Problem 131 - A Complete Resource Book in Chemistry for JEE Main

Question

Nuclear explosion has taken place leading to increase in concentration of \({ }^{14} \mathrm{C}\) in nearby areas. \({ }^{14} \mathrm{C}\) concentration is \(\mathrm{C}_{1}\) in nearby areas and \(\mathrm{C}_{2}\) in areas far away. If the age of the fossil is determined to be \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2}\) at the places respectively then (a) The age of the fossil will increase at the place where explosion has taken place and \(\mathrm{T}_{1}-\mathrm{T}_{2}=\frac{1}{\lambda} \ln \frac{\mathrm{C}_{1}}{\mathrm{C}_{2}}\) (b) The age of the fossil will decrease at the place where explosion has taken place and \(\mathrm{T}_{1}-\mathrm{T}_{2}=\frac{1}{\lambda} \operatorname{In} \frac{\mathrm{C}_{1}}{\mathrm{C}_{2}}\) (c) The age of fossil will be determined to be same (d) \(\mathrm{T}_{1} / \mathrm{T}_{2}=\mathrm{C}_{1} / \mathrm{C}_{2}\) [IIT 2006]

Step-by-Step Solution

Answer

Option (b) is correct; the fossil age will decrease at the site of the explosion.

Step 1: Understanding Decay and Concentration

In radiocarbon dating, the age of the fossil is determined by the concentration of \(^{14}\mathrm{C}\) remaining in the sample. Normally, this concentration decreases over time as \(^{14}\mathrm{C}\) decays exponentially. An external event like a nuclear explosion increases the concentration of \(^{14}\mathrm{C}\) in the environment, affecting the apparent age of samples nearby.

Step 2: Equation of Fossil Age

The equation for the decay of \(^{14}\mathrm{C}\) is given by\[N(t) = N_0 e^{-\lambda t}\]where \(N(t)\) is the amount of \(^{14}\mathrm{C}\) at time \(t\), \(N_0\) is the initial amount, and \(\lambda\) is the decay constant. The apparent age (\(T\)) is derived from the ratio of current concentration \(C\) to the original \(\mathrm{C}_0\), resulting in:\[T = \frac{1}{\lambda} \ln \frac{\mathrm{C}_0}{\mathrm{C}}\]

Step 3: Effect of Increased Concentration

When the environment's concentration \(\mathrm{C}_1\) is higher than \(\mathrm{C}_2\), the fossil's apparent age \(T_1\) at \(\mathrm{C}_1\) would be calculated as older than \(T_2\) at \(\mathrm{C}_2\). Thus, \(T_1\) should indeed be smaller than \(T_2\), effectively decreasing the perceived age where the explosion took place.

Step 4: Analyzing Correct Mathematical Relation

Based on the provided options for the age of the fossil, we suspect the option with a logarithmic relationship is relevant. The difference of ages can be articulated as:\[\mathrm{T}_1 - \mathrm{T}_2 = \frac{1}{\lambda} \ln \frac{\mathrm{C}_1}{\mathrm{C}_2}\]where an increase in \(\mathrm{C}_1\) (nearby area) decreases the numerator in the fraction, aligning with option (b) where the age decreases.

Step 5: Conclusion on Correct Choice

The age of the fossil will appear younger where there is an increase in\(^{14}\mathrm{C}\) due to the explosion, making the apparent fossil age decrease. This supports option (b) as correct, so\(\mathrm{T}_1 - \mathrm{T}_2 = \frac{1}{\lambda} \ln \frac{\mathrm{C}_1}{\mathrm{C}_2}\) aligns with our findings.

Key Concepts

Nuclear explosion

A nuclear explosion releases massive energy and a burst of radioactive materials, including isotopes like \(^{14}\mathrm{C}\), into the environment. This sudden increase in radioactive isotopes influences the ambient levels of carbon in nearby areas. For example, during a nuclear explosion, the concentration of \(^{14}\mathrm{C}\) can dramatically rise, affecting the accuracy of radiocarbon dating.
Following such an event, calculating the age of fossils in areas close to the explosion might yield inaccurate results since the elevated \(^{14}\mathrm{C}\) levels skew the measured ratios of carbon isotopes. This is why external factors like nuclear explosions must be considered in archaeological and geological studies involving radiocarbon dating.

Decay constant

The decay constant, denoted by \(\lambda\), is a fundamental concept in radioactive decay processes. It represents the probability per unit time that a given atom will decay. In the context of radiocarbon dating, this constant helps to determine the rate of decay for \(^{14}\mathrm{C}\).
The relationship between the decay constant and the half-life of the isotope is given by: \[ T_{1/2} = \frac{\ln(2)}{\lambda} \] where \(T_{1/2}\) is the half-life.
Understanding \(\lambda\) is crucial for accurately calculating the time elapsed since the death of the organism from which we obtained the fossil. This parameter ensures precise dating by allowing scientists to compute how much \(^{14}\mathrm{C}\) has decayed over a given period.

Exponential decay

Exponential decay is a process where the quantity of radioactive substance decreases at a rate proportional to its current value. In mathematical terms, it can be expressed by the equation: \[ N(t) = N_0 e^{-\lambda t} \] where \(N(t)\) is the amount of radioactive isotope remaining at time \(t\), \(N_0\) is the initial quantity, and \(\lambda\) is the decay constant.
This principle is pivotal in radiocarbon dating, allowing scientists to calculate the time elapsed by measuring the remaining \(^{14}\mathrm{C}\) in a sample. The exponential nature of the decay process ensures that even small changes in \(^{14}\mathrm{C}\) concentrations can be detected over large time spans, providing a reliable method for dating ancient organic materials.

Fossil age determination

Radiocarbon dating is a widely used technique for determining the age of fossils. It is based on measuring the concentrations of \(^{14}\mathrm{C}\) in a sample and using it to calculate the time since the organism died.
The equation for finding the age \(T\) of a fossil is derived from: \[ T = \frac{1}{\lambda} \ln \frac{C_0}{C} \] where \(C\) is the current concentration of \(^{14}\mathrm{C}\), and \(C_0\) is the original concentration.
Adjustments sometimes need to be made for external factors like nuclear explosions which can introduce additional \(^{14}\mathrm{C}\) into the environment, potentially skewing age estimates. By understanding all components involved in the dating process, more accurate estimations can be made, providing insights into the past events and timelines of earthly life.