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).