SOLUTION: 3. An isotope of sodium, Na^24, has a half-life of 15 hours. How many hours will it take to decay to 2^(-4) grams, if the initial mass is 4 grams?

Question 314580: 3. An isotope of sodium, Na^24, has a half-life of 15 hours. How many hours will it take to decay to 2^(-4) grams, if the initial mass is 4 grams?
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
3. An isotope of sodium, Na^24, has a half-life of 15 hours. How many hours will it take to decay to 2^(-4) grams, if the initial mass is 4 grams?
.
Exponential growth/decay formula:
A = Ne^(rt)
where
A is amount after time t
N is the initial amount
r is the rate of growth/decay
t is time
.
Since we know half-life is 15 hours
Let N = initial amount
then
A must by N/2 giving you
N/2 = Ne^(15r)
Solving for r:
1/2 = e^(15r)
ln(1/2) = 15r
ln(1/2)/15 = r
.
Our formula is now:
A = Ne^(tln(1/2)/15)
We can now use the above to answer:
How many hours will it take to decay to 2^(-4) grams, if the initial mass is 4 grams?
2^(-4) = 4e^(tln(1/2)/15)
2^(-4)/4 = e^(tln(1/2)/15)
ln(2^(-4)/4) = tln(1/2)/15
ln(2^(-4)/4)/(ln(1/2)/15) = t
ln(.015625)/(ln(1/2)/15) = t
90 hours = t

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