Problem 37 - Fundamentals of Physics

Question

The radionuclide \({ }^{64} \mathrm{Cu}\) has a half-life of \(12.7 \mathrm{~h}\). If a sample contains \(5.50 \mathrm{~g}\) of initially pure \({ }^{64} \mathrm{Cu}\) at \(t=0,\) how much of it will decay between \(t=14.0 \mathrm{~h}\) and \(t=16.0 \mathrm{~h} ?\)

Step-by-Step Solution

Answer

0.23 grams of \\({}^{64}Cu\\) decayed.

Step 1: Determine Initial Decay Constant

The decay constant \( \lambda \) is calculated using the formula \( \lambda = \frac{\ln(2)}{\text{half-life}} \). The half-life provided is 12.7 hours, so\[ \lambda = \frac{\ln(2)}{12.7} \approx 0.0546 \text{ hr}^{-1}. \]

Step 2: Calculate Remaining Material at t = 14.0 h

We use the formula \( N(t) = N_0 e^{-\lambda t} \), where \( N_0 = 5.50 \, \text{g} \). At \( t = 14.0 \, \text{h} \):\[ N(14) = 5.50 \, \text{g} \times e^{-0.0546 \times 14} \approx 3.38 \, \text{g}. \]

Step 3: Calculate Remaining Material at t = 16.0 h

Using the same decay formula, at \( t = 16.0 \, \text{h} \):\[ N(16) = 5.50 \, \text{g} \times e^{-0.0546 \times 16} \approx 3.15 \, \text{g}. \]

Step 4: Determine Amount Decayed Between t = 14.0 h and t = 16.0 h

The amount that decayed is the difference between the material present at \( t = 14.0 \, \text{h} \) and \( t = 16.0 \, \text{h} \):\[ \Delta N = N(14) - N(16) = 3.38 \, \text{g} - 3.15 \, \text{g} = 0.23 \, \text{g}. \]

Step 5: Conclusion

The amount of \( {}^{64} \text{Cu} \) that decayed between \( t = 14.0 \, \text{h} \) and \( t = 16.0 \, \text{h} \) is 0.23 grams.

Key Concepts

Understanding Half-Life

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms to decay. It acts as a clock to measure how quickly a substance loses its radioactivity. A key point is that the half-life remains constant, no matter how much substance you start with or how much is left. For example, with a half-life of 12.7 hours, in 12.7 hours half of the ^{64}Cu in the sample will have decayed.
This concept is useful in many fields, from archaeology to medicine.

• Archaeologists use it to date artifacts through carbon dating.
  
• Medical professionals use it to calculate safe dosages of radiopharmaceuticals.

Every radioactive isotope has its own unique half-life. This characteristic makes half-life calculations critical when working with different radionuclides.

The Decay Constant

The decay constant, denoted by \( \lambda, \) is a probability measure of the decay process. It describes how frequently the individual atoms decayed.
Calculated using the formula \( \lambda = \frac{\ln(2)}{\text{half-life}}, \) it links to the half-life by showing the decay rate.
For ^{64}Cu with a half-life of 12.7 hours, this gives a decay constant of approximately 0.0546 \( \text{hr}^{-1}. \)

• A larger decay constant means a faster decay rate.
  
• This measure is critical for predicting how much of a substance remains over time.
  

Understanding the decay constant helps in comprehending how rapidly a substance loses its radioactivity over a given period.

Exploring Exponential Decay

Exponential decay describes how the quantity of a radioactive substance decreases over time. The substance reduces at a rate proportional to its current value, creating a curve that represents rapid early loss, slowing down over time.
The mathematical representation is \( N(t) = N_0 e^{-\lambda t}, \) where \( N_0 \) is the initial amount.

• The equation uses the decay constant \( \lambda \) to predict remaining material after a time \( t. \)
• In the exercise, you see this calculation predicting the amounts left at different times.
• Exponential decay is foundational in understanding how substances lose mass and radioactivity.

This concept applies to any radioactive substance and is vital for applications like nuclear power generation or dating ancient biological materials.

Radionuclide Basics

A radionuclide, also known as a radioactive isotope, is an atom with an unstable nucleus. This instability causes the nucleus to release energy in the form of radiation until it becomes stable. \(^{64}Cu \) is an example where it emits radiation until it transforms into a stable form.
Key characteristics include:

• Radionuclides occur naturally or can be artificially produced.
  
• Each one has a unique decay mode, resulting in different types of radiation.
• They are used in medicine, industry, and scientific research.
  

Understanding radionuclides helps in comprehending phenomena such as background radiation and the workings of nuclear reactors. It aids in solving the exercise by acknowledging how much of a radionuclide like \(^{64}Cu \) decays in a given time.