A researcher made160mg of radioactive sodium(Na24) and found that there was only 20 mg left after 45 hours. a). What is the half-life of Na24?

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lemjay eNotes educator| Certified Educator

Since the amount of NA24 is decreasing, let's apply the formula of exponential decay, which is:

`y = ae^(kt)`

where y- amount of substance at time t, a - amount of substance at the start and k - rate of decay.

To solve for the half-life of NA24, we have to determine k first. So, substitute y=20, a=160 and t=45 to the formula.


To simplify the equation, divide both sides by 160.



To bring down k, take the natural logarithm of both sides.

`ln0.125 = lne^(45k)`

Then at the right side of the equation,  apply the power property of logarithm which is ` ln x^m= m*ln x ` .


Note that ln e= 1.

`ln 0.125=45k*1`


And divide both sides by 45.



Now that we know the value of k, let's solve proceed to solve for half-life of NA24.

Note that  half-life means the time when y = a/2. So to determine t, substitute y = a/2 and k=-0.046 to the formula of exponential decay and follow the steps above.

`y =ae^(kt)`


Divide both sides by a.



Take the natural logarithm of both sides.

`ln(1/2)=ln e^(-0.046t)`

Then, apply the power property of logarithm.




And, divide both sides by -0.046.



Hence, the half-life of NA24 is 15.068 hours.

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