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

The half-life \( T_{50} \) of element Y is approximately 6.45 years (option a).

Step 1: Understanding Radioactive Equilibrium

In radioactive equilibrium, the rate of production of a daughter element (Y) from a parent element (X) is equal to the rate of decay of the daughter element (Y). Therefore, the decay constants and the atoms' ratio are related by \( N_X \lambda_X = N_Y \lambda_Y \), where \( N_X \) and \( N_Y \) are the number of atoms of X and Y, respectively, and \( \lambda_X \) and \( \lambda_Y \) are the decay constants of X and Y.

Step 2: Using the Atoms Ratio

We are given the ratio \( N_X : N_Y = 3.1 \times 10^9 : 1 \). This implies \( \frac{N_X}{N_Y} = 3.1 \times 10^9 \). Substituting this in the equilibrium condition gives: \( 3.1 \times 10^9 \lambda_X = 1 \times \lambda_Y \) or \( \lambda_Y = 3.1 \times 10^9 \times \lambda_X \).

Step 3: Relation between Decay Constant and Half-life

The decay constant \( \lambda \) is related to the half-life \( T_{50} \) by the equation \( \lambda = \frac{0.693}{T_{50}} \). Substituting this into the previous relation, we get: \( \frac{0.693}{T_{50, Y}} = 3.1 \times 10^9 \times \frac{0.693}{T_{50, X}} \).

Step 4: Calculate \( T_{50, Y} \)

Rearranging the equation gives: \( T_{50, Y} = \frac{T_{50, X}}{3.1 \times 10^9} \). We are given \( T_{50, X} = 2 \times 10^{10} \) years. Substituting into the equation yields: \( T_{50, Y} = \frac{2 \times 10^{10}}{3.1 \times 10^9} \approx 6.45 \) years.

Step 5: Final Answer

Thus, the half-life \( T_{50} \) of element \( Y \) is approximately 6.45 years.

Key Concepts

Decay Constant

The decay constant, often represented by the symbol \( \lambda \), is a crucial concept in understanding radioactive decay. It signifies the probability of decay of a nucleus in a given amount of time. The decay constant is directly related to how quickly a radioactive substance will decrease over time. A larger decay constant means a faster rate of decay. It is an exponent in the equation that describes radioactive decay over time, given by \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the quantity that still remains and \( N_0 \) is the initial quantity. This constant helps to define the behavior of the substance as it transforms into another substance, deteriorating at a predictable rate. For students, it's key to understand that the decay constant provides a direct connection to the substance’s rate of decay, making it foundational for calculations in radioactive processes.

Half-Life Calculation

The half-life of a radioactive element, denoted as \( T_{50} \), is the time it takes for half of any given quantity of the substance to decay. This is a pivotal concept because it provides a time scale for the rate of decay.

• The half-life is expressed in units of time, like years, days, or even seconds, depending on the speed of decay.
• The relation between the half-life and the decay constant \( \lambda \) is given by the equation \( \lambda = \frac{0.693}{T_{50}} \).
• This equation highlights that the half-life is inversely proportional to the decay constant, meaning if one increases, the other decreases.

Understanding this relationship is essential for deducing one when you know the other, as seen in the problem where the half-life of element \( Y \) is calculated using the given half-life of element \( X \) and their respective decay constants.

Radioactive Decay Series

A series of radioactive decays can be referred to as a radioactive decay series. It is a sequence of successive transformations from one unstable nucleus to another until a stable nucleus is achieved. In nature, these series are common with heavy elements such as uranium and thorium that decay into lead, or other stable elements. Each step in the series can have its own decay constant and half-life, contributing to the overall time it takes for a parent isotope to transform into a stable daughter isotope.

• In radioactive equilibrium, an important concept within decay series, the rate of decay of the parent is equal to the rate of production and decay of the daughter isotope.
• This equilibrium condition links the decay constants and can effectively determine the half-life of intermediates in the series.

This understanding aids in solving exercises related to radioactive decay by utilizing the relationships between the decay constants and the ratios of the isotopes involved. The calculations reflect the physical transformations during the decay process, allowing a clearer insight into the isotope behaviors over time.