A 10 kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.

In en.wikipedia type: "Half-life"

and read text, and retype formulas.

So:

N(t)=N0*e^(-lambda*t)

N(t)=N(17)=3

N0=10

t=17

3=10*e^(-lambda*17) Divide with 10

3/10=e^(-lambda*17)

e^(-lambda*17)=3/10

-lambda*17=ln(3/10)Divide with 17

-lamba=ln(0.3)/17

lamda=-ln(0.3)/17

lamda=-(-1,203972804326)/17

lamda=1,203972804326/17

lamda=0,07082193

To find the decay constant of the isotope, we can use the exponential decay formula:

N(t) = N0 * e^(-λt),

where N(t) is the quantity of the isotope at time t, N0 is the initial quantity, λ is the decay constant, and e is the mathematical constant approximately equal to 2.71828.

In this case, we know the initial quantity N0 is 10 kg, and after 17 years, the quantity N(t) is 3 kg. Plugging these values into the exponential decay formula, we get:

3 = 10 * e^(-λ * 17).

To solve for the decay constant (λ), we need to isolate it. Divide both sides of the equation by 10:

3 / 10 = e^(-λ * 17).

Take the natural logarithm (ln) of both sides of the equation to remove the exponential term:

ln(3 / 10) = -λ * 17.

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

λ = -ln(3 / 10) / 17.

Using a calculator, we can evaluate the right side of the equation to find the decay constant λ.