A radioactive element decays exponentially according to the function Q(t) = Q0ekt
.If 100 mg of the element decays to 25 mg in 12 days, find the half-life of the element.
please help
NEvermind got it its 6
100 * 1/2 * 1/2 = 25
12 days = 2 half lives
OR
1/4 = e^(12k)
ln(1/4) = 12 k
[ln(1/4)] / 12 = k
ln(1/2) = k t
t = ln(1/2) / {[ln(1/4)] / 12} = 6
To find the half-life of the element, we need to find the value of k in the exponential decay function Q(t) = Q0e^(kt). Given that 100 mg of the element decays to 25 mg in 12 days, we can substitute these values into the equation to solve for k.
Q(t) = Q0e^(kt)
25 mg = 100 mg * e^(k * 12 days)
Now, divide both sides of the equation by 100 mg to simplify it:
0.25 = e^(k * 12 days)
To isolate the exponent, we can take the natural logarithm (ln) of both sides:
ln(0.25) = ln(e^(k * 12 days))
Using the property of logarithms, ln(e^x) = x, the equation becomes:
ln(0.25) = k * 12 days
Now, solve for k by dividing both sides of the equation by 12 days:
k = ln(0.25) / 12
Using a calculator or a math software, find the value of ln(0.25) and divide it by 12 to get the value of k.
Once you have the value of k, you can find the half-life by using the formula for half-life, t1/2 = ln(2) / k.
Substitute the value of k that you found into the formula, and calculate ln(2) to find the half-life of the radioactive element.