Problem 30 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Four vessels \(1,2,3\) and 4 contain respectively, 10 g-atom \(\left(t_{1 / 2}=10 \mathrm{~h}\right), 1 \mathrm{~g}\) -atom \(\left(t_{1 / 2}=5 \mathrm{~h}\right), 5\) g-atom \(\left(t_{1 / 2}=2 \mathrm{~h}\right)\) and 2 g-atom \(\left(t_{1 / 2}=1 \mathrm{~h}\right)\) of different radioactive nuclides. In the beginning, the maximum radioactivity would be exhibited by the vessel (a) 4 (b) 3 (c) 2 (d) 1

Step-by-Step Solution

Answer

In the beginning, the maximum radioactivity would be exhibited by the vessel 3.

Step 1: Understand Radioactivity

The radioactivity of a sample is directly proportional to the number of radioactive atoms present and inversely proportional to the half-life (\(t_{1/2}\)) of the nuclide. Radioactivity (\(R\)) can be expressed as \(R \propto \frac{N}{t_{1/2}}\), where \(N\) is the number of g-atom.

Step 2: Calculate the Initial Radioactivity for Each Vessel

Since the proportionality constant is the same for each nuclide, we can compare the radioactivity of the vessels by calculating \(\frac{N}{t_{1/2}}\) directly. For vessel 1: \(\frac{10}{10} = 1\), for vessel 2: \(\frac{1}{5} = 0.2\), for vessel 3: \(\frac{5}{2} = 2.5\), and for vessel 4: \(\frac{2}{1} = 2\).

Step 3: Compare the Results

Looking at the calculated ratios, vessel 3 has the highest value of \(\frac{N}{t_{1/2}}\), which implies that it has the maximum radioactivity at the beginning.

Key Concepts

Half-Life of Nuclides

Understanding the half-life of a radioactive nuclide is key to many areas of physical chemistry and nuclear physics. The half-life, symbolized as t_1/2, is the time required for half the atoms of a radioactive sample to decay. This concept illustrates that radioactive decay is an exponential process, meaning that it is not linear but decreases by half in set periods.

For example, if a nuclide has a half-life of 2 hours, it means that every 2 hours, half of the remaining atoms will have decayed. After the first half-life, 50% of the atoms remain, after the second, 25%, and this pattern continues. In the context of our problem, calculating the activity of the nuclide depends heavily on both its initial quantity N and its half-life. The nuclide with the shortest half-life does not necessarily have the highest radioactivity, a common misconception. Instead, the ratio of the number of atoms to the half-life, as the solution step illustrates, determines the initial radioactivity.

Calculating Radioactivity

The process of calculating radioactivity, which is the emission of particles and energy from an unstable atomic nucleus, involves knowing not just the quantity of radioactive material but also its decay rate. As we break down in the exercise, the radioactivity R is inversely proportional to the half-life; the shorter the half-life, the more radioactive decays per unit of time, assuming the same quantity of atoms.

The exercise correctly tackles the problem by comparing ratios of number of grams-atom over their respective half-lives. By representing radioactivity with a simple mathematical ratio, students can directly compare the relative activity of different samples without needing complex calculations. It is important to emphasize that radioactivity is not about just one factor, but the interplay between quantity and decay rate, which is concisely shown in the steps provided.

Concepts in Physical Chemistry

Physical chemistry is a branch of chemistry that deals with the quantitative relationships and theories that underpin the behavior of matter at the atomic and molecular level. Concepts like radioactivity highlight the physics behind chemical changes—in this case, nuclear reactions rather than chemical bonds. The principles used to solve our exercise are grounded in the physicochemical understanding of atomic stability and kinetics.

Moreover, understanding radioactive decay is crucial for applications ranging from radiometric dating to nuclear energy and medicine. In educational settings, it is essential to present this interrelation between quantities, properties, and rates in a manner that underscores the intrinsic physics within chemistry. Exercises like these allow students to practice that interdisciplinary thinking and to appreciate the atomic scale at which both chemical and physical processes drive the observable world around us.