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Given that the rate of decay of a radioactive substance is directly proportional to the amount of the substance present. Let the present amount of the radioactive substance be N.

Therefore,$\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}\mathbf{鈭�}\mathit{N}$

Given that the only undiscovered isotopes of the two unknown elements hohum and inertium are radioactive. Hohum decays into inertium with a decay constant of 2/yr, and inertium decays into the non-radioactive isotope of bunkum with a decay constant of 1/yr. An initial mass of 1 kg of hohum is put into a non-radioactive container, with no other source of hohum, inertium, or bunkum. We have to find the mass of each of the three elements in the container after t years.

Given,

$$\begin{gathered} \frac{dN}{dt}鈭�N \\ \frac{dN}{dt}=-位N \end{gathered}$$

Where, $位$is the constant of proportionality.

$$\begin{gathered} \frac{dN}{N}=-位 \\ 鈭�\frac{dN}{N}=-位鈭�dt \\ lnN=-位t+ln{N}_0 \end{gathered}$$

where, N₀ is an arbitrary constant.

role="math" localid="1664197583386" $$\begin{gathered} \mathit{l}\mathit{n}\mathit{N}\mathbf{-}\mathit{l}\mathit{n}{\mathit{N}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\mathit{位}\mathit{t} \\ \mathbf{}\mathbf{}\mathbf{}\mathit{l}\mathit{n}\left(\frac{N}{N_0}\right)\mathbf{=}\mathbf{-}\mathit{位}\mathit{t} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\frac{\mathbf{N}}{{\mathbf{N}}_{\mathbf{0}}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{位}\mathbf{t}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{N}\mathbf{=}{\mathit{N}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{位}\mathbf{t}}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{路}\left(1\right) \end{gathered}$$

One will use this formula to solve the question.

Let the initial mass of Hohum, which is to be put in the container be

i.e., N₀= 1 kg

Also, given that Hohum decays into inertium with a decay constant of 2/yr,

i.e., $位$=2

Let $N_1$be the mass of hohum in the container after t years.

Therefore, from equation (1),

role="math" localid="1664197555945" $$\begin{gathered} {\mathit{N}}_{\mathbf{1}}\mathbf{=}{\mathit{N}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{位}\mathbf{t}} \\ {\mathit{N}}_{\mathbf{1}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{路}\left(2\right) \end{gathered}$$

Thus, the amount of Hohum (Hh) in the container isrole="math" localid="1664197435094" ${\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathit{k}\mathit{g}$
.

Here, N₀= 2 kg

Also, given thatinertium decays into the non-radioactive isotope of bunkum with a decay constant of 1/yr i.e., $位$=1

Let N₂ be the mass of Inertium in the container after t years.

Thereafter, from equation (1),

role="math" localid="1664197538397" $$\begin{gathered} {\mathit{N}}_{\mathbf{2}}\mathbf{=}\mathbf{2}\left(e^{-t}-e^{-2t}\right) \\ {\mathit{N}}_{\mathbf{2}}\mathbf{=}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathbf{-}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}} \end{gathered}$$

Hence, the amount of Inertium (It) in the container is role="math" localid="1664197517736" $\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathbf{-}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}$kg.

Let be the mass of Bunkum in the container after t years.

Therefore,

$$\begin{gathered} {\mathit{N}}_{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{-}{\mathit{N}}_{\mathbf{1}}\mathbf{-}{\mathit{N}}_{\mathbf{2}} \\ {\mathit{N}}_{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{-}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{-}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathbf{+}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}} \\ {\mathit{N}}_{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{-}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{t}} \end{gathered}$$

So, the amount of Bunkum (Bu) in the container is $\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{-}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{t}}$ kg.