Problem 82 - A Complete Resource Book in Chemistry for JEE Main

Question

The decay constant of \(C^{14}\) is \(2.31 \times 10^{-4}\) year \(^{-1}\). Its half life is (a) \(2 \times 10^{3} \mathrm{yrs}\) (b) \(2.5 \times 10^{3} \mathrm{yrs}\) (c) \(3 \times 10^{3} y r s\) (d) \(3.5 \times 10^{3} \mathrm{yrs}\)

Step-by-Step Solution

Answer

The half-life of \(C^{14}\) is \(3 \times 10^{3} \text{ yrs}\), so the answer is (c).

Step 1: Understand the Half-Life Formula

The formula to find the half-life \( t_{1/2} \) is related to the decay constant \( \lambda \) by the equation: \[ t_{1/2} = \frac{0.693}{\lambda} \] where \( \lambda \) is the decay constant.

Step 2: Insert Known Values

We know from the problem statement that the decay constant \( \lambda \) is \( 2.31 \times 10^{-4}\) year\(^{-1}\). Let's substitute this value into the half-life formula:\[ t_{1/2} = \frac{0.693}{2.31 \times 10^{-4}} \]

Step 3: Calculate the Half-Life

Perform the division to get the half-life. Calculating this gives:\[ t_{1/2} = \frac{0.693}{2.31 \times 10^{-4}} = 3000 \text{ years} \]

Step 4: Compare with Given Options

The calculated half-life is 3000 years. Checking this against the given options, we find that:\( (c) \ 3 \times 10^{3} \text{ yrs} \) matches the calculated value.

Key Concepts

Decay Constant

In the world of nuclear chemistry, understanding radioactive decay is fundamental. At the heart of this process is the decay constant, often symbolized by the Greek letter \( \lambda \). The decay constant provides a measure of the probability of decay of a radioactive nucleus per unit time. It is a crucial factor in determining the rate at which a radioactive substance transforms. Each radioactive isotope has its own unique decay constant, which remains unchanged regardless of external conditions.

Key concepts about decay constant include:

• The decay constant \( \lambda \) is expressed in the units of inverse time, such as year\(^{-1}\), making it straightforward to use in calculations related to decay processes.
• A higher decay constant value means a faster decay rate, while a lower value suggests that the substance decays more slowly.

Decay constant plays a critical role in calculating the half-life of a material, allowing chemists and physicists to predict how long it will take for a given quantity of the substance to reduce by half.

Half-Life

Half-life is a concept that connects closely with the decay constant. It refers to the time required for half of the radioactive nuclei in a sample to decay. The half-life \( t_{1/2} \) is calculated using the formula:\[ t_{1/2} = \frac{0.693}{\lambda} \]where \( \lambda \) is the decay constant. This formula derives from the natural logarithm of 2, indicating a consistent fractional decay over time.

Important points about half-life:

• Half-life remains constant for a stable isotope and doesn't change with temperature, pressure, or chemical state.
• It provides an easy way to compare the stability and decay speed of different isotopes.
• Understanding half-life helps in fields ranging from archaeology, through carbon dating, to managing nuclear waste.

Through practical problems, like calculating the half-life of \(C^{14}\), we see the half-life as an invaluable tool for both theoretical and applied sciences.

Nuclear Chemistry

Nuclear chemistry is a vast field that focuses on the chemical changes in atomic nuclei. It delves into how atomic nuclei interact, transform, and decay, as well as the energies involved in these processes. Key aspects of nuclear chemistry:

• Radioactive decay, which involves the breakdown of unstable nuclei, releasing particles and energy.
• Transmutation, the process where one element changes into another due to nuclear reactions.
• The study of isotopes, which are variants of elements with different numbers of neutrons, impacting nuclear structure and stability.

In nuclear chemistry, concepts like decay constant and half-life are essential for predicting how different isotopes behave over time. As a scientific discipline, nuclear chemistry extends into practical applications, including energy production in nuclear reactors, medical imaging and treatment, and understanding the universe's elemental composition.