Question

The activity of radioactive sample is measured as 9750 counts per minute at \(\mathrm{t}=0\), and 975 counts per minute at \(\mathrm{t}=5\) minutes. The decay constant approximately is : (1) \(0.230\) per minute (2) \(0.461\) per minute (3) \(0.691\) per minute (4) \(0.922\) per minute

Short answer

The decay constant is approximately \( 0.461 \) per minute.

Step-by-step solution

Understand the problem

A radioactive sample's activity is given at two different times. Activity at \( \mathrm{t}=0 \) minutes is 9750 counts per minute, and at \( \mathrm{t}=5 \) minutes it is 975 counts per minute. The goal is to find the decay constant (\( \lambda \)).

Use the exponential decay formula

The radioactive decay formula is \[ A(t) = A_0 e^{-\lambda t} \], where \( A(t) \) is the activity at time \( t \), \( A_0 \) is the initial activity, and \( \lambda \) is the decay constant.

Set up the equation at \( t = 5 \)

When \( t = 5 \) minutes, \[ 975 = 9750 e^{-5 \lambda} \]

Solve for the decay constant (\( \lambda \))

Rearrange and solve the equation for \( \lambda \): \[ \frac{975}{9750} = e^{-5 \lambda} \] which simplifies to \[ 0.1 = e^{-5 \lambda} \]. Take the natural logarithm on both sides: \[ \ln(0.1) = -5 \lambda \], so \[ \lambda = -\frac{\ln(0.1)}{5} \].

Calculate the value

Using the natural logarithm value \[ \ln(0.1) \approx -2.3026 \], we get \[ \lambda = -\frac{-2.3026}{5} \approx 0.461 \].

Key concepts

Exponential Decay Formula

A foundational concept in understanding radioactive decay is the exponential decay formula. This formula describes how the activity of a radioactive sample decreases over time. The formula is written as \[ A(t) = A_0 e^{-\lambda t} \] where:
* \( A(t) \) is the activity at time \( t \)
* \( A_0 \) is the initial activity at time \( t = 0 \)
* \( \lambda \) is the decay constant.
This formula is very useful because it shows that the decrease in activity follows an exponential pattern. It helps us understand that the activity doesn't just decrease linearly, but instead decreases by a factor of \( e^{-\lambda t} \) as time progresses. This means the rate of decay depends on the current amount, leading to a gradually slowing decline in activity.

Natural Logarithm

In solving exponential decay problems, the natural logarithm (denoted as \( \ln \)) is essential. The natural logarithm is the inverse function of the exponential function \( e^x \). This means that \( \ln(e^x) = x \) and \( e^\ln(x) = x \).
Let's look at how \ln is used to solve for the decay constant:
When we encounter the equation \( 0.1 = e^{-5\lambda} \), we can't solve for \( \lambda \) directly. Instead, we take the natural logarithm of both sides:
\[ \ln(0.1) = \ln(e^{-5\lambda}) \]
Since \( \ln(e^x) = x \), this simplifies to:
\[ \ln(0.1) = -5\lambda \]
Now, we can solve for \( \lambda \) easily:
\[ \lambda = -\frac{\ln(0.1)}{5} \]
This step demonstrates how powerful logarithms are in transforming multiplicative relationships into additive ones, making the equations much simpler to solve.
Additionally, understanding that \( \ln(0.1) \approx -2.3026 \) is important for quick calculations.

Activity Measurement

Measuring the activity of a radioactive sample is crucial in many fields, from medicine to nuclear physics. The activity is typically measured in counts per minute (cpm), which represents the number of decay events detected per minute.
In the given problem, we know the activity at two points in time: 9750 cpm at \( t=0 \) minutes and 975 cpm at \( t=5 \) minutes.
These measurements tell us how rapidly the sample is decaying.
By comparing the activity at different times, we can use the exponential decay formula to find the decay constant, which is a measure of how quickly the radioactive material is decaying. \
Here’s a quick summary of the steps involved:

• Note the initial and subsequent activities.
• Use the exponential decay formula to set up an equation.
• Solve for the decay constant using the natural logarithm.

Understanding activity measurement and its role in such problems allows for accurate calculations and interpretations, which are essential in real-world applications.