Problem 69 - Master Resource Book in JEE Main Physics

Question

Question No. 69 to 76 are Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choice from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion If the half-life of a radioactive substance is 40 days then \(25 \%\) substance decays in 20 days. Reason \(\quad N=N_{0}\left(\frac{1}{2}\right)^{n}\) where \(n=\frac{\text { time elapsed }}{\text { half-life period }}\)

Step-by-Step Solution

Answer

(d) Assertion is false, but the Reason is true.

Step 1: Interpret the Assertion

The assertion states that for a substance with a half-life of 40 days, 25% of it decays in 20 days. This means that after 20 days, 75% of the substance should remain because 25% is gone.

Step 2: Understand the Reason

The reason provides a formula: \[ N = N_0 \left(\frac{1}{2}\right)^n \] where \[ n = \frac{\text{time elapsed}}{\text{half-life period}} \] This explains how the amount of substance remaining after a certain period can be calculated, depending on the half-life.

Step 3: Calculate the Decay Using the Formula

Using the reasoning formula, determine the fraction of the original substance remaining:1. Calculate \( n \): \[ n = \frac{20}{40} = 0.5 \]2. Calculate \( N \): \[ N = N_0 \left(\frac{1}{2}\right)^{0.5} \] This simplifies to: \[ N = N_0 \times \left(\frac{1}{\sqrt{2}}\right) \approx 0.707 \times N_0 \] This means that approximately 70.7% of the original substance remains after 20 days.

Step 4: Compare Calculation with Given Assertion

The calculation shows that approximately 70.7% of the substance remains, meaning that roughly 29.3% has decayed. This does not match the assertion, which incorrectly claims that only 25% decays in 20 days.

Step 5: Determine the Correct Answer

Both the assertion and the reason are related to decay of a radioactive substance. However, since the assertion is based on the wrong percentage of decay, it is false.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs and forms a new element as the nucleus of the atom changes. It may seem complex, but understanding this is crucial in fields like physics and environmental science.

During the decay process, radioactive elements emit particles and energy. The main thing that makes a substance radioactive is the instability of its atomic nucleus. This instability results from an imbalance in the number of protons and neutrons.

Here's what's interesting about radioactive decay:

• It's a random process but follows a predictable pattern over time.
• There are several types of decay processes like alpha, beta, and gamma decay.
• It can lead to changes in both the type of energy released and the kind of element formed.

Remember, despite being random at the atomic level, radioactive decay can be accurately described using statistical methods in large-scale scenarios.

Half-Life Calculation

Half-life is a concept used to describe the rate at which a radioactive substance decays. It is the time required for half of the radioactive nuclei in a sample to undergo decay.

The half-life of a material helps us predict how much of that material will remain over time. This is particularly useful in fields such as nuclear medicine, archaeology, and carbon dating. The concept may sound intimidating, but it’s very straightforward when broken down.

Here's a basic rundown on how to understand and calculate half-life:

• Identify the half-life period, which is usually provided.
• Use the formula \[ N = N_0 \left(\frac{1}{2}\right)^n \] where \(N_0\) is the original amount, and \(N\) is the remaining amount.
• The term \(n\) is determined by dividing the elapsed time by the half-life period.

To visualize: If you start with 100 units of a radioactive substance, and after one half-life, 50 units will be left. After another half-life, you'll have 25 units remaining.

Physics Problem Solving

When tackling physics problems, especially those involving concepts like radioactive decay and half-life, there's a systematic approach that can simplify the process:

1. **Interpret the Problem**: Understand what is being asked. In this context, you should define terms like half-life and how they relate to the problem.

2. **Analyze the Given Data**: Determine what information is provided and how it can be used. Differentiate between what is directly stated and what needs calculation.

3. **Apply the Right Equations**: In our context, meaning using the appropriate formula(s) to determine values like \(n\), which represents the number of half-lives over a given period.

4. **Calculate**: Perform the calculations. Take it step-by-step, and double-check each phase to avoid errors.

• Calculate \( n = \frac{\text{time elapsed}}{\text{half-life period}} \)
• Use the equation \( N = N_0 \left(\frac{1}{2}\right)^n \) to find the remaining quantity.

5. **Verify**: Compare your calculated results with the logical expectations. If calculations do not match logical outcomes, reassess your approach.

Using this technique, whether for classroom exercises or real-world applications, systematic problem-solving helps clarify the path to a solution.