Problem 92 - Master Resource Book in JEE Main Physics

Question

If the mass of a radioactive sample is doubled, the activity of the sample and the disintegration constant of the sample are respectively. (a) Increases, remains the same (b) Decreases, increases (c) Decreases, remains same (d) Increases, decreases

Step-by-Step Solution

Answer

(a) Increases, remains the same

Step 1: Understand Radioactive Activity

The activity of a radioactive sample, denoted as \( A \), is the rate at which the radioactive material decays. It is directly proportional to the number of radioactive nuclei present in the sample. Thus, if the mass of the sample doubles, the number of radioactive nuclei also doubles, causing the activity to increase. Therefore, the activity increases when the mass is doubled.

Step 2: Examine the Disintegration Constant

The disintegration constant, also known as the decay constant, denoted by \( \lambda \), is a characteristic of each radioactive isotope and is independent of the sample's mass. The decay constant remains the same regardless of how much sample you have, since it is a property intrinsic to the radioactive isotope itself.

Step 3: Combine Observations

Based on the understanding of activity and decay constant, when the mass of a sample is doubled, the activity increases, but the disintegration constant remains unchanged.

Key Concepts

Activity of Radioactive Sample

The activity of a radioactive sample is a crucial concept in understanding how radioactive materials decay over time. Activity refers to the rate at which a radioactive sample disintegrates, meaning how many radioactive nuclei decay per unit of time. This rate is measured in becquerels (Bq), where one becquerel corresponds to one disintegration per second.
To envision this, think of activity as a measure of how 'active' a sample is, in terms of its emission of radioactive particles. The more nuclei that are present to decay, the higher the activity observed.
One important thing to note is the direct relationship between the number of radioactive nuclei present in the sample and its activity.

• If the number of radioactive nuclei doubles, the activity also doubles.
• This is because more nuclei are available to undergo decay over time.

Such a principle explains why when a radioactive sample's mass is doubled, its activity increases proportionally. The mass increase doubles the number of radioactive nuclei available for decay, leading to increased activity.

Disintegration Constant

The disintegration constant plays an essential role in the study of radioactive decay. It is an inherent property of the radioactive isotope, unique to each. This constant is represented by the symbol \( \lambda \) and determines the rate at which the nuclei of a given isotope decay.
The disintegration constant is significant because it remains unchanged, regardless of the sample size, amount, or mass. This intrinsic nature allows the disintegration constant to act as a fingerprint of sorts for radioactive isotopes.

• The value of \( \lambda \) represents the probability per time unit that a given nucleus will decay.
• Lower values of \( \lambda \) mean slower decay, while higher values indicate faster decay.

So, if we were to double the mass of a radioactive sample, the disintegration constant persists unaffected. This constancy is crucial when calculating the remaining radioactive material over a given period and determining its future activity.

Radioactive Nuclei

Radioactive nuclei are the atomic units in radioactive materials that are unstable and prone to decay. These nuclei undergo transformations as they emit particles or radiation to achieve more stable configurations.
Radioactivity involves transformations such as alpha decay, beta decay, and gamma emission, each releasing different particles and energy.
It helps to remember that the number of radioactive nuclei present sets the stage for the overall activity of the sample. The more nuclei there are, the higher the chances of decay happening per second, thereby increasing activity.

• Each decay transforms a radioactive nucleus into a different, usually more stable, nucleus.
• Over time, this leads to a decrease in the number of radioactive nuclei, lowering the sample's activity.

Understanding radioactive nuclei and their decay processes are crucial for managing and using radioactive materials safely in industries like medical imaging, nuclear power, and archaeological dating.