Question

Nuclear disarmament could be accomplished if weapons were not “replenished.” The tritium in warheads decays to helium with a half-life of 12.26 yr and must be replaced or the weapon is useless. What fraction of the tritium is lost in 5.50 yr?

Short answer

In 5.50 years, 0.2673 percent of the tritium is lost.

Step-by-step solution

Definition of Nuclear disarmament

The act of lowering or removing nuclear weapons is known as nuclear disarmament. It might also be the final state of a nuclear-free world, in which all nuclear weapons have been eradicated.

Calculating fraction of the tritium

The change in the number of nuclei (N) divided by the change in time is the decay rate, or activity (A), of a radioactive sample (t).

$A=\frac{-4N}{t}$

We put N_tequal to $\frac{1}{2}{N}_0$ and solve for t_1/2 to get the half-life (t_1/2).

$\ln \frac{N_0}{\frac{1}{2}{N}_0}=k{t}_{\frac{1}{2}}$

Rearranging the formula that was derived,

$t_{\frac{1}{2}}=\frac{\ln 2}{k}$

$N_t=N_0{e}^{-kt}$or$\ln \frac{N_t}{N_0}=-kt$ is an expression for calculating the number of nuclei left after a given period t.

Calculating fraction of tritium

We must use the half-life formula to find the constant (k) in this situation.

$t_{\frac{1}{2}}=\frac{\ln 2}{k}\to k=\frac{\ln 2}{t_{\frac{1}{2}}}$

$k=\frac{\ln 2}{t_{\frac{1}{2}}}=\frac{\ln 2}{12.26yr}=\frac{0.05653729}{yr}$

We can solve for the value of fraction lost using the expression for calculating the number of nuclei remaining.

$\ln \frac{N_t}{N_0}=-kt$

$\ln \frac{N_t}{N_0}=-\left(\frac{0.05653729}{yr}\right)(5.50yr)$

$\ln \frac{N_t}{N_0}=-0.3110$

$\frac{N_t}{N_0}=e^{-0.3110}$

$\frac{N_t}{N_0}=0.7327$

$Fractionlost=1-0.7327=0.2673$