Problem 87 - A Complete Resource Book in Chemistry for JEE Main

Question

The radioisotope of hydrogen has a half-life of \(12.33\) y. What is the age of an old bottle of wine, whose \({ }_{1} \mathrm{H}^{3}\) radiation is \(10 \%\) of that present in a new bottle of wine? (a) 41 years (b) \(123.3\) years (c) \(1.233\) years (d) 410 years

Step-by-Step Solution

Answer

The age of the old bottle of wine is 41 years.

Step 1: Understand the Problem

We are given that the radiation of the radioisotope of hydrogen (tritium) in an old bottle of wine is 10% of that in a new bottle, and we need to calculate the age of the old bottle. We know the half-life of tritium is \(12.33\) years.

Step 2: Use the Decay Formula

To find the age of the wine bottle, we use the radioactive decay formula:\[ N = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]where \(N\) is the remaining quantity of the isotope, \(N_0\) is the initial quantity, \(t\) is the time elapsed, and \(t_{1/2}\) is the half-life.

Step 3: Set Up the Equation

We are given \( \frac{N}{N_0} = 0.10 \) or 10%. Substituting this into the decay formula gives:\[ 0.10 = \left( \frac{1}{2} \right)^{\frac{t}{12.33}} \]

Step 4: Solve for Time \(t\)

Take the natural logarithm of both sides to solve for \(t\):\[ \ln(0.10) = \ln\left(\left( \frac{1}{2} \right)^{\frac{t}{12.33}}\right) \]Using the logarithm power rule:\[ \ln(0.10) = \frac{t}{12.33} \times \ln\left(\frac{1}{2}\right) \]Solve for \(t\):\[ t = \frac{\ln(0.10)}{\ln(0.5)} \times 12.33 \]

Step 5: Calculate the Result

Calculate \( t \) by evaluating the expression:\[ t = \frac{\ln(0.10)}{\ln(0.5)} \times 12.33 \approx 41 \text{ years} \]

Key Concepts

Half-Life Calculation

In the world of radioactive decay, the term **half-life** plays a crucial role. It refers to the time required for half of the radioactive substance in a sample to decay. This concept is especially helpful when dealing with isotopes like tritium in our exercise. Imagine you have a certain amount of a radioactive substance. After one half-life period, only half of the original amount remains.

To calculate the age of an object using half-life, you can apply the decay formula:

• \(N = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \(N\) is the remaining substance, \(N_0\) is the original amount, \(t\) is the time elapsed, and \(t_{1/2}\) is the half-life.
• For the exercise, where tritium has a half-life of 12.33 years, this equation allowed us to find how much time it took for the radiation to drop to 10% of its original level.

Understanding half-life calculations is key to dating ancient objects, helping archaeologists, and researchers measure the age of artifacts accurately.

Tritium Decay

Tritium, or \( {}^{3}_{1} \text{H} \), is a radioactive isotope of hydrogen. It occurs naturally in the environment and can also be produced by nuclear reactions. Its half-life of 12.33 years means it's suitable for studying processes that happen over decades.

In tritium decay, nuclei of tritium spontaneously change by emitting a beta particle. This transformation reduces its quantity over time. As shown in the exercise, even though only 10% of tritium was left in the wine bottle, we could use its known decay properties to accurately determine how many years had passed since the wine was produced.

Knowing about tritium decay is not only useful for scientific exercises like this but is critical in fields such as

• environmental monitoring, where tracking tritium can help assess contamination levels, and
• hydrology, for tracking the movements of water masses.

Radiocarbon Dating

Radiocarbon dating is a scientific method used to determine the age of an artifact or fossil. It relies on the principle of radioactive decay and the concept of half-life, often involving isotopes like carbon-14, but in some cases, other suitable isotopes like tritium can be used. This dating technique is invaluable for archaeologists and geologists when dating organic materials from ancient times.

By measuring the remaining amount of the radioactive isotope and factoring in its decay rate (half-life), scientists can backtrack and estimate when the organism was alive. In our exercise, this method of understanding decay was applied to calculate the age of a bottle of wine based on the amount of tritium left.

Radiocarbon dating is reliable because:

• It considers known decay rates, precisely balancing the calculations.
• Standard tables and formulas provide a consistent method of determining the elapsed time.
• Its use is widely accepted in fields such as archaeology and earth sciences.

Mastering radiocarbon dating principles allows for a better understanding of how scientists piece together Earth's history and reconstruct past climates and ecosystems.