How long ago was the ratio$\frac{{}^{\mathbf{235}}\mathbf{U}}{{}^{\mathbf{238}}\mathbf{U}}$in natural uranium deposits equal to 0.15?

a) The ratio of the uranium deposits initially, $\frac{N_5\left(0\right)}{N_8\left(0\right)}=0.15$.

b) The ratio of the uranium deposits now $\frac{N_5\left(t\right)}{N_8\left(t\right)}=0.0072$.

c) Disintegration constant of ${}^{235}\mathrm{U}{\mathrm{\lambda }}_{\frac{1}{2}\left({}^{235}\mathrm{U}\right)}=\frac{9.85\times {10}^{-10}}{\mathrm{y}}$.

d) Disintegration constant of ${}^{235}\mathrm{U}{\mathrm{\lambda }}_{\frac{1}{2}\left({}^{238}\mathrm{U}\right)}=1.55\times {10}^{-10}$.

Being a highly unstable radioactive nuclide, the uranium isotopes undergo a decay process. This decay results in the disintegration of the deposits. Now, using the decay equation, determine the required value of the time taken for the decay.

Formula:

The amount of undecayed nuclei in a sample is as follows:

$N=N_0{e}^{-\lambda t}$ ….. (i)

Now, using equation (i) and substitute the given data as per required, the time taken for the initially present uranium deposits to decay can be given as follows:

$$\begin{gathered} \frac{N_5\left(t\right)}{N_8\left(t\right)}=\frac{N_5\left(0\right)}{N_8\left(0\right)}{e}^{-\left({\lambda }_5-{\lambda }_8\right)t} \\ t=\frac{1}{{\lambda }_8-{\lambda }_5}\mathrm{In}\left[\left(\frac{{\mathrm{N}}_5\left(\mathrm{t}\right)}{{\mathrm{N}}_8\left(\mathrm{t}\right)}\right)\left(\frac{{\mathrm{N}}_8\left(\mathrm{t}\right)}{{\mathrm{N}}_5\left(\mathrm{t}\right)}\right)\right] \end{gathered}$$

Solve further as:

$$\begin{gathered} t=\frac{1}{\left(1.55-9.85\right)\times {10}^{-10}/y}\mathrm{In}\left[\left(0.0072\right){\left(0.15\right)}^{-1}\right] \\ \mathrm{t}=3.6\times {10}^9\mathrm{y} \end{gathered}$$

Hence, the required value of the time is $3.6\times {10}^9\mathrm{y}$.