Problem 39 - Master Resource Book in JEE Main Physics

Question

The ratio of molecular mass of two radioactive substances is \(3 / 2\) and the ratio of their decay constants is \(4 / 3\). Then, the ratio of their initial activity per mole will be (a) 2 (b) \(4 / 3\) (c) \(8 / 9\) (d) \(9 / 8\)

Step-by-Step Solution

Answer

The ratio of their initial activity per mole is \(\frac{8}{9}\), option (c).

Step 1: Understanding The Given Ratios

Firstly, identify the known ratios: the ratio of molecular masses is given as \(\frac{3}{2}\) and the ratio of decay constants is \(\frac{4}{3}\).

Step 2: Understanding Activity Formula

The initial activity \(A_0\) per mole of a substance is given by \(A_0 = \lambda \times N\), where \(\lambda\) is the decay constant and \(N\) is Avogadro's number, which depends on the molecular mass.

Step 3: Expressing in Terms of Ratios

For two substances with masses \(M_1\) and \(M_2\) and decay constants \(\lambda_1\) and \(\lambda_2\), the activities are \(A_0^1 = \lambda_1 \times N/M_1\) and \(A_0^2 = \lambda_2 \times N/M_2\).

Step 4: Formulate Ratio of Activities

The ratio of activities is \(\frac{A_0^1}{A_0^2} = \frac{\lambda_1/M_1}{\lambda_2/M_2} = \frac{\lambda_1 M_2}{\lambda_2 M_1}\).

Step 5: Plug In Ratios

Substituting the given ratios, \(\frac{\lambda_1}{\lambda_2} = \frac{4}{3}\) and \(\frac{M_1}{M_2} = \frac{3}{2}\), into the formula, you get \(\frac{A_0^1}{A_0^2} = \frac{4}{3} \times \frac{2}{3} = \frac{8}{9}\).

Step 6: Final Calculation and Answer

Simplifying gives the final ratio \(\frac{A_0^1}{A_0^2} = \frac{8}{9}\). Thus, the correct answer is option (c) \(\frac{8}{9}\).

Key Concepts

Decay Constant

In radioactive decay, the decay constant, denoted by \( \lambda \), is a crucial parameter that describes the rate at which a radioactive substance undergoes decay. It specifies the probability of a nucleus decaying per unit time, essentially quantifying how quickly a radioactive material loses its radioactivity.

• The larger the decay constant, the faster the decay process is.
• Decay constant is expressed in inverse time units, such as \( \text{s}^{-1} \) or \( \text{min}^{-1} \).

Understanding decay constant helps predict the activity of a radioactive substance at any given time, vital for applications ranging from nuclear medicine to radiometric dating. In solving problems involving decay, knowing the decay constant allows us to calculate how many atomic nuclei decay over a specific period, thereby determining the material's radioactivity level.

Molecular Mass

Molecular mass, or molar mass, is the mass of one mole of a substance, typically measured in atomic mass units (u) or grams per mole (g/mol). In the context of radioactive decay, it plays a significant role because it helps determine the number of atoms or nuclei present in a sample.

• A lower molecular mass indicates a higher number of atoms per gram of the material.
• In radioactive substances, the molecular mass impacts the initial activity because it is directly linked to the number of decaying nuclei.

When calculating activity per mole, the molecular mass completes the equation by relating number of moles to Avogadro's number, providing the number of atoms available for decay, essential for precise activity measurements.

Initial Activity Per Mole

Initial activity per mole, expressed as \( A_0 \), is the activity of a radioactive substance when measured per mole of its material. It is the product of the decay constant and Avogadro's number divided by the molecular mass. This concept simplifies comparing radioactivity across different substances regardless of their quantity or form.
Consider the formula:\[ A_0 = \lambda \times \frac{N}{M} \]where \( N \) is Avogadro's number and \( M \) is the molecular mass.

• High initial activity per mole indicates a substance with a fast decay and more radioactive nuclei per mole.
• It shows how radioactive a substance inherently is without needing to consider the sample size.

This concept is pivotal for chemists and physicists who wish to understand and predict the behavior of radioactive materials on a standard molar basis, making it easier to design experiments and safety guidelines for handling.