Understanding Radioactive Decay Half-Life

The question asks about the factors that influence the half-life of a radioactive material during radioactive decay. Let's break down what half-life means and how it's determined.

What is Half-Life?

Half-life (often denoted as $t_{1/2}$) is the time required for a radioactive substance to decay to half of its initial amount. It's a fundamental property of a specific radioactive isotope.

Radioactive Decay and Kinetics

Radioactive decay is a first-order process. This means the rate of decay is directly proportional to the number of radioactive nuclei present at that time. Mathematically, this can be represented as:

$ \frac{dN}{dt} = -kN $

Where:

• $N$ is the number of radioactive nuclei at time $t$.
• $t$ is time.
• $k$ is the decay constant, a characteristic constant for each radionuclide.
• The negative sign indicates that the number of nuclei decreases over time.

Deriving the Half-Life Formula

By integrating the rate equation, we get the integrated rate law for first-order reactions:

$ N_t = N_0 e^{-kt} $

Where:

• $N_t$ is the number of nuclei remaining at time $t$.
• $N_0$ is the initial number of nuclei (at $t=0$).

The half-life ($t_{1/2}$) is the time when $N_t = \frac{N_0}{2}$. Substituting this into the integrated rate law:

$ \frac{N_0}{2} = N_0 e^{-kt_{1/2}} $

Divide both sides by $N_0$:

$ \frac{1}{2} = e^{-kt_{1/2}} $

Take the natural logarithm of both sides:

$ \ln\left(\frac{1}{2}\right) = -kt_{1/2} $

$ -\ln(2) = -kt_{1/2} $

Solving for $t_{1/2}$:

$ t_{1/2} = \frac{\ln(2)}{k} $

Analyzing the Half-Life Dependence

From the derived formula, $ t_{1/2} = \frac{\ln(2)}{k} $, we can see:

• Dependence on Initial Concentration: The formula for $t_{1/2}$ does not include $N_0$ (initial concentration or amount). This means the half-life is independent of the initial concentration of the radioactive material. Whether you start with 1 gram or 100 grams, it will take the same amount of time for half of it to decay.
• Dependence on Decay Constant: The half-life ($t_{1/2}$) is inversely proportional to the decay constant ($k$). This means if the decay constant $k$ is large (implying rapid decay), the half-life $t_{1/2}$ will be short. Conversely, if $k$ is small (slow decay), $t_{1/2}$ will be long.

Evaluating the Options

Based on our analysis:

• Option 1 is incorrect because half-life is independent of initial concentration.
• Option 2 is incorrect because half-life is independent of initial concentration.
• Option 3 is incorrect because half-life is independent of initial concentration (and final concentration is also dependent on time/initial amount).
• Option 4 correctly states that half-life is independent of the initial concentration and inversely proportional to the decay constant.

Therefore, the half-life of a radioactive material during radioactive decay is independent of the initial concentration and inversely proportional to the decay constant.