Question

A radioactive isotope is being formed at a constant rate \(\mathrm{K} .\) At \(\mathrm{t}=0\), the number of active nuclei is \(\mathrm{N}_{0}\). The decay constant of isotope is \(\lambda .\) The number of active nuclei. (1) first increase then decreases (2) goes on increasing (3) goes on decreasing (4) is \(\frac{\mathrm{K}}{\lambda}\) after a time \(\mathrm{t}>>\frac{1}{\lambda}\)

Short answer

(4) is \frac{K}{\lambda}\ after a time t>>1/\lambda.

Step-by-step solution

Understand the problem

Determine the behavior of the number of active nuclei of a radioactive isotope being formed at a constant rate \(\text{K}\) and having an initial number of active nuclei \(N_0\) with a decay constant \(\lambda\). Analyze what happens to the number of active nuclei over time.

Set up the differential equation

Write the differential equation for the number of active nuclei \(N(t)\) considering both formation at a constant rate \(K\) and decay with decay constant \(\lambda\). The equation is: \[ \frac{dN(t)}{dt} = K - \lambda N(t) \]

Solve the differential equation

Solve the differential equation. The general solution will have the form: \[ N(t) = \frac{K}{\lambda} + \left(N_0 - \frac{K}{\lambda}\right)e^{-\lambda t} \]

Analyze the behavior over time

Examine the solution to determine the behavior: \[N(t) = \frac{K}{\lambda} + \left(N_0 - \frac{K}{\lambda}\right)e^{-\lambda t}\] Initially (at t=0): \[N(0) = N_0\] As t becomes very large (t → ∞): \[N(t) = \frac{K}{\lambda}\], since \(e^{-\lambda t}\) approaches zero.

Determine the correct option

From the analysis, it can be seen that initially the number of active nuclei will increase until it stabilizes at \[\frac{K}{\lambda} \] after a long time (\(t >> \frac{1}{\lambda}\)). Therefore, the correct option is: \[ \text{Option 4: is } \frac{K}{\lambda} \text{ after a time } t>>\frac{1}{\lambda} \]

Key concepts

Radioactive Decay

When we discuss radioactive decay, we're looking at how unstable atomic nuclei lose energy by emitting radiation. This process changes the nuclei into a different state or different elements over time. Think of it like a burning candle that slowly melts away.

Radioactive isotopes decay at predictable rates, characterized by their decay constant. Not all isotopes decay the same way; each isotope has a unique decay rate.

For example, Carbon-14 decays very slowly and is used in dating ancient artifacts. In contrast, isotopes like Iodine-131 decay rapidly and are used in medical treatments. Understanding radioactive decay helps in various fields such as archaeology, medicine, and nuclear energy.

Differential Equations in Radioactive Decay

Differential equations are mathematical tools essential for modeling radioactive decay. They describe the rate at which the number of radioactive nuclei changes over time.

In our problem, a differential equation helps us understand how the number of active nuclei changes with time when both formation and decay are happening. The differential equation is:
\[ \frac{dN(t)}{dt} = K - \lambda N(t) \]
This equation means that the rate of change in the number of active nuclei (\