Problem 41 - Master Resource Book in JEE Main Physics

Question

A radioactive substance of half-life 6 min is placed near a Geiger counter which is found to register 1024 particles per minute. How many particles per minute will it register after 42 min? (a) 4 per min (b) 8 per min (c) 5 per min (d) 7 per \(\min\)

Step-by-Step Solution

Answer

After 42 min, it registers 8 particles per minute (Option B).

Step 1: Understand Half-Life

The half-life of a substance is the time required for it to reduce to half its initial quantity. Here, the substance has a half-life of 6 minutes. This means that every 6 minutes, the quantity of the substance, and thus the number of particles registered, will halve.

Step 2: Calculate Number of Half-Lives

To find out how many half-lives have passed in 42 minutes, divide the total time by the half-life: \( n = \frac{42}{6} = 7 \). Therefore, 7 half-lives have passed.

Step 3: Apply the Half-Life Formula

The formula to determine the remaining quantity after a certain number of half-lives is: \( Q = Q_0 \left(\frac{1}{2}\right)^n \). Here, \( Q_0 = 1024 \) particles per minute and \( n = 7 \). Substitute these values into the formula to find \( Q \).

Step 4: Perform the Calculation

Using the formula, calculate the quantity: \( Q = 1024 \times \left(\frac{1}{2}\right)^7 = 1024 \times \frac{1}{128} = 8 \).

Step 5: Verify and Conclude

After calculating, we find that the number of particles registered per minute after 42 minutes is 8 per minute. This matches option (b).

Key Concepts

Half-Life Calculation

The concept of half-life is central to understanding radioactive decay. Half-life refers to the time it takes for half of a radioactive substance to decay. For the exercise given, the half-life of the substance is 6 minutes. This means every 6 minutes, the particle count detected by the Geiger counter reduces to half its previous count. Suppose initially, there are 1024 particles per minute; after 6 minutes, this number will drop to 512 particles per minute. To determine how many half-lives have passed in 42 minutes, we simply divide the total elapsed time by the given half-life. - Total time: 42 minutes - Half-life duration: 6 minutes By calculating, we get that 7 half-lives have occurred. Using the half-life concept, we can further calculate how many particles remain after a specific time using exponential decay formulas.

Geiger Counter Usage

A Geiger counter is an instrument used to measure radiation levels. In this context, it is utilized to detect the number of radioactive particles being emitted per minute. The Geiger counter registers the count of particles, which helps in determining the rate of decay of a radioactive substance. Here are some key points about using this tool:

• The Geiger counter provides real-time data on particle detection.
• It is calibrated to provide a count rate, which is usually displayed as particles per minute.
• Accurate measurements depend on the proper setup and calibration of the device.

By constantly measuring this data over specific intervals, you can track how quickly the substance is losing particles, a direct indicator of its radioactive decay over half-lives.

Exponential Decay Formula

Radioactive decay follows an exponential decay pattern, which can be expressed mathematically. This means that the quantity of a radioactive substance decreases exponentially over time. The exponential decay formula is given as:\[ Q = Q_0 \left( \frac{1}{2} \right)^n \]Where:- \( Q \) is the remaining quantity of the substance after a time \( t \),- \( Q_0 \) is the initial quantity,- \( n \) is the number of half-lives that have elapsed.Applying this formula, if we begin with 1024 particles per minute and 7 half-lives elapse, we substitute these values in to calculate the remaining particles:\[ Q = 1024 \times \left( \frac{1}{2} \right)^7 = 8 \]This calculation shows that only 8 particles per minute remain after 42 minutes, which confirms the solution to the problem in the exercise. Understanding this formula allows you to predict how much of a radioactive substance will remain after any given time.