Problem 81 - Master Resource Book in JEE Main Physics

Question

The normal activity of living carbon containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive \({ }_{6}^{14} \mathrm{C}\) present with the stable carbon isotope \({ }_{6}^{12} \mathrm{C}\). when the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life \((5730 \mathrm{yr})\) of \({ }_{6}^{14} \mathrm{C}\) and the measured activity, the age of the specimen can be approximately estimated. This is the principle of \({ }_{6}^{14} \mathrm{C}\) dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilization. (a) \(5224 \mathrm{yr}\) (b) \(4224 \mathrm{yr}\) (c) \(8264 \mathrm{yr}\) (d) \(6268 \mathrm{yr}\)

Step-by-Step Solution

Answer

The age of the Indus-Valley civilization is approximately 5224 years, option (a).

Step 1: Understand the Concept of Radioactive Decay

Radioactive carbon (^{14}C) undergoes decay over time. Living organisms maintain a specific amount of this isotope, indicated by a decay rate of 15 decays per minute per gram of carbon. After death, this decay rate drops as the organism no longer exchanges carbon with the atmosphere.

Step 2: Use the Decay Law Formula

The activity of carbon in a dead organism decreases as it no longer receives new ^{14}C. We use the radioactive decay formula to find age: \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] Here, \(N(t)\) is the current activity (9 decays/min), \(N_0\) is the initial activity (15 decays/min), \(t\) is the elapsed time, and \(T_{1/2}\) is the half-life (5730 years).

Step 3: Solve for the Age of the Specimen

First, express the ratio of current to initial activity: \[ \frac{N(t)}{N_0} = \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]Plug in the values:\[ \frac{9}{15} = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \]Simplify to find:\[ 0.6 = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \]

Step 4: Calculate the Time (t)

To find \(t\), take the logarithm of both sides:\[ \log(0.6) = \frac{t}{5730} \times \log \left( \frac{1}{2} \right) \]Solve for \(t\):\[ t = \frac{5730 \times \log(0.6)}{\log(0.5)} \]Calculate the values using logarithms.

Step 5: Final Calculation and Interpretation

Using the logarithms:\[ \log(0.6) \approx -0.2218 \] \[ \log(0.5) \approx -0.3010 \]Plug these logs into the equation:\[ t \approx \frac{5730 \times (-0.2218)}{-0.3010} \]\[ t \approx 5224 \text{ years} \]

Step 6: Conclude and Choose the Correct Answer

Based on the calculation, the approximate age of the Indus-Valley civilization specimen is 5224 years. Thus, option (a) is correct.

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics. It refers to a process where an unstable atomic nucleus loses energy by emitting radiation. Over time, the radioactive isotopes change to a more stable form. This process is completely natural and continues without external influence until the isotope reaches a stable state.

For instance, Carbon-14 (\(^{14}\text{C}\)), a common isotope used in archaeology, undergoes decay. In living organisms, Carbon-14 levels remain stable due to the constant exchange with the atmosphere. This balance results in a predictable decay rate, around 15 decays per minute per gram of carbon. When an organism dies, the Carbon-14 decay begins without replenishment, leading to a decrease in activity.

The predictable nature of this decay allows scientists to determine the age of archaeological specimens, such as those from ancient civilizations.

Half-life Calculation

The concept of half-life is crucial to understanding radioactive decay. Half-life refers to the time required for half of the radioactive atoms in a sample to decay. It's a constant value, specific to each radioactive isotope, and forms the basis of radiocarbon dating.

For Carbon-14, the half-life is approximately 5730 years. This means that in 5730 years, half of the initial amount of Carbon-14 in a sample will have decayed to Nitrogen-14. Understanding this constant helps archaeologists estimate the time since an organism stopped exchanging carbon with the environment when it died.

The decay formula used is:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\]where \(N(t)\) is the number of decays per minute at time \(t\), \(N_0\) is the initial number, and \(T_{1/2}\) is the half-life. By using logarithms, the exact time since death, or the age of the specimen, can be calculated.

Indus-Valley Civilization Age Estimation

Estimating the age of the Indus-Valley Civilization using radiocarbon dating gives fascinating insight into ancient human history. The civilization, known for its advanced urban planning and architecture, thrived around regions now part of modern-day Pakistan and northwest India.

With a specimen showing an activity of 9 decays per minute per gram of carbon compared to a living organism's 15 decays, we can use the radioactive decay and half-life principles to find its age. By inserting these values into the decay formula and solving the equation, we determine the time passed since the organism ceased living.

In this case, using a calculated relationship:\[ \frac{9}{15} = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \]we simplify and solve for \(t\).This computation helps us estimate that the age of that particular Indus-Valley specimen is approximately 5224 years, offering invaluable data on the timelines of their civilization.