Problem 73 - Master Resource Book in JEE Main Physics

Question

Half-life of radium is \(1600 \mathrm{yr}\). Its average life is (a) \(3200 \mathrm{yr}\) (b) \(4800 \mathrm{yr}\) (c) \(2308 \mathrm{yT}\) (d) \(4217 \mathrm{yr}\)

Step-by-Step Solution

Answer

The average life of radium is approximately 2308 years.

Step 1: Understanding Half-Life

The half-life of a radioactive substance is the time taken for half of its atoms to decay. For radium, the half-life is given as 1600 years.

Step 2: Formula for Average Life

The average life (also known as mean lifetime) of a radioactive substance is given by the formula: \[ \text{Average Life} = \text{Half-Life} \times \frac{1}{\ln(2)} \]This can be simplified to:\[ \text{Average Life} = \frac{\text{Half-Life}}{0.693} \]

Step 3: Substitute the Given Values

Substitute the half-life of radium into the formula:\[ \text{Average Life} = \frac{1600}{0.693} \]

Step 4: Calculate the Average Life

Perform the division to find the average life:\[ \text{Average Life} \approx 2308 \text{ years} \]

Key Concepts

Half-Life

Half-life is an essential concept in understanding radioactive decay. It refers to the period it takes for one-half of the atoms in a radioactive sample to disintegrate. Imagine you start with 100 atoms of a certain radioactive material. After one half-life, only 50 of those original atoms remain; the rest have decayed into another element or isotope. This process continues until all the atoms have transformed, occurring in predictable half-life periods.
Key points on half-life:

• It is constant for a given material, unaffected by external factors like temperature or pressure.
• Half-life helps predict how long a substance will remain active or dangerous.
• It's crucial for dating archaeological finds, medical uses, and nuclear power management.

In summary, half-life provides a clear measure of the stability and rate of decay of a radioactive substance, making it a fundamental concept in nuclear physics.

Average Life

The average life, or mean lifetime, of a radioactive material provides another perspective on its decay process. This measure considers how long an atom remains in its original state before decaying. Unlike half-life, average life includes all the atoms from the very beginning of the decay process.
The formula for average life is interconnected with half-life:

• The mathematical relationship is expressed as: \[ \text{Average Life} = \frac{\text{Half-Life}}{0.693} \]
• This equation derives from the natural logarithm associated with decay calculations, specifically the number \( \ln(2) \), approximated as 0.693.

Average life is often slightly longer than the half-life, offering a broader view of the decay trend over time.

Mean Lifetime

Mean lifetime is simply another term for average life, illustrating the same concept using different terminology. Although 'half-life' and 'mean lifetime' are closely linked, they are distinct in explaining the behavior of radioactive substances.
Understanding mean lifetime:

• It offers a mean value of how long particles remain undecayed, thus useful in statistical analysis.
• Applying the formula for average life, you divide the half-life by 0.693, reflecting constant exponential decay rates.

These terms, whether labeled as `average life` or `mean lifetime`, deliver insights into the quantitative nature of decay and enable scientists to compare various elements effectively. Each serves as a tool for comprehensively predicting and understanding radioactive behavior over time.