Problem 24 - A Complete Resource Book in Chemistry for JEE Main

Question

The half lives of two radioactive nuclides \(\mathrm{A}\) and \(\mathrm{B}\) are 1 and 2 min respectively. Equal weights of \(\mathrm{A}\) and \(\mathrm{B}\) are taken separately and allowed to disintegrate for \(4 \mathrm{~min}\). What will be the ratio of weights of \(\mathrm{A}\) and \(\mathrm{B}\) disintegrated? (a) \(1: 2\) (b) \(1: 1\) (c) \(1: 3\) (d) \(5: 4\)

Step-by-Step Solution

Answer

The ratio of weights of A and B disintegrated is 5:4 (option d).

Step 1: Understand the Problem

We are given two nuclides, A and B, with half-lives of 1 minute and 2 minutes respectively. We need to find the ratio of disintegrated weights after 4 minutes, given equal initial weights.

Step 2: Determine Remaining Amounts

For a nuclide, the remaining amount after time \( t \) is given by \( N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} \), where \( N_0 \) is the initial amount and \( T_{1/2} \) is the half-life. For A, \( T_{1/2} = 1 \) minute, and for B, \( T_{1/2} = 2 \) minutes.

Step 3: Calculate Remaining Weight of A

Using the formula for nuclide A over 4 minutes: \[ N_A(4) = N_0 \left(\frac{1}{2}\right)^{4/1} = N_0 \times \frac{1}{16} \] Thus, the remaining weight of A is \( \frac{N_0}{16} \).

Step 4: Calculate Disintegrated Weight of A

The disintegrated weight of A is given by \( N_0 - \text{remaining weight} \). Thus: \[ \, W_{diss,A} = N_0 - \frac{N_0}{16} = \frac{15N_0}{16} \]

Step 5: Calculate Remaining Weight of B

Using the formula for nuclide B over 4 minutes: \[ N_B(4) = N_0 \left(\frac{1}{2}\right)^{4/2} = N_0 \times \frac{1}{4} \] So, the remaining weight of B is \( \frac{N_0}{4} \).

Step 6: Calculate Disintegrated Weight of B

The disintegrated weight of B is given by \( N_0 - \text{remaining weight} \). Thus: \[ W_{diss,B} = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

Step 7: Calculate Ratio of Disintegrated Weights

The ratio of weights disintegrated is \( \frac{W_{diss,A}}{W_{diss,B}} \): \[ \text{Ratio} = \frac{\frac{15N_0}{16}}{\frac{3N_0}{4}} = \frac{15}{12} = \frac{5}{4} \] Therefore, the ratio of weights of A to B disintegrated is 5:4.

Key Concepts

Half-life

The term 'half-life' refers to the time required for half of the radioactive nuclei in a sample to undergo decay. This concept is crucial in understanding how radioactive materials change over time.
It helps scientists predict how long it will take for a material to become harmless. Each radioactive nuclide, a type of atom, has a unique half-life that does not change with time or environmental conditions.

• A shorter half-life means the substance decays quickly.
• A longer half-life indicates the substance remains active for a longer period.

In exercises involving radioactive decay, knowing the half-life allows you to calculate how much of a substance remains after a given period. For instance, in the provided exercise, nuclide A has a half-life of 1 minute, which means after virtually any 1-minute period, only half of it remains. Understanding this principle is key to solving the exercise correctly.

Disintegration ratio

The disintegration ratio is a helpful way to compare how different substances decay over time. In the context of radioactive decay, it is the ratio of the amounts of two substances that have decayed over a specific time period.
To calculate the disintegration ratio, one must find how much of each substance has decayed and then compare these amounts using a ratio.

• The formula used is: \( Disintegrated\,Amount = Initial\,Amount - Remaining\,Amount \).
• In the solution provided, the disintegration ratio of nuclide A to nuclide B is found to be 5:4.
• This means that for every 5 parts of nuclide A that decayed, 4 parts of nuclide B decayed in the same time.

The disintegration ratio gives insight into the relative stability and decay characteristics of different nuclides when they are subjected to the same conditions.

Radioactive nuclides

Radioactive nuclides, also known as radioisotopes or radionuclides, are atoms that have excess nuclear energy, making them unstable. This instability leads to the process known as radioactive decay, whereby the nuclide transforms into a different element or a different state, releasing radiation in the form of particles or electromagnetic waves.
Key aspects of radioactive nuclides include:

• Each nuclide has a distinct half-life, influencing its rate of decay.
• During decay, nuclides may emit alpha particles, beta particles, or gamma rays, all of which have different implications for the decay process and stability.
• The decay process continues until a stable nuclide is formed.

Understanding the properties and behaviors of radioactive nuclides is essential for applications ranging from medical diagnostics to nuclear energy production. The exercise provided gives a practical example of how two different nuclides, A and B, behave over time, reflecting their unique decay rates.