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 manydays will it be safe to enter the room? \(\quad\) [2007] (a) 300 days (b) 10 days (c) 100 days (d) 1000 days

Step-by-Step Solution

Answer

It takes approximately 100 days for the activity to reach a safe level.

Step 1: Understand Half-Life Concept

The half-life of a substance is the time it takes for half of the radioactive atoms to decay. This problem involves a half-life of 30 days.

Step 2: Use the Exponential Decay Formula

Radioactive decay is modeled by the formula: \[ A(t) = A_0 \times \left( \frac{1}{2} \right)^{t/T} \] where \( A(t) \) is the activity at time \( t \), \( A_0 \) is the initial activity, and \( T \) is the half-life period (30 days).

Step 3: Identify the Initial and Final Activity Levels

The initial activity \( A_0 \) is 10 times the permissible value. We need \( A(t) \), the activity at time \( t \), to equal the permissible value. Thus, \[ \frac{A_0}{10} = A_0 \times \left( \frac{1}{2} \right)^{t/30} \].

Step 4: Solve for Time (t)

Substitute and solve for \( t \):\[ \frac{A_0}{10} = A_0 \times \left( \frac{1}{2} \right)^{t/30} \]\[ \frac{1}{10} = \left( \frac{1}{2} \right)^{t/30} \]Taking logarithms on both sides, we have:\[ \log_{10} \frac{1}{10} = \frac{t}{30} \log_{10} \frac{1}{2} \]Solve for \( t \):\[ t = 30 \times \frac{\log_{10} \frac{1}{10}}{\log_{10} \frac{1}{2}} \]Calculating yields:\[ t \approx 100 \text{ days} \]

Step 5: Answer the Question

After calculating, we find that approximately 100 days are needed for the activity to reach a safe level. Therefore, the correct answer is option (c) 100 days.

Key Concepts

Half-Life

A half-life is a crucial concept when studying radioactive decay. It is the time required for half of the radioactive atoms in a substance to undergo decay. In the given problem, the radioactive element has a half-life of 30 days. This means that every 30 days, the amount of radioactive material is reduced by half.

Consider this practical scenario: If you start with 100 grams of a substance, in 30 days, only 50 grams will remain. After another 30 days, that amount will reduce to 25 grams, and the cycle continues.

• This steady decrease helps predict how long it will take for a material to become safe.
• It shows how gradually a radioactive element becomes less hazardous.

Understanding half-life helps in planning safe handling and storage of radioactive materials. It is vital in industries ranging from nuclear power to medical applications where radioactive substances are used for treatment.

Exponential Decay

Radioactive decay is often described as "exponential decay." This is because the rate of decay of the radioactive substance is proportional to its current quantity. In simpler terms, as the radioactive material diminishes, the speed at which it decays slows down too. This phenomenon is beautifully captured in the exponential decay formula:

• If you begin with a large quantity, it decays rapidly.
• As the quantity lessens, decay happens more slowly.

The formula used is: \[ A(t) = A_0 \times \left( \frac{1}{2} \right)^{t/T} \] where:

• \( A(t) \) is the activity at time \( t \)
• \( A_0 \) is the initial activity
• \( T \) is the half-life period

The exponential nature ensures that we can predict when the material reaches a safe activity level. This provides valuable information on when spaces contaminated with radioactive material can be regained safely.

Radioactive Safety

Understanding and calculating radioactive safety is vital whenever dealing with radioactive spills or materials. Safety is determined by the permissible activity level, which is the maximum radioactive decay rate considered safe for human exposure. In this problem, the initial activity is ten times higher than this safe level. This mandates calculating precisely how long it will take for the material to decay to a non-hazardous level.

Using half-life and decay calculations, we can determine this timeline. For example:

• If the initial activity exceeds safety thresholds, it requires a structured decay process to drop below these limits.
• By applying the decay formula, we ascertain the duration until the activity is safe.

Such calculations help ensure environmental and personal safety, crucial in preventing radiation exposure and its harmful effects on health.

Activity Calculation

Activity calculation is a critical step in regulating the safety of environments exposed to radioactive decay. The activity level refers to how much radiation is emitted at a given moment. It hinges on the initial quantity and the time elapsed, factoring in the exponential decay.

To find when the activity is safe, we equate the final activity to the permissible level as shown:

• If initial activity is \( A_0 \), safe activity is \( \frac{A_0}{10} \).
• Using this, we solve: \[ \frac{1}{10} = \left( \frac{1}{2} \right)^{t/30} \]

Calculating the time \( t \) by taking logarithms and substituting values provides precision in safety assessments. It tells us the expected waiting period before considering the area or material safe. This calculation is indispensable for industries that must rigorously manage and mitigate risks associated with radioactive substances.