A 10 kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.
To find the decay constant of the isotope, we can use the formula for exponential decay:
N(t) = N₀ * e^(-kt)
Where:
- N₀ is the initial quantity of the substance
- N(t) is the quantity of the substance at time t
- k is the decay constant
- e is the base of natural logarithms (approximately 2.71828)
In this case, we are given that the initial quantity N₀ is 10 kg and the quantity after 17 years N(t) is 3 kg. Let's substitute these values into the equation:
3 kg = 10 kg * e^(-k * 17 years)
To find the value of k, we need to solve this equation. We can do this by isolating k and using logarithms to get rid of the exponential:
e^(-k * 17 years) = 3 kg / 10 kg
Taking the natural logarithm (ln) on both sides of the equation:
ln(e^(-k * 17 years)) = ln(3 kg / 10 kg)
Using the property of logarithms that ln(e^x) = x:
-17k years = ln(3 kg / 10 kg)
Now, we can solve for k by dividing both sides by -17 years:
k = ln(3 kg / 10 kg) / -17 years
Using a calculator, we can evaluate the right side of the equation:
k ≈ -0.065 years⁻¹
Therefore, the decay constant of the isotope is approximately -0.065 years⁻¹.