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

The room will be safe to enter after approximately 100 days (option c).

Step 1: Understand the Concept of Half-life

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

Step 2: Initial and Final Values

We are given that the initial activity of the radioactive element is ten times the permissible value. We need to find the time it takes for the activity to reduce to 1 (the permissible value), from 10 times the permissible value.

Step 3: Apply the Formula for Exponential Decay

The decay of a radioactive substance is given by the formula \( N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \), where \( N \) is the remaining activity, \( N_0 \) is the initial activity, \( T_{1/2} \) is the half-life, and \( t \) is the time. Here, \(N_0\) is 10 and \(N\) is 1.

Step 4: Solve for Time \( t \)

Set up the equation: \( 1 = 10 \times \left(\frac{1}{2}\right)^{\frac{t}{30}} \). Divide both sides by 10 to get \( \frac{1}{10} = \left(\frac{1}{2}\right)^{\frac{t}{30}} \). Take logarithms on both sides to solve for \( t \): \( \log\left(\frac{1}{10}\right) = \frac{t}{30} \log\left(\frac{1}{2}\right) \).

Step 5: Calculate \( t \)

Using the values: \( \log\left(\frac{1}{10}\right) = -1 \) and \( \log\left(\frac{1}{2}\right) \approx -0.3010 \), calculate \( t \):\[\frac{-1}{-0.3010} = \frac{t}{30}\] Solving this gives \( t = 30 \times 3.322 \approx 99.67 \).

Step 6: Round to the Nearest Option

The calculated value \( t \approx 100 \) days, which matches option (c). Therefore, the room will be safe to enter after approximately 100 days.

Key Concepts

Half-life

The concept of half-life is fundamental to understanding radioactive decay. It refers to the amount of time it takes for half of a radioactive substance to decay into a more stable form. Every radioactive element has a unique half-life, which remains constant over time. In this problem, the half-life is given as 30 days. This means that every 30 days, half of the substance will have decayed.
Understanding half-life is crucial because it allows us to predict how long it will take for a substance to decay to a safe level. When dealing with safety or disposal of radioactive materials, knowing the half-life helps determine how long a substance will remain hazardous. For instance, if a substance has a half-life of 30 days, it will need multiple half-life periods to become significantly less radioactive.

Exponential Decay

Exponential decay refers to the process by which a quantity decreases at a rate proportional to its current value. In the case of radioactive substances, this means that the activity decreases exponentially over time. This type of decay can be modeled using the exponential decay formula:

• \( N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \)
• Here, \( N \) is the final activity, \( N_0 \) is the initial activity, \( T_{1/2} \) is the half-life, and \( t \) is the time elapsed.

This formula helps in determining how the quantity of a radioactive element will diminish over time. As seen in the problem, the initial activity was 10 times the permissible amount. Using the exponential decay formula, we can calculate the time required to reduce activity to a safe level.

Radiation Safety

Radiation safety is a vital aspect whenever dealing with radioactive materials. This means ensuring that exposure to radiation is minimized for both individuals and the environment.

• Understanding the safe levels of radiation is crucial to determine when an environment is safe for occupancy after contamination.
• Permissible values or limits are set based on scientific research to ensure human health is not compromised by radiation exposure.
• In this scenario, the activity must decrease to a level that is considered safe, which is typically one times the permissible value.

By knowing the half-life and applying the decay formula, we can predict when a specific level of safety will be achieved, ensuring that entry into the room will not pose significant health risks.

Logarithmic Calculation

In the context of radioactive decay, logarithmic calculations become important when solving for time. The process involves using logarithms to rearrange the exponential decay equation into a format that allows for solving unknown values mathematically.

• One key step involves recognizing the necessity to take the logarithm of both sides of an equation to solve for time.
• In the given solution, the step \( \log\left(\frac{1}{10}\right) = \frac{t}{30} \log\left(\frac{1}{2}\right) \) simplifies solving for \( t \) by allowing for straightforward algebraic manipulation.
• Utilizing \( \log(\text{base 10}) \) or other logarithm bases facilitates the calculation.

Mastering these calculations ensures you can predict how quickly radioactive activity will drop to safe levels, making it an essential mathematical tool in nuclear science and radiation protection.