Internal energy of a decaying atom (simple energy problem)

In summary: When you hit the Create Thread button in the HW forums, the HW template with items 1., 2., and 3. appears in the editing box. Don't delete this template. Fill it out with your HW question, any relevant formulas or equations, and your attempts at solutions.
  • #1
mnphys
10
0
This is my first time here. Sorry if the formatting is a bit off. This is one of a few questions I am having difficulty with. I am not asking for any answers; and I have definitely done the work - multiple times - but have not been able to find the correct answer.

Q) A uranium-238 atom can break up into a thorium-234 atom and a particle called an alpha particle, α -4. The numbers indicate the inertias of the atoms and the alpha particle in atomic mass units (1 amu = 1.66 × 10−27 kg ). When an uranium atom initially at rest breaks up, the thorium atom is observed to recoil with an x component of velocity of -2.9 × 105 m/s.

- How much of the uranium atom's internal energy is released in the breakup? (Express your answer to three significant digits and include the appropriate units.)

Relevant Equations: k = 1/2mv2

- I found the velocity of the object to be -2.9 x 105 m/s (given)
- I found the mass of the thorium atom to be 234 * 1.66 x 10-27 kg = 388.44 x 10-27 kg
- I found v2 to be (-2.9 x 105 m/s)2 = 8.41 x 1010 m2/s2
- Multiply: 1/2 * 388.44 x 10-27 kg * 8.41 x 1010 m2/s2 = 1633.3902 x 10-17 joules
- Reduce this to three significant digits: 163 x 10-16 joules

Where did I go wrong? The program says that answer is not correct, but I went over it several times and can't find my mistake. Any help is appreciated. Please do not give me the answer. I doubt anyone here would, but that would violate my conscience (among other things!)
 
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  • #2
mnphys said:
This is my first time here. Sorry if the formatting is a bit off. This is one of a few questions I am having difficulty with. I am not asking for any answers; and I have definitely done the work - multiple times - but have not been able to find the correct answer.

Q) A uranium-238 atom can break up into a thorium-234 atom and a particle called an alpha particle, α -4. The numbers indicate the inertias of the atoms and the alpha particle in atomic mass units (1 amu = 1.66 × 10−27 kg ). When an uranium atom initially at rest breaks up, the thorium atom is observed to recoil with an x component of velocity of -2.9 × 105 m/s.

- How much of the uranium atom's internal energy is released in the breakup? (Express your answer to three significant digits and include the appropriate units.)

Relevant Equations: k = 1/2mv2

- I found the velocity of the object to be -2.9 x 105 m/s (given)
- I found the mass of the thorium atom to be 234 * 1.66 x 10-27 kg = 388.44 x 10-27 kg
- I found v2 to be (-2.9 x 105 m/s)2 = 8.41 x 1010 m2/s2
- Multiply: 1/2 * 388.44 x 10-27 kg * 8.41 x 1010 m2/s2 = 1633.3902 x 10-17 joules
- Reduce this to three significant digits: 163 x 10-16 joules

Where did I go wrong? The program says that answer is not correct, but I went over it several times and can't find my mistake. Any help is appreciated. Please do not give me the answer. I doubt anyone here would, but that would violate my conscience (among other things!)
First of all, you should use the HW template when posting HW problems at PF. That's one of the rules.

Second, v2 has units of (m/s)2, not joules.

Third, although you are given the recoil velocity of the Th-234 nucleus, what happens to the alpha particle? Does it just sit there, after the U-238 nucleus has emitted it?
 
  • #3
Sorry, I thought I was using the template? I posted the question first, then the equations, then the attempt. If I didn't follow the rules I apologize.

I know nothing about the alpha particle's speed; all the information I was given was posted there. I thought it odd that they didn't mention anything but wasn't sure what to do about it.
 
  • #4
mnphys said:
Sorry, I thought I was using the template? I posted the question first, then the equations, then the attempt. If I didn't follow the rules I apologize.
When you hit the Create Thread button in the HW forums, the HW template with items 1., 2., and 3. appears in the editing box. Don't delete this template. Fill it out with your HW question, any relevant formulas or equations, and your attempts at solutions.
I know nothing about the alpha particle's speed; all the information I was given was posted there. I thought it odd that they didn't mention anything but wasn't sure what to do about it.
You're not given the speed of the alpha particle, but you are told that the U-238 nucleus was at rest before the alpha was emitted, and you are given the speed at which the Th-234 nucleus recoils. Using the Conservation of Momentum, you should be able to figure out the speed of the alpha.
 
  • #5
Thank you! I will be sure to follow that template if I have to ask any more questions. And your help was critical in solving this problem; given the mass of both particles and the velocity of one particle it's a simple matter of finding the velocity of the second particle and then calculating the kinetic energy of both particles, followed by adding them together.

So in assuming that the second particle was at rest, I made a critical error that violated the conservation of momentum. Lesson learned!
 
  • #6
Glad everything worked out.
 

FAQ: Internal energy of a decaying atom (simple energy problem)

1. What is internal energy of a decaying atom?

The internal energy of a decaying atom refers to the energy that is released during the process of radioactive decay. This energy is in the form of radiation and is typically measured in electron volts (eV).

2. How is internal energy of a decaying atom calculated?

The internal energy of a decaying atom can be calculated using the formula E = mc^2, where E is the energy, m is the mass of the atom, and c is the speed of light. This formula was derived by Albert Einstein and is commonly known as the mass-energy equivalence equation.

3. What factors affect the internal energy of a decaying atom?

The internal energy of a decaying atom is affected by several factors, including the type of radioactive material, the rate of decay, and the half-life of the material. Additionally, external factors such as temperature and pressure can also influence the release of internal energy.

4. Why is the internal energy of a decaying atom important?

The internal energy of a decaying atom is important because it plays a crucial role in nuclear reactions and power generation. It is also used in various medical applications, such as cancer treatment, and is essential in understanding the behavior of radioactive materials in the environment.

5. How does the release of internal energy from a decaying atom affect its stability?

The release of internal energy from a decaying atom can affect its stability by changing its atomic structure and causing it to become more or less stable. This is because the energy released during decay can cause the atom to lose or gain particles, which can alter its stability and eventually lead to a stable state.

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