Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

If a substance with half-life of 3 days is taken to another place in 12 days. What is the amount of substance left now? (a) \(1 / 8\) (b) \(1 / 32\) (c) \(1 / 4\) (d) \(1 / 16\)

Step-by-Step Solution

Answer

(d) \(1 / 16\)

Step 1: Identify the Number of Half-Lives

The half-life of the substance is 3 days. We need to determine how many half-lives have passed in 12 days. Calculate this by dividing the total time (12 days) by the half-life (3 days): \[ \text{Number of half-lives} = \frac{12}{3} = 4 \] So, 4 half-lives have passed.

Step 2: Calculate the Remaining Amount of Substance

With each half-life, the amount of the substance reduces by half. Starting with a full amount \( (1) \), after 4 half-lives, the amount remaining is: \[ 1 \times \left(\frac{1}{2}\right)^4 \] Calculate: \[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \] Thus, \( \frac{1}{16} \) of the substance remains.

Step 3: Identify the Correct Answer

We found that \( \frac{1}{16} \) of the substance remains after 12 days. Compare this with the options given: (a) \(\frac{1}{8}\) (b) \(\frac{1}{32}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{16}\) The correct answer is (d) \(\frac{1}{16}\).

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept that describes a process where the quantity of a substance decreases at a rate proportional to its current value. In simple terms, it means that the substance diminishes slowly at first but then decreases more rapidly over time. This pattern is often seen in processes like radioactive decay, where the number of atoms of a radioactive substance reduces over time.

For any process exhibiting exponential decay, the rate of decrease is characterized by a constant, called the decay constant. We can calculate the remaining amount of a substance using the exponential decay formula: \ \( A(t) = A_0 \times e^{-\lambda t} \) \ Here:

• \( A(t) \) is the amount remaining at time \( t \).
• \( A_0 \) is the initial amount.
• \( \lambda \) is the decay constant.
• \( t \) is time.

This relationship highlights how the initial quantity and decay constant impact how rapidly the substance decreases over time.

Nuclear Chemistry

Nuclear chemistry is a branch of chemistry that focuses on the processes, properties, and reactions of nuclei within atoms. It's deeply intertwined with concepts like radioactive decay, fission, fusion, and the strong nuclear force.

Nuclear chemistry plays a significant role in various technologies and scientific developments. For instance, it is crucial in medical imaging techniques such as PET scans, which utilize radioactive tracers to view internal structures of the body. It’s also central to understanding the processes within stars, where nuclear reactions fuel their cores.

The field explores how unstable isotopes break down over time, which leads us to our next topic, radioactive decay. Understanding how and why certain elements decay helps us harness nuclear power, while also guiding safe handling and storage practices for radioactive materials.

Radioactive Decay Law

The radioactive decay law is a theoretical framework that describes how unstable nuclei lose energy by emitting radiation. This process occurs until the nuclei reach a stable state, resulting in the transformation into a different element or a more stable isotope.

The law is mathematically expressed through a first-order differential equation that dictates the decay process. The amount of radioactive substance decreases over time according to: \ \( N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/T} \) \ Here:

• \( N(t) \) is the number of undecayed atoms at time \( t \).
• \( N_0 \) is the initial number of atoms.
• \( T \) is the half-life of the substance.

The concept of half-life is critical here, representing the time required for half of the radioactive atoms in a sample to decay. This property is unique to each radioactive isotope and aids in calculations involving how much of a substance remains over time, such as in the given exercise.