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

Question

The decay constant of \({ }_{6} \mathrm{C}^{14}\) is \(2.31 \times 10^{-4}\) year \(^{-1} .\) Its half life is (a) \(2 \times 10^{3}\) yrs (b) \(2.5 \times 10^{3} \mathrm{yrs}\) (c) \(3 \times 10^{3} \mathrm{yrs}\) (d) \(3.5 \times 10^{3}\) yrs

Step-by-Step Solution

Answer

The half-life of \({ }_{6} \mathrm{C}^{14}\) is \(3 \times 10^{3}\) yrs, choice (c).

Step 1: Understanding Half-Life Formula

The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei to decay. The relationship between decay constant (C14) and half-life () is given by the formula: \[ T_{1/2} = \frac{0.693}{\lambda} \] where \( \lambda \) is the decay constant.

Step 2: Substitute the Given Decay Constant

We are given that the decay constant \( \lambda \) for \({ }_{6} \mathrm{C}^{14}\) is \(2.31 \times 10^{-4}\) year\(^{-1}\). 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 calculation: \[ T_{1/2} = \frac{0.693}{2.31 \times 10^{-4}} = \frac{0.693}{0.000231} \approx 3000 \] Thus, the half-life of \({ }_{6} \mathrm{C}^{14}\) is approximately 3000 years.

Step 4: Select the Correct Answer

From the calculation, the closest value is option (c) which is "\(3 \times 10^{3}\) yrs".

Key Concepts

Half-Life Calculation

Radioactive decay involves the transformation of unstable atomic nuclei, and a key aspect is calculating the half-life. Half-life is the duration needed for half of the radioactive isotopes in a sample to transform. One common formula to find half-life is

• \( T_{1/2} = \frac{0.693}{\lambda} \)
  

This formula relates half-life \( T_{1/2} \) to the decay constant \( \lambda \), providing a powerful tool for understanding how quickly an isotope transforms.

Using this formula, we can explore the specific case of Carbon-14 with a decay constant of \( 2.31 \times 10^{-4} \) year\(^{-1}\). By substituting into the formula,

• \( T_{1/2} = \frac{0.693}{2.31 \times 10^{-4}} \)
  

This calculation results in a half-life of approximately 3000 years, establishing how long it takes for half of the Carbon-14 in a given sample to decay.

Understanding half-life helps scientists date ancient objects and describe the stability of different isotopes.

Decay Constant

The decay constant, represented by \( \lambda \), is a probability value that reflects how likely a nucleus is to decay within a given time. This constant is measured in units of inverse time, often year\(^{-1}\), which provides a sense of how fast or slow the decay process occurs.

In the context of our exercise, the decay constant for Carbon-14 is given as \( 2.31 \times 10^{-4} \) year\(^{-1}\). This tells us that Carbon-14 is relatively slow to disintegrate, which is crucial for its application in dating archaeological finds. The comfortable, gradual rate ensures it takes thousands of years for a significant portion to transform.

Calculating with this decay constant involves substituting it into the half-life formula, showing us that:

• The calculation reveals the half-life of Carbon-14 is about 3000 years.
  

This slow decay makes it an excellent candidate for studying fossils and understanding historical timelines.

Radioactive Isotopes

Radioactive isotopes, such as Carbon-14, consist of atoms that are not stable, and therefore decay over time. This decay involves the emission of radiation, transforming the isotope into a different element or isotope. This natural process can impact various areas from dating historical artifacts to medicine.

Carbon-14 is a well-known radioactive isotope, often used in radiocarbon dating to determine the age of once-living materials up to about 50,000 years old. The presence of naturally occurring radioactive isotopes like Carbon-14 in living organisms allows for this dating. Once an organism dies, the Carbon-14 begins to decay at a known rate, represented by the decay constant.

The uniqueness of radioactive isotopes:

• They offer insights into the historical age of objects.
  
• Medical and industrial applications utilize their properties for diagnoses and material analysis.
  

Overall, radioactive isotopes play a crucial role in science, providing tools for research, historical analysis, and practical applications.