Question

The time of decay for a nuclear reaction is given by \(t=4 t_{1 / 2}\). The relation between the mean life \((T)\) and time of decay \((t)\) is given by : (a) \(2 T \ln 2\) (b) \(4 T \ln 2\) (c) \(2 T^{4} \ln 2\) (d) \(\frac{1}{T^{2}} \ln 2\)

Short answer

The relationship between mean life (T) and time of decay (t) is (b) 4T ln(2).

Step-by-step solution

Understanding the Given Data

We are given the decay time of a reaction as a multiple of its half-life, i.e., t = 4t_{1/2}. The goal is to find a relationship between mean life (T) and decay time (t). The mean life is the average time a nucleus remains before it decays, mathematically it is given by the reciprocal of the decay constant (λ). The relation between half-life and the decay constant is given by t_{1/2} = (ln(2))/λ.

Express Mean Life in Terms of Decay Constant

Since T is the reciprocal of the decay constant, we write T = 1/λ. The relationship between half-life and the decay constant allows us to substitute the decay constant in the expression for T. Thus we get T = 1/λ = t_{1/2}/ln(2).

Find the Relationship Between t and T

Now we can express t as a multiple of T by substituting the expression of t_{1/2} in terms of T. Substituting, we get t = 4t_{1/2} = 4(T ln(2)), which shows the direct proportionality between decay time and mean life.

Key concepts

Half-Life

Half-life, represented as \( t_{1/2} \), is a term used in nuclear physics to describe the time required for half of the radioactive nuclei in a sample to undergo decay. It's a constant for any given radioactive substance, making it a vital tool for scientists to predict how a substance behaves over time.

Imagine placing 100 radioactive atoms on a table; if these atoms have a half-life of one year, you would only have 50 atoms left at the end of the first year, 25 after the second, and so on. The concept of half-life helps us grasp how quickly a substance loses its radioactivity and is key for topics in physical chemistry, especially when discussing nuclear reactions and stability.

Mean Life

Mean life, represented by \( T \), is the average lifetime of a radioactive particle before it decays. Unlike half-life which addresses when half of the sample has decayed, mean life considers the full lifespan of all particles.

A good analogy is to think about the mean life as the average time a group of light bulbs would last—a few may burn out quickly, some might last far longer, but the mean life is their average shining duration. In our earlier example of radioactive atoms, while half-life tells us when half the atoms would decay, mean life tells us the average time each atom lasted before decaying. Calculating mean life is crucial in fields like medical imaging and radiation therapy, where understanding the duration of a substance's activity is essential.

Decay Constant

The decay constant, \( \lambda \), is intrinsic to each radioactive substance and denotes the probability of decay per unit time for a single nucleus. It's an essential parameter because it governs the exponential decay rate of the substance.

To visualize this concept, imagine every radioactive atom as a ticking time bomb with a differently set timer. The decay constant is like the likelihood of each bomb exploding per second. The lower the decay constant, the more stable the substance, and it will take longer to decay. Conversely, a high decay constant indicates a substance that decays rapidly. In mathematical terms, it's inversely related to the mean life \( T = 1/\lambda \). Understanding the decay constant is key to mastering nuclear chemistry equations and is a fundamental topic for students preparing for competitive exams like JEE in physical chemistry.

Physical Chemistry JEE

Physical Chemistry in the context of competitive exams like the Joint Entrance Examination (JEE) covers a wide range of topics, including nuclear reactions and their kinetics. Grasping concepts like half-life, mean life, and decay constant is crucial for JEE aspirants, as these topics bridge the gap between theoretical chemistry and practical applications.

Questions in the JEE often require not just rote memorization but a deep understanding and ability to apply these concepts to solve complex problems. By focusing on the principles of radioactive decay and the equations that describe them, students can gain a competitive edge in the examination. It's also valuable for future studies as these concepts lay the foundation for understanding real-world phenomena involving radioactivity, such as carbon dating, nuclear medicine, and energy production.