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.