The half-life of strontium-90 is 28 years. How long will it take a 60 mg sample to decay to a mass of 15 mg?
56 years I reckon
To calculate the time it takes for a radioactive substance to decay to a certain mass, you need to use the formula for exponential decay:
N = N₀ * (1/2)^(t / t₁/₂),
where:
N is the final amount of the substance,
N₀ is the initial amount of the substance,
t is the time elapsed,
t₁/₂ is the half-life of the substance.
In this case, the initial mass (N₀) is 60 mg, and the final mass (N) is 15 mg. The half-life of strontium-90 (t₁/₂) is 28 years. We can rearrange the formula to solve for t:
15 mg = 60 mg * (1/2)^(t / 28 years).
Divide both sides of the equation by 60 mg:
(1/4) = (1/2)^(t / 28 years).
To remove the base-2 exponent, we can take the logarithm of both sides of the equation. Let's use the natural logarithm, ln:
ln(1/4) = ln((1/2)^(t / 28 years)).
Using a logarithmic property, we can bring the exponent down:
ln(1/4) = (t / 28 years) * ln(1/2).
Evaluate the natural logarithm of 1/4 and the natural logarithm of 1/2:
ln(1/4) = -ln(4) = -ln(2^2) = -2 * ln(2),
ln(1/2) = -ln(2).
Plug these values back into the equation:
-2 * ln(2) = (t / 28 years) * -ln(2).
Cancel out the common factor of -ln(2):
-2 = (t / 28 years).
Multiply both sides of the equation by 28 years:
-2 * 28 years = t.
The result is:
t = -56 years.
However, since time cannot be negative, it seems there might be an error or misconception in the question or calculations. Can you provide any additional information or clarify the problem?
60 mg will decay to 30mg in 28 years.
Similarly, 30mg will decay to 15 mg in another 28 years.
Figure out the answer.