A sample of a radioactive isotope had an initial mass of 800 mg in the year 2000 and decays exponentially over time. A measurement in the year 2005 found that the sample's mass had decayed to 190 mg. What would be the expected mass of the sample in the year 2009, to the nearest whole number?
To find the expected mass of the sample in the year 2009, we need to determine the decay constant of the isotope.
The decay constant, denoted by λ, can be found using the formula:
initial mass = final mass * e^(λ * time)
Where e is the base of the natural logarithm and is approximately 2.71828.
Let's plug in the given values to solve for the decay constant:
800 mg = 190 mg * e^(λ * 5)
Dividing both sides by 190 mg gives:
800 mg / 190 mg = e^(λ * 5)
Approximately:
4.21053 = e^(λ * 5)
Taking natural logarithm of both sides gives:
ln(4.21053) = λ * 5
Using a calculator, we find:
λ ≈ 0.39379
Now that we have the decay constant, we can find the expected mass of the sample in the year 2009.
Using the same formula, we have:
expected mass = final mass * e^(λ * time)
Substituting the values:
expected mass = 190 mg * e^(0.39379 * 9)
For simplicity, let's round the decay constant to four decimal places:
expected mass = 190 mg * e^(0.3938 * 9)
Approximately:
expected mass ≈ 190 mg * e^(3.5442)
Using a calculator, we find:
expected mass ≈ 190 mg * 34.4717102
expected mass ≈ 6559.3251399 mg
Rounding to the nearest whole number:
expected mass ≈ 6559 mg
Therefore, the expected mass of the sample in the year 2009 would be approximately 6559 mg.