How long will it take for a sample of&nbsp;100&nbsp;mg to decay to 98 mg with a half life of 240 days?

Question

How long will it take for a sample of&amp;nbsp;100&amp;nbsp;mg to decay to 98 mg with a half life of 240 days?

alenahelenash · Accepted Answer

To solve this problem, we can use the equation for radioactive decay:

N (t) = N_{0} e^{- λ t}

where

N (t)

is the amount of radioactive substance at time

t

,

N_{0}

is the initial amount,

λ

is the decay constant, and

t

is time.
We know that the half-life

T_{1 / 2}

of the substance is 240 days. This means that after one half-life, the amount of substance remaining will be half of the initial amount:

N (240) = \frac{1}{2} N_{0}

We can rearrange this equation to solve for the decay constant:

λ = \frac{\ln 2}{T_{1 / 2}}

Substituting the given values, we get:

λ = \frac{\ln 2}{240} \approx 0.00289

d a y^{- 1}

Now we can use this value of

λ

to find how long it will take for the sample to decay from 100 mg to 98 mg:

N (t) = N_{0} e^{- λ t}

Substituting

N_{0} = 100

mg and

N (t) = 98

mg, we get:

98 = 100 e^{- 0.00289 t}

Dividing both sides by 100 and taking the natural logarithm of both sides, we get:

\ln (\frac{98}{100}) = - 0.00289 t

Simplifying, we get:

t \approx 239.8

days
Therefore, it will take approximately 239.8 days for the sample to decay from 100 mg to 98 mg with a half-life of 240 days.

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