Problem 5 - Applied Physics

Question

Find the percent of a sample of \({ }_{55}^{124} \mathrm{Cs}\) that will decay in the next \(10.0 \mathrm{~s}\). Its half life is \(30.8 \mathrm{~s}\).

Step-by-Step Solution

Answer

About 20% of the sample will decay in the next 10 seconds.

Step 1: Understand the Problem

We need to find out what percent of the sample of \({ }_{55}^{124} \mathrm{Cs}\) will decay in 10.0 seconds given its half-life is 30.8 seconds.

Step 2: Recall the Exponential Decay Formula

Exponential decay can be represented by the formula: \[ N(t) = N_0 \times e^{-\lambda t} \] where \(N(t)\) is the remaining quantity at time \(t\), \(N_0\) is the initial quantity, \(\lambda\) is the decay constant, and \(t\) is time.

Step 3: Find the Decay Constant

The decay constant \(\lambda\) can be found using the relation \( \lambda = \frac{\ln 2}{t_{1/2}} \), where \(t_{1/2}\) is the half-life.Thus, \( \lambda = \frac{\ln 2}{30.8} \approx 0.0225 \, \text{s}^{-1} \).

Step 4: Calculate Remaining Quantity After 10 Seconds

Using the decay formula: \[ N(10) = N_0 \times e^{-0.0225 \times 10} \]This simplifies to \[ N(10) = N_0 \times e^{-0.225} \].

Step 5: Calculate the Decayed Quantity

The decayed quantity in 10 seconds is \( N_0 - N(10) = N_0 - N_0 \times e^{-0.225} \)Simplifying gives \( N_0 \times (1 - e^{-0.225}) \).

Step 6: Find the Percent Decayed

The percent that decayed is given by \[ \left( 1 - e^{-0.225} \right) \times 100\% \] Calculating gives approximately 20\%.

Key Concepts

Half-Life Calculation

The concept of half-life is essential to understanding how substances decay over time. It is defined as the time it takes for half of a sample of a radioactive substance to decay.
This is a crucial property of radioactive materials like { }_{55}^{124} ext{Cs} .
In our exercise, the half-life is given as 30.8 seconds, meaning it takes this duration for half of the cesium sample to decay. The half-life helps us estimate how quickly a radioactive sample transforms. It is a reliable measure because it's constant for given isotopes under identical conditions, regardless of the initial quantity.

• To determine decay over another time period, simply repeat using the half-life.
• Each subsequent half-life period results in half of the remaining material decaying.

Mastering half-life provides insight into the timeline for radioactive decay and is essential for various applications, such as medical treatments and historical artifact dating.

Exponential Decay Formula

The exponential decay formula is a mathematical representation of how radioactive substances decrease over time. This formula is N(t) = N_0 \times e^{-\lambda t}, where:

• N(t) represents the quantity remaining at time t.
• N_0 is the initial quantity.
• \lambda is the decay constant.

This formula predicts how much of a substance remains after a certain duration. It models decay in real time, so can be used dynamically for different time intervals.To use this formula in an exercise like ours:

1. Determine the decay constant \(\lambda\).
2. Input desired time t.
3. Compute using the formula steps, simplifying to find N(t).

Here, it enabled computing the remaining { }_{55}^{124} ext{Cs} after 10 seconds. Understanding this formula's structure cross-applies to all situations involving exponential decay, making it a robust concept to grasp.

Decay Constant

The decay constant, denoted as \(\lambda\), is a critical factor in calculating radioactive decay, embodying a unique rate for each substance. It provides a measure of how rapidly a sample decays.
This constant is derived by the formula \[ \lambda = \frac{\ln 2}{t_{1/2}} \]where \(t_{1/2}\) is the half-life. It essentially connects the half-life to the decay process.Calculating \(\lambda\) allows you to normalize the decay equation for any scenario:

• It shows how the decay rate changes depending on the material.
• Smaller \(\lambda\) values indicate slower decay.
• Larger \(\lambda\) values correlate with faster decay.

In our example,\(\lambda\approx0.0225 \, \text{s}^{-1}\) makes it possible to substitute into the decay formula.
Understanding this constant is vital, as it quantifies the specific rate at which radioactive decay occurs, serving as a baseline to predict how much remains after any given time.