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:

N(t) = N0 * e^(-λt)

Where:
N(t) = amount of radioactive isotope remaining after time t
N0 = initial amount of radioactive isotope
λ = decay constant
t = time

Given that the initial amount N0 is 10 kg and the remaining amount N(t) after 17 years is 3 kg, we can substitute these values into the equation:

3 = 10 * e^(-17λ)

Now we can solve for the decay constant λ. Dividing both sides of the equation by 10, we get:

0.3 = e^(-17λ)

Next, take the natural logarithm (ln) of both sides to remove the exponential:

ln(0.3) = -17λ

Finally, divide both sides of the equation by -17 to solve for the decay constant λ:

λ = ln(0.3) / -17

Using a calculator, we can find the decay constant to be approximately -0.0789.

To find the decay constant (λ) of the radioactive isotope, we can use the formula:

m(t) = m0 * e^(-λt),

where:
m(t) is the mass of the isotope at time t,
m0 is the initial mass of the isotope,
e is the base of the natural logarithm, and
t is the time elapsed.

We have the initial mass (m0) as 10 kg and the mass after 17 years (m(17)) as 3 kg. Plugging in these values into the equation, we get:

3 = 10 * e^(-λ * 17).

To find the decay constant (λ), we need to solve for it.

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 isolate λ:
ln(3/10) = -λ * 17.

3. Divide both sides of the equation by -17:
ln(3/10) / -17 = λ.

Using a calculator, we can determine the value of λ by evaluating the expression on the right side of the equation:

λ ≈ -0.14517.

Therefore, the decay constant (λ) of the isotope is approximately -0.14517.