# Polonium-210 has a half life of 139 days? a) if you have 500 mg substance approximately how many milligrams will you have left in 30 days? b) if you have 500 mg substance approximately how many...

Polonium-210 has a half life of 139 days?

a) if you have 500 mg substance approximately how many milligrams will you have left in 30 days?

b) if you have 500 mg substance approximately how many days will you have only 100 mg?

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As you know, radioactive decay is an exponetial decay.

The equation for exponential decay is given by,

`(dN)/(dt) = -lambdaN`

where N - Number of activity or amount at time t and `lambda` is decay constant.

Rearranging thae above equation you get,

`(dN)/N = -lambdadt`

Integrating this gives,

`ln(N) = -lambdat+C`

where C is a constant.

Now we have to find `lambda` and C by using the given data.

at t = 0, N = 500, then,

ln(500) = C

therefore the equation changes to,

`ln(N) = -lambdat+ln(500)`

`ln(N/500) = -lambdat`

The half-time of this is 139 days, That means at 139 days N will be 250 mili grams.

`ln(250/500) = -lambda * 139`

`ln(1/2) = -lambda * 139`

`ln(2) = lambda *139`

`lambda = ln(2)/139 = 0.6931/139`

therefore the equation changes to,

`ln(N/500) = -(0.6931/139)t`

a) How many left after 30 days??

`ln(N/500) = -(0.6931/139)*30`

`ln(N/500) = -0.1496`

`N/500 = e^(-0.1496)`

`N/500 = 0.861`

N = 430.5

**Therefore after 30 days 430.5 mg will be left.**

b)How many days to get to 100 mg,

`ln(100/500) = - (0.6931/139) * t`

`ln(1/5) = -(0.6931/139) *t`

-1.6094 =-(0.6931/139) *t

t = 322.76 days.

**The days required to pass to have only 100 mg is 322.76 days.**