A 10 kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.
One post is enough, please.
To find the decay constant of the isotope, we need to use the formula for radioactive decay:
N(t) = N0 * e^(-λt)
Where:
N(t) = current quantity of the isotope after time t
N0 = initial quantity of the isotope
λ = decay constant
t = time in years
Given that the initial quantity N0 is 10 kg, and the current quantity N(t) is 3 kg after 17 years, we can substitute these values into the formula:
3 kg = 10 kg * e^(-λ * 17 years)
To solve for the decay constant (λ), we need to rearrange the equation:
e^(-λ * 17 years) = 3 kg / 10 kg
Now, take the natural logarithm (ln) of both sides to isolate the exponential term:
ln(e^(-λ * 17 years)) = ln(3 kg / 10 kg)
Using the property ln(e^x) = x, we can simplify the equation:
-17λ = ln(3/10)
Finally, solving for the decay constant (λ):
λ = ln(3/10) / -17 years
Using a calculator, the value of the natural logarithm of (3/10) is approximately -1.20397. Since the time is given in years, the decay constant (λ) is:
λ ≈ (-1.20397) / 17 years
Therefore, the decay constant of the isotope is approximately -0.07088 per year.