A radioactive substance decays from 85mg to 22.1mg in years according to the exponential decay model

Question

A radioactive substance decays from 85mg to 22.1mg in years according to the exponential decay model y=ae^-bx, where a is the initial amount and y is the amount remaining after years.

 

Find the b-value.  

Round to the nearest hundredth, if needed.

b=    ?   

Use the EXACT b-value to write the exponential decay model for this substance with initial amount 85 mg, then use that model to find the half-life.

Find the half-life.

Round to the nearest hundredth of a year, if needed.

half-life ≈ __?__years

Answer 1

Using the radioactive model provided (I'll use N(t) = y as the amount available after some number of years = t and N₀ = a as the initial amount)...

N(t) = N₀e^-bt

We know the initial amount was 85 mg --> N₀. e also know that some amount of time later we have 22.1 mg remaining. So...

N(t)/N₀ = e^-bt --> ln(N(t)/N₀) = -bt --> t = -ln(N(t)/N₀) / b or b = -ln(N(t)/N₀) / t

But we don't know the time in years (as specified in the problem) or the b-value yet. If we know the time it takes to get to 22.1mg, then we can compute b. Without the time to get to 22.1mg we cannot compute the b-value.

We can solve for the half-life in terms of the b-value to determine the number of years to reduce the sample from the initial amount to half its value...

N(t) = 1/2 * N₀ at the half-life...

N(t)/N₀ = 1/2 = e^-bt' (Here the t' is the time need to reduce the initial sample to half)

-ln(2)/b = t' (half-life time once we figure out b)

However, 85/2 = 42.5mg and 42/5 = 21.25mg so we know there has been only approximately two "half-life" periods to get very close to 22.1mg remaining. With the b value we would have calculated from above, we can compute the half-life time period.