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Chapter 33: Problem 44

For measuring the activity of a radioactive sample, a count rate meter is used. At certain observation, count rate meter recorded 5050 counts per minute but after 10 minutes later, the count rate showed 2300 counts per minute. The disintegration constant \((\lambda)\) is : (a) \(0.065\) per min (b) \(0.078\) per min (c) \(0.24\) per min (d) \(0.868\) per min

Short Answer

Expert verified
(b) \(0.078\) per min

Step by step solution

01

Understand the decay formula

Radioactive decay can be described by the formula \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the amount of radioactive substance at time \( t \), \( N_0 \) is the initial amount, and \( \lambda \) is the disintegration constant. This formula models the exponential decay of radioactive material over time.
02

Identify known values

In this problem, we are given that \( N_0 = 5050 \) counts/min and \( N(t) = 2300 \) counts/min after \( t = 10 \) minutes. We need to use these values to find the disintegration constant \( \lambda \).
03

Set up the decay equation

Start with the equation \( N(t) = N_0 e^{-\lambda t} \). Substitute the known values to get:\[ 2300 = 5050 e^{-10\lambda} \]
04

Solve for \(\lambda\)

First, divide both sides of the equation by 5050:\[ \frac{2300}{5050} = e^{-10\lambda} \]Calculate the ratio:\[ 0.4554 = e^{-10\lambda} \]Take the natural logarithm on both sides:\[ \ln(0.4554) = -10\lambda \]Solve for \( \lambda \):\[ \lambda = -\frac{\ln(0.4554)}{10} \]Computing this gives:\[ \lambda \approx 0.078 \text{ per minute} \]
05

Verify the correct answer

Compare the calculated \( \lambda \) with the given options. The closest value to the calculated \( 0.078 \text{ per minute} \) is option (b) \( 0.078 \text{ per minute} \). Thus, the disintegration constant is correctly identified as option (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Disintegration Constant
The disintegration constant, represented by \( \lambda \), is a crucial parameter in radioactive decay processes. It offers a measure of how quickly a radioactive substance undergoes decay. Essentially, \( \lambda \) tells us the probability of disintegration of a radioactive atom per unit of time. In practical terms, a larger disintegration constant means the substance decays faster, losing its radioactivity more quickly. Conversely, a smaller \( \lambda \) indicates a slower decay process, where the radioactive substance remains active for a longer period. In the context of our exercise, knowing the initial count rate and the count rate at a later time allows us to calculate \( \lambda \), helping us to understand how quickly the material decays over the given period. The calculation of \( \lambda \) involved taking the natural logarithm of the ratio of count rates and dividing by time. This process provides a clear and quantitative insight into the decay characteristics of the sample.
Exponential Decay Formula
The exponential decay formula is fundamental to understanding how radioactive substances diminish over time. This formula is expressed as \( N(t) = N_0 e^{-\lambda t} \), where:
  • \( N(t) \) is the amount of remaining radioactive substance at time \( t \).
  • \( N_0 \) is the initial quantity of the substance.
  • \( \lambda \) represents the disintegration constant.
  • \( t \) is the time that has elapsed.
The formula captures the concept of exponential decay, meaning the quantity decreases at a rate proportional to its current value. This relationship is seen in nature with radioactive elements where half-lives are often used to describe the rate of decay. Applying this formula in practice involves substituting the known values into the equation and solving for the unknown variables. In our exercise, by knowing the initial and later counts, we examined how the substance decayed to find \( \lambda \), demonstrating the principle of decay over the specified time period.
Count Rate Meter
A count rate meter is an essential tool in monitoring and measuring the activity of radioactive materials. It records the number of disintegrations (or decays) per unit of time, often in counts per minute or counts per second. This measurement provides direct observational data from which calculations about the decay process and the disintegration constant can be derived. In our exercise, the count rate meter initially recorded 5050 counts per minute and then 2300 counts per minute after 10 minutes. These readings are vital in applying the exponential decay formula, allowing us to solve for the disintegration constant. Overall, a count rate meter is a fundamental instrument in radiation measurements and provides the accurate and real-time data necessary for understanding radioactive decay behavior.

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Most popular questions from this chapter

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