Question

A radioactive isotope is being produced at a constant rate \(\mathrm{d} N / \mathrm{d} t=R\) in an experiment. The isotope has a half-life, \(t_{1 / 2}\). After a time \(t \gg t_{1 / 2}\), the number of active nuclei will become constant. The value of this constant is (a) \(R\) (b) \(\underline{1}\) (c) \(R / \lambda\) (d) \(\lambda / R\)

Short answer

\(\frac{R}{\lambda}\)

Step-by-step solution

Understand Half-Life and Decay Constant

First, recognize that the half-life (\(t_{1/2}\)) is related to the decay constant (\textbackslash lambda) by the equation \(\lambda = \frac{\ln(2)}{t_{1/2}}\). The decay constant represents the probability per unit time that a nucleus will decay.

Establish the Differential Equation for Radioactive Decay

Using the decay constant \(\lambda\), the rate at which nuclei decay over time is given by \(\frac{\mathrm{d} N}{\mathrm{d} t} = -\lambda N\), where \(N\) is the number of active nuclei.

Combine Production and Decay Rates

In the experiment, nuclei are being produced at a constant rate \(R\), while decaying at a rate of \(\lambda N\). Combining these rates, we get \(\frac{\mathrm{d} N}{\mathrm{d} t} = R - \lambda N\).

Apply the Steady-State Condition

For a time \(t \gg t_{1/2}\), the number of active nuclei becomes constant, implying \(\frac{\mathrm{d} N}{\mathrm{d} t} = 0\). Using the equation from Step 3, we get \(0 = R - \lambda N_{\mathrm{steady-state}}\).

Solve for the Constant Number of Active Nuclei

Rearranging the equation from Step 4 for \(N_{\mathrm{steady-state}}\), we find \(N_{\mathrm{steady-state}} = \frac{R}{\lambda}\). Therefore, the constant number of active nuclei in the steady state is \(\frac{R}{\lambda}\).

Key concepts

Understanding the Half-Life of Radioactive Isotopes

The concept of half-life is critical in understanding the behavior of radioactive isotopes. Half-life is defined as the time required for half of the radioactive atoms in a sample to decay. This simple definition has profound implications for the stability and longevity of radioactive substances.

A radioactive isotope with a short half-life will rapidly decay, quickly reducing the amount of the radioactive material. Conversely, a radioactive isotope with a long half-life decays more slowly and therefore remains active for a longer period. The half-life allows scientists and engineers to predict how long a radioactive isotope will continue to emit radiation, which is vital for applications in medicine, industry, and environmental studies.

It's important to note that the half-life is a constant value for a given isotope and does not change regardless of the quantity of the material or its form. In other words, whether you have a gram or a kilogram of a radioactive substance, the half-life will be the same. This characteristic makes it a reliable tool for dating archaeological finds and for gauging the safety of radioactive materials.

The Role of the Radioactive Decay Constant

The decay constant, typically denoted by the Greek letter lambda\textbf{(}\(\lambda\)), is an intrinsic property of each radioactive isotope. It represents the probability per unit time that a single nucleus will decay. The decay constant is a way of quantifying the rate at which a radioactive substance undergoes radioactive decay.

The relationship between the half-life and the decay constant is given by the equation: \[lambda = \frac{\ln(2)}{t_{1/2}}\]Here, \(\ln(2)\) is the natural logarithm of 2, which comes from the fact that half of the substance decays after one half-life. A higher decay constant implies that a substance decays more quickly, with a shorter half-life, while a lower decay constant means the substance is more stable and decays more slowly. When dealing with nuclear reactions, understanding the decay constant is crucial for predicting how much radioactive material will remain after a certain period.

Differential Equations in Radioactive Decay

Differential equations play a significant role in modeling the process of radioactive decay. These equations describe the rate at which changes occur in the system, which is crucial for understanding the dynamics of radioactive substances.

In the context of radioactive decay, the fundamental differential equation is:\[\frac{\mathrm{d} N}{\mathrm{d} t} = -\lambda N\]Where \(N\) represents the number of undecayed nuclei at a certain time, and \(\lambda\) is the decay constant. The negative sign indicates that the number of radioactive nuclei decreases over time due to decay.

When production and decay processes are happening simultaneously, as in the situation described in the exercise, the differential equation is modified to:\[\frac{\mathrm{d} N}{\mathrm{d} t} = R - \lambda N\]This equation reflects the reality that while new radioactive nuclei are produced at rate \(R\), others decay at a rate proportional to the current number of active nuclei. Finding the steady-state solution involves setting the derivative of \(N\) with respect to time equal to zero, reflecting a state where the production and decay rates are balanced. In this state, the number of active nuclei remains constant—such a concept is crucial for applications in pharmacokinetics and environmental science.