.
So, `t=30s`, `N=(1-1/2)N_o=N_o/2`, substituting them into the equation gives us
`cancelN_o/2=cancelN_oe^{-30k}`

Applying ln (log base e) to both sides of the equation, I get
`-ln2=-30lambda`
So that means `lambda=ln2/30` (further simplifying can be a hassle, so I'll keep it like this)

So the equation in total is `N=N_oe^{-ln2/30t}`

Now, we're trying to find out how many seconds pass by until `94%` of the total mass originally taken is degraded into something else. So, `N=(1-94/100)N_o=6/100N_o`
Substituting that into the equation leads us to
`6/100cancelN_o=cancelN_oe^{-ln2/30t}`
`3/50=e^{-ln2/30t`
Apply ln on both sides give us
`ln(3/50)=-ln2/30t`
You better get your calculator out here, it's turning out to be a tad windy.
`-2.813=-0.693/30t`

Now, the rest is up to you, solve this and save the human race from nuclear annihilation!