Problem 72 - Master Resource Book in JEE Main Physics

Question

A radioactive isotope has a half-Ifie of \(T\) years. How long will it take the activity to reduce to (a) \(4.5 \mathrm{~T}\) and \(7.5 \mathrm{~T}\) (b) \(9.5 \mathrm{~T}\) and \(5 \mathrm{~T}\) (c) \(5 \mathrm{~T}\) and \(9.5 \mathrm{~T}\) (d) \(5 \mathrm{~T}\) and \(6.65 \mathrm{~T}\)

Step-by-Step Solution

Answer

(a) 4.5T and 7.5T years; (b) 9.5T and 5T years; (c) 5T and 9.5T years; (d) 5T and 6.65T years.

Step 1: Understand the Problem

We need to calculate how long it takes for a radioactive isotope's activity to reduce to given fractions of its initial state, based on its half-life.

Step 2: Half-life Formula

The general formula for radioactive decay is given by: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \]where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is time elapsed, and \(T\) is the half-life.

Step 3: Rearrange for Time

We need to solve for \(t\) which gives us:\[ \frac{N(t)}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{T}} \]Take the logarithm on both sides to rearrange for \(t\):\[ \frac{t}{T} = \log_{0.5}\left(\frac{N(t)}{N_0}\right) \]\[ t = T \cdot \log_{0.5}\left(\frac{N(t)}{N_0}\right) \]

Step 4: Handling Multiple Durations Simultaneously

Each sub-question gives you two time values in terms of \(T\). Let's analyze each scenario individually to find the time in years for activity reduction.

Step 5: Evaluate (a) for \(4.5T\) and \(7.5T\)

Using the simplified formula \(t = mT\) (for each case) where \(m\) is the multiplier, it is straightforward. Since \(m\) implies direct substitution:- For \(4.5T\), it takes \(4.5T\) years,- For \(7.5T\), it takes \(7.5T\) years.

Step 6: Evaluate (b) for \(9.5T\) and \(5T\)

Again, substitute the given multipliers directly:- For \(9.5T\), it takes \(9.5T\) years,- For \(5T\), it takes \(5T\) years.

Step 7: Evaluate (c) for \(5T\) and \(9.5T\)

Direct substitution of the given multipliers:- For \(5T\), it takes \(5T\) years,- For \(9.5T\), it takes \(9.5T\) years.

Step 8: Evaluate (d) for \(5T\) and \(6.65T\)

Substitute the multiplier directly for each case:- For \(5T\), it takes \(5T\) years,- For \(6.65T\), it takes \(6.65T\) years.

Key Concepts

Half-Life Calculation

Half-life is a crucial concept in radioactive decay, defined as the time required for a substance to reduce to half of its initial quantity. This is essential for understanding how quickly a radioactive isotope will diminish in activity over time. To determine the time it takes for a certain reduction in activity, the half-life formula is employed.

• The formula used is: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \]where:
  • \(N(t)\) is the remaining quantity after time \(t\),
  • \(N_0\) is the initial quantity,
  • \(t\) is the time elapsed,
  • \(T\) is the half-life.

For calculations involving multiple time scenarios, such as determining when a radioisotope reduces to specific fractions of its original activity, understanding half-life allows us to calculate the exact times needed for these reductions.

Logarithmic Functions in Physics

Logarithms provide a way to manage the exponential nature of radioactive decay and are essential in simplifying and solving decay-related problems. In this context, logarithmic transformation helps rearrange the decay equation to isolate the time variable. This reorganization enables easy computation of the time required for various degrees of activity reduction.

• By taking the logarithm of both sides of the half-life formula, we get:\[ \log_{0.5}\left(\frac{N(t)}{N_0}\right) = \frac{t}{T} \]
• Rearranging for \(t\) gives:\[ t = T \cdot \log_{0.5}\left(\frac{N(t)}{N_0}\right) \]
• This equation allows us to compute the time \(t\) for different levels of remaining activity as multiples of the half-life \(T\).

Logarithmic functions simplify the process by reducing complex exponential expressions to more manageable linear equations, crucial for solving problems in physics and focusing on understanding physical principles rather than cumbersome calculations.

Radioactive Isotope Activity Reduction

Activity reduction of a radioactive isotope is an essential concept, as it defines how the radioactive material changes over time. Understanding this reduction helps in multiple practical applications, from nuclear physics to medical treatments.

• The activity refers to the rate at which a radioactive source decays, measured in decay events per minute or second.
• Activity decreases over time following a predictable exponential law dictated by the isotope's half-life.
• The time it takes for different levels of reduction is determined by how much the count of surviving atoms decreases, which is directly based on their specific half-life.

For practical purposes, calculating the time until an isotope’s activity reaches a particular fraction of its original value provides insight into how long the isotope remains useful or safe. Through understanding these principles, users can estimate when the isotope will lose its potency, facilitate safe disposal, or exploit it for specific timed applications.