Question

A radioactive nuclide is produced at a constant rate of \(\alpha\).per second. It's decay constant is \(\lambda\). If \(N_{0}\) be the no. of nuclei at time \(t=0\), then max. no. of nuclei possible are : (a) \(N_{0}\) (b) \(\alpha / \lambda\) (c) \(N_{0}+\frac{\alpha}{\lambda}\) (d) \(\frac{\lambda}{\sigma}+N_{0} s\).

Short answer

\(\frac{\alpha}{\lambda}\)

Step-by-step solution

- Understand the production and decay process

Recognize that the rate of change of the number of nuclei at any time depends on the production rate \(\alpha\) and the decay rate, which is \(\lambda\) times the current number of nuclei. The differential equation describing the process can be written as \(\frac{dN}{dt} = \alpha - \lambda N\).

- Write the differential equation

Set up the differential equation that accounts for both production and decay of nuclei, \(\frac{dN}{dt} = \alpha - \lambda N\). This equation states that the change in the number of nuclei over time is equal to the production rate minus the decay, which is proportional to the number of existing nuclei.

- Solve the differential equation

Solve the differential equation to find \(N(t)\). This requires integrating the equation, usually by separation of variables or another method appropriate for first-order linear differential equations. The general solution will be in the form of \(N(t) = C\cdot e^{-\lambda t} + \frac{\alpha}{\lambda}\), where \(C\) is a constant determined by the initial condition.

- Apply the initial condition

Use the initial condition \(N_0\) to find the constant \(C\). Since at \(t=0\), \(N(t) = N_0\), we substitute these values into the general solution to obtain \(N_0 = C + \frac{\alpha}{\lambda}\). Solving for \(C\) gives \(C = N_0 - \frac{\alpha}{\lambda}\).

- Determine the maximum number of nuclei

The maximum number of nuclei \(N_{max}\) will occur when the production and decay rates are equal, that is, when \(\frac{dN}{dt} = 0\). Setting \(\alpha - \lambda N = 0\) and solving for \(N\) gives us \(N_{max} = \frac{\alpha}{\lambda}\). At this point, no net increase in the number of nuclei occurs since production and decay balance each other.

Key concepts

Nuclear Chemistry

Nuclear chemistry deals with the study of changes that occur within the nuclei of atoms. This field encompasses a vast range of topics including the processes of radioactivity, nuclear reactions, fission, and fusion. Radioactive decay, an intrinsic property of certain nuclides, is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves.

In nuclear chemistry, the stability of a nucleus is determined by its binding energy, which is the energy required to disassemble a nucleus into its constituent protons and neutrons. Nuclides with insufficient binding energy are prone to radioactive decay until they reach a more stable state. Understanding the principles behind nuclear reactions and decay processes is crucial for the development of applications such as nuclear energy, medical diagnostics, and treatments, as well as in the study of the universe's formation and the age of materials.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation, leading to the emission of alpha particles, beta particles, or gamma rays. This process can be described mathematically through decay laws that quantify the rate at which an unstable isotope transforms into a stable one.

The decay constant \(\lambda\) is a probability factor that represents the likelihood of a single nucleus decaying per unit time. The half-life of a radioactive isotope, which is the time taken for half of the nuclei in a sample to decay, is another fundamental concept in understanding radioactive decay. It is inversely proportional to the decay constant.

Differential Equations in Chemistry

Differential equations play a pivotal role in modeling a multitude of dynamic processes in chemistry, including reaction rates and radioactive decay. These equations are mathematical representations of the relationships between changing quantities and their rates of change. In the context of nuclear chemistry, a first-order linear differential equation like \(\frac{dN}{dt} = \alpha - \lambda N\) is used to describe the behavior of a radioactive substance considering both its production and natural decay processes.

These equations require specific mathematical techniques for their solutions, such as separation of variables, integrating factors, or the use of characteristic equations. The solutions to these equations give insight into how the quantity of interest changes over time, allowing chemists to predict future behavior of the system and understand the underlying kinetics of the reactions.

JEE Physical Chemistry Problems

The Joint Entrance Examination (JEE) is an engineering entrance assessment conducted in India, where physical chemistry is a crucial component. Students aspiring to excel need a deep understanding of concepts such as radioactive decay and their mathematical descriptions.

Problems like the radioactive nuclide decay, which involve concepts of both nuclear chemistry and differential equations, highlight the importance of interdisciplinary knowledge in solving JEE physical chemistry questions. These questions not only test the students' mastery of theory but also their ability to solve complex problems and apply concepts effectively using mathematics. Preparing for these problems demands an integrated approach, studying theoretical concepts in tandem with problem-solving strategies and frequently practicing with various question types to become adept at recognizing patterns and selecting appropriate methods for finding solutions.