Q 3.3-27E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. The only undiscovered isotopes of the two unknown elements hohum and inertium (symbols Hh and It) are radioactive. Hohum decays into inertium with a decay constant of 2/yr, and inertium decays into the nonradioactive isotope of bunkum (symbol Bu) with a decay constant of 1/yr. An initial mass of 1 kg of hohum is put into a non-radiaoctive container, with no other source of hohum, inertium, or bunkum. How much of each of the three elements is in the container after t yr? (The decay constant is the constant of proportionality in the statement that the rate of loss of mass of the element at any time is proportional to the mass of the element at that time.)

 

Step-by-Step Solution

Answer

The amount of Hohum in the container after t years is $e^{-2t}kg$.

The amount of Inertium in the container after t years is $2e^{-t}-2e^{-2t} kg$

The amount of Bunkum in the container after t years is$1+e^{-2t}-2e^{-t} kg$

Step 1: Analyzing the given statement

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, $\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}\mathbf{\propto }\mathit{N}$

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.

Step 2: Determining the formula with the help of the given proportionality relation, to solve the question

Given, 

$$\begin{gathered} \frac{dN}{dt}\propto N \\ \frac{dN}{dt}=-\lambda N \end{gathered}$$

 Where, $\lambda$ is the constant of proportionality.

 $$\begin{gathered} \frac{dN}{N}=-\lambda \\ \int \frac{dN}{N}=-\lambda \int dt \\ lnN=-\lambda t+ln{N}_0 \end{gathered}$$

where, N₀ is an arbitrary constant.

$$\begin{gathered} \mathit{l}\mathit{n}\mathit{N}\mathbf{-}\mathit{l}\mathit{n}{\mathit{N}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\mathit{\lambda }\mathit{t} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathit{l}\mathit{n}\left(\frac{N}{N_0}\right)\mathbf{=}\mathbf{-}\mathit{\lambda }\mathit{t} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\frac{\mathbf{N}}{{\mathbf{N}}_{\mathbf{0}}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{\lambda }\mathbf{t}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{N}\mathbf{=}{\mathit{N}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{\lambda }\mathbf{t}}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(1\right) \end{gathered}$$ 

One will use this formula to solve the question.

Step 3: Determining the mass of Hohum

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., $\lambda$=2

 

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

 

Therefore, from equation (1),

$$\begin{gathered} {\mathit{N}}_{\mathbf{1}}\mathbf{=}{\mathit{N}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{\lambda }\mathbf{t}} \\ {\mathit{N}}_{\mathbf{1}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(2\right) \end{gathered}$$ 

Thus, the amount of Hohum (Hh) in the container is  ${\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathit{k}\mathit{g}$
.

 

Step 4: Determining the mass of Inertium

Here,  N₀= 2 kg

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

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

Thereafter, from equation (1),

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

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

 

Step 4: Determining the mass of Bunkum

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.