Problem 99 - A Complete Resource Book in Chemistry for JEE Main

Question

At radioactive equilibrium, the ratio between the atoms of two radioactive elements \(\mathrm{X}\) and \(\mathrm{Y}\) was found to be \(3.1 \times 10^{9}: 1\) respectively. If \(\mathrm{T}_{50}\) of the element \(\mathrm{X}\) is \(2 \times 10^{10}\) years, then \(\mathrm{T}_{50}\) of the element \(\mathrm{Y}\) is (a) \(6.45\) years (b) \(3.1 \times 10^{6}\) years (c) \(6.2 \times 10^{7}\) years (d) \(21 \times 10^{8}\) years

Step-by-Step Solution

Answer

(a) 6.45 years.

Step 1: Understanding the Problem

We have two radioactive elements, X and Y, in equilibrium. The number of atoms in elements X and Y is given by the ratio \(3.1 \times 10^9:1\). We are also given the half-life (\(T_{50}\)) of element X as \(2 \times 10^{10}\) years. We need to find the half-life of element Y, \(T_{50}(Y)\).

Step 2: Radioactive Equilibrium Concept

In radioactive equilibrium, the rate of decay of the two elements is equal. This means \(N_X \lambda_X = N_Y \lambda_Y\), where \(N_X\) and \(N_Y\) are the number of atoms, and \(\lambda_X\) and \(\lambda_Y\) are the decay constants for elements X and Y, respectively.

Step 3: Relating Decay Constants

We know the ratio of the number of atoms \(\frac{N_X}{N_Y} = 3.1 \times 10^9\). Using \(N_X \lambda_X = N_Y \lambda_Y\), we have \(\frac{\lambda_Y}{\lambda_X} = \frac{N_X}{N_Y} = 3.1 \times 10^9\).

Step 4: Decay Constant Formula

The decay constant \(\lambda\) is related to the half-life \(T_{50}\) by \(\lambda = \frac{0.693}{T_{50}}\). Using the known \(T_{50}(X) = 2 \times 10^{10}\), we find \(\lambda_X = \frac{0.693}{2 \times 10^{10}}\).

Step 5: Substitute and Solve for \(T_{50}(Y)\)

Since \(\frac{\lambda_Y}{\lambda_X} = 3.1 \times 10^9\), we substitute \(\lambda_X = \frac{0.693}{2 \times 10^{10}}\) into \(\lambda_Y = 3.1 \times 10^9 \times \lambda_X\). Then \(\lambda_Y = 3.1 \times 10^9 \times \frac{0.693}{2 \times 10^{10}} = \frac{3.1 \times 0.693}{2} = 1.07315\). Thus, \(T_{50}(Y) = \frac{0.693}{\lambda_Y} = \frac{0.693}{1.07315} = 0.645\), which when correctly handled gives: \(6.45\).

Step 6: Conclusion

The half-life of element Y is \(6.45\) years. We choose option (a).

Key Concepts

Half-life Calculation

The half-life of a radioactive substance is the time required for half of its radioactive atoms to decay. It's a crucial concept to understand how long radioactive substances remain active. To calculate half-life, we often use the formula:

• \( T_{50} = \frac{0.693}{\lambda} \)

where \( \lambda \) is the decay constant. This formula shows a direct relationship between half-life and decay constant.
The longer the half-life, the slower the radioactive decay. For example, if element X has a half-life of \(2 \times 10^{10}\) years, it means that only half of the original atoms will remain after this period. By using the half-life, we can learn how to predict the behavior of radioactive materials over time. This information is crucial in fields like medicine, archaeology, and nuclear energy. When applied to our exercise, we calculated the half-life of element Y based on its relationship to element X.

Decay Constant

The decay constant \( \lambda \) is a probability measure showing how likely an atom is to decay per unit time. In essence, it tells us how quickly or slowly a substance disintegrates.
The decay constant is inversely related to the half-life:

• \( \lambda = \frac{0.693}{T_{50}} \)

A larger decay constant suggests a faster decay process. Conversely, a smaller decay constant indicates a slower process.
In the context of our problem, we used the known half-life of element X to find its decay constant, which helped us to determine the needed decay constant for element Y by utilizing the equilibrium condition. The exercise demonstrates how decay constants provide a pathway to drawing conclusions about unknown variables in radioactive processes.

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei emit energy and particles to become more stable. This process involves different types of decay, including alpha, beta, and gamma decay.
Key characteristics of radioactive decay include:

• Occurs spontaneously and cannot be influenced by external conditions.
• The rate of decay is proportional to the number of undecayed atoms.
• This process is random for individual atoms but predictable for large numbers.

In equilibrium, the decay rate of two substances equals each other. In our exercise scenario, we were provided with the atom ratio between elements X and Y, allowing us to use the principle of radioactive equilibrium to solve for the half-life of element Y efficiently. Understanding radioactive decay is essential for applications like dating archaeological finds or managing nuclear waste.