Problem 102 - Master Resource Book in JEE Main Physics

Question

The half-life of a radioactive substance is \(20 \mathrm{~min}\). The approximate time interval \(\left(t_{2}-t_{1}\right)\) between the time \(t_{2}\) when \(\frac{2}{3}\) of it has decayed and time \(t_{1}\) when \(\frac{1}{3}\) of it had decayed is (a) \(14 \mathrm{~min}\) (b) \(20 \mathrm{~min}\) (c) \(28 \mathrm{~min}\) (d) \(7 \mathrm{~min}\)

Step-by-Step Solution

Answer

The time interval \(t_2 - t_1\) is 20 minutes.

Step 1: Understand the Decay Formula

The radioactive decay of a substance can be described by the formula \( N = N_0 \cdot (1/2)^{t/T_{1/2}} \), where \(N\) is the remaining quantity of the substance, \(N_0\) is the initial quantity, \(T_{1/2}\) is the half-life, and \(t\) is the time elapsed.

Step 2: Determine Time for \(\frac{1}{3}\) Decay

If \(\frac{1}{3}\) of the substance has decayed, then \(\frac{2}{3}\) remains. Using the formula \( \frac{2}{3}N_0 = N_0 \cdot \left(\frac{1}{2}\right)^{t_1/20} \), solve for \(t_1\). Simplifying results in \( \left(\frac{1}{2}\right)^{t_1/20} = \frac{2}{3} \) which means \(t_1 = 20 \log_{1/2}(\frac{2}{3})\).

Step 3: Determine Time for \(\frac{2}{3}\) Decay

When \(\frac{2}{3}\) has decayed, \(\frac{1}{3}\) remains. Solve \( \frac{1}{3}N_0 = N_0 \cdot \left(\frac{1}{2}\right)^{t_2/20} \). Thus, \( \left(\frac{1}{2}\right)^{t_2/20} = \frac{1}{3} \). Solving gives \(t_2 = 20 \log_{1/2}(\frac{1}{3})\).

Step 4: Find \(t_{2} - t_{1}\)

The time interval \( (t_2 - t_1) \) is calculated as \( 20 \log_{1/2}(\frac{1}{3}) - 20 \log_{1/2}(\frac{2}{3}) \). This simplifies to \( 20 (\log_{1/2}(\frac{1}{3}) - \log_{1/2}(\frac{2}{3})) \).

Step 5: Simplify Using Logarithm Properties

By logarithmic properties, \(\log_{1/2}(\frac{1/3}) - \log_{1/2}(\frac{2/3}) = \log_{1/2}(\frac{1/3} \times \frac{3/2}) = \log_{1/2}(\frac{1}{2}) = 1\).

Step 6: Calculate Final Time Interval

Thus, the time interval \(t_2 - t_1 = 20 \cdot 1 = 20\) minutes.

Key Concepts

Understanding Half-Life

Half-life is a fundamental aspect of radioactive decay. It refers to the time required for half of a radioactive substance to decay. This concept is crucial for understanding how the amount of a radioactive material decreases over time. Knowing the half-life helps predict how long it will take for radioactive material to become non-hazardous.
For example, if a substance has a half-life of 20 minutes, after 20 minutes only half of the original amount will remain. After another 20 minutes, only a quarter of the original amount will be left, and so on. This exponential decrease makes it easy to estimate how much material will be left after several half-lives. Understanding this allows us to accurately calculate remaining quantities at any point in time.

The Role of Logarithmic Functions in Radioactive Decay

Logarithmic functions play an essential role in solving problems related to radioactive decay. They are especially useful when working with exponential decay equations. Using logarithms helps simplify the calculations and make understanding decay processes less complex.
When we want to solve for time using the decay equation, we often encounter variables in the exponents. Here, logarithms are vital as they can help bring the exponents down to the base level, making it easier to solve. For instance, if we need to find the time when a certain fraction of a substance remains, we use logarithms to do so, transforming equations into solvable forms. This technique is indispensable when determining specific decay times or intervals.

Explaining the Decay Formula

The decay formula for radioactive substances is given by \( N = N_0 \cdot (1/2)^{t/T_{1/2}} \). This establishes a relationship between the remaining amount of a substance and time. Here:

• \(N\) is the remaining quantity.
• \(N_0\) is the initial quantity.
• \(T_{1/2}\) is the half-life.
• \(t\) is the time elapsed.

This formula tells us how much of a substance remains after a certain period, based on its half-life and the elapsed time. By substituting known values into the formula, we can solve for unknown quantities, such as how long it has taken for a certain fraction to decay. This versatility makes it a powerful tool in various fields like physics and chemistry that deal with radioactive materials.

Calculating Time Intervals in Radioactive Decay

Calculating time intervals between different levels of decay involves understanding the basic decay principles and using logarithmic functions. We need these skills to determine how long it takes for a substance to transition between specified decay stages.
In the given exercise, to find the interval between when two-thirds of the substance decays and when only one-third remains, we use the decay formula: first to find \(t_1\) and \(t_2\) separately using logarithms, then calculate the difference \(t_2 - t_1\). This approach demonstrates how decay progresses and allows us to solve specific decay timeline problems effectively. This calculation helps in accurately predicting durations for safety assessments and various scientific applications.