Problem 147 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive element gets spilled over the floor of a room. Its half-life period is 30 days. If the initial activity is ten times the permissible value, after how many days will it be safe to enter the room? (a) 300 days (b) 10 days (c) 100 days (d) 1000 days

Step-by-Step Solution

Answer

It will be safe to enter the room after approximately 100 days.

Step 1: Understanding Half-Life

The half-life of a substance is the time it takes for half of it to decay. For this exercise, the half-life of the radioactive element is given as 30 days.

Step 2: Setting Up Exponential Decay Equation

The decay of the substance can be modeled using the equation \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \), where \( N(t) \) is the amount remaining at time \( t \), \( N_0 \) is the initial amount, and \( T \) is the half-life. For this question, initially, it is 10 times the permissible activity.

Step 3: Solving for Time

We need to find \( t \) when \( N(t) = \text{permissible level} \). Using the equation, \( \left( \frac{1}{2} \right)^{\frac{t}{30}} = \frac{1}{10} \). Taking the logarithm of both sides gives \( \frac{t}{30} \log\left(\frac{1}{2}\right) = \log\left(\frac{1}{10}\right) \).

Step 4: Calculating Logarithms and Time

Calculate the logarithms: \( \log\left(\frac{1}{10}\right) = -1 \) and \( \log\left(\frac{1}{2}\right) \approx -0.3010 \). Substitute these into the equation to solve for \( t \): \( \frac{t}{30} \times (-0.3010) = -1 \), giving \( t = \frac{-1}{-0.3010} \times 30 \approx 100 \) days.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This process results in the transformation of the original element into a different element or a different isotope of the same element.
Key characteristics of radioactive decay include:

• It is spontaneous, meaning it occurs without any external influence.
• It is random, and we cannot predict when a particular atom will decay.
• Decay rates are specific to each radioactive element and are constant over time.

The half-life of a radioactive element is the time it takes for half of the remaining radioactive nuclei to decay. In the context of the exercise, knowing the half-life helps us determine how quickly the radioactive material decreases to a safe level.

Exponential Decay Equation

The exponential decay equation models how quantities decrease over time. It is strongly linked to processes like radioactive decay, where things diminish at a rate proportional to their current value.
The standard form of this equation is:

• \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \)

Here,

• \( N(t) \) is the remaining quantity at time \( t \).
• \( N_0 \) is the initial amount.
• \( T \) is the half-life period.

In the given problem, the task is to find out when the initial activity, which is ten times the permissible level, becomes safe (equal to the permissible level). Using this equation helps calculate the days required for such transformation.

Logarithmic Calculations

Logarithmic calculations are crucial in solving exponential decay problems, especially when trying to find the time it takes for a substance to decay to a certain level.
Logarithms help linearize exponential equations, making it easier to solve them. For example, given the equation from the exercise:

• \( \left( \frac{1}{2} \right)^{\frac{t}{30}} = \frac{1}{10} \)

We apply logarithms to both sides to solve for \( t \):

• \( \frac{t}{30} \log\left(\frac{1}{2}\right) = \log\left(\frac{1}{10}\right) \)

This transforms a complex exponential problem into an algebraic form. Calculating these logarithms \( \log\left(\frac{1}{2}\right) \approx -0.3010 \) and \( \log\left(\frac{1}{10}\right) = -1 \) allows us to solve for \( t \), resulting in the needed time for the substance to be safe.