Problem 82 - Master Resource Book in JEE Main Physics

Question

Half-life of radioactive substance is 140 days. Initially, is \(16 \mathrm{~g}\). Calculate the time for this substance when it reduces to \(1 \mathrm{~g}\) (a) 140 days (b) 280 days (c) 420 days (d) 560 days

Step-by-Step Solution

Answer

(d) 560 days

Step 1: Understanding the Half-Life Concept

The half-life of a radioactive substance is the time it takes for half of the substance to decay. For this exercise, the half-life is given as 140 days.

Step 2: Determine Remaining Fraction of Substance

Initially, we have 16g and want it to reduce to 1g. The fraction of the substance remaining is \( \frac{1}{16} \) because \( \frac{1 \text{ g}}{16 \text{ g}} = \frac{1}{16} \).

Step 3: Express Remaining Fraction Using Half-Lives

In terms of half-lives, \( \left( \frac{1}{2} \right)^n = \frac{1}{16} \), which simplifies to \( 2^{-n} = 2^{-4} \). Hence, \( n = 4 \).

Step 4: Calculate Time Using Half-Lives

Each half-life is 140 days. If 4 half-lives are needed for the substance to reduce to 1g, the total time is \( 4 \times 140 \) days.

Step 5: Final Calculation

Multiply to find the total time: \( 4 \times 140 = 560 \) days.

Key Concepts

Half-life calculation

The half-life of a radioactive substance is a crucial concept in understanding how these substances change over time. It refers to the time it takes for half of the amount of a radioactive material to decay or transform into a different element or isotope.
For instance, if you start with 16 grams of a radioactive substance, after one half-life, only 8 grams would be left; after two half-lives, only 4 grams would remain, and so on.
This predictable reduction process allows us to make precise calculations about the time it will take for a substance to decrease to a certain amount.

• For the exercise at hand, the half-life is set at 140 days.
• To find when the substance will reduce from 16g to 1g, we need to check how many half-lives it takes to reach this level.

By understanding that after each half-life the mass is halved, we can calculate the total elapsed time by multiplying the number of half-lives required with the duration of each half-life.

Radioactive substances

Radioactive substances are materials that spontaneously emit radiation as they decay into more stable forms. This decay process releases energy in the form of alpha, beta, or gamma particles.
These substances have both beneficial and harmful effects, depending on their use and exposure levels.

• In medicine, they help trace or treat certain conditions.
• In nuclear power, they generate energy.
• However, excessive exposure can be hazardous to health.

The exercise involves a radioactive substance with a known half-life, allowing us to predict its reduction over time. Proper safety measures always have to be considered when working with such materials to avoid negative consequences. Thus, understanding radioactive decay is essential for applications and handling these substances safely in various fields.

Exponential decay

Exponential decay is a mathematical concept used to describe processes where a quantity decreases at a rate proportional to its current value. In essence, the amount diminishes quickly at first and more slowly over time.
Radioactive decay is a prime example, modeled by exponential functions.
For a given radioactive substance, the relationship between time and remaining substance can be represented through the formula:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/H} \]Here,

• \( N(t) \) is the remaining amount at time \( t \),
• \( N_0 \) is the initial amount,
• and \( H \) is the half-life.

Through exponential decay calculations, we can efficiently determine the time at which the material's mass reduces to a specific value, such as from 16g to 1g in the given exercise. This makes the concept useful in various practical situations.