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^(-λt)
where:
N(t) is the quantity of the substance at time t
N₀ is the initial quantity of the substance
λ is the decay constant
t is the time
In this case, we are given:
N₀ = 10 kg (initial quantity)
N(t) = 3 kg (quantity after 17 years)
t = 17 years
Substituting these values into the formula, we get:
3 = 10 * e^(-λ * 17)
To find the value of the decay constant λ, we need to solve this equation for λ.
1. Divide both sides of the equation by 10:
3/10 = e^(-λ * 17)
2. Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential:
ln(3/10) = ln(e^(-λ * 17))
ln(3/10) = -λ * 17
3. Divide both sides of the equation by -17:
ln(3/10) / -17 = λ
Using a calculator, we can evaluate the left side of the equation to find the value of λ:
λ ≈ -0.089
Therefore, the decay constant of the isotope is approximately -0.089 (per year).