Kinetic energy of daughter nucleus and alpha particle from alpha decay

In summary, the conversation discusses the process of alpha decay, where a nucleus X decays into a daughter nucleus Y and an alpha particle. The equations of conservation of momentum and kinetic energy are used to determine the kinetic energy of the decay products, with the Q value representing the available energy. The expressions for the kinetic energy of the alpha particle and the daughter nucleus are given by E^{\alpha}_{k}=\frac{M_{Y}}{(M_{Y}+M_{\alpha})}Q and E^{Y}_{k}=\frac{M_{\alpha}}{(M_{Y}+M_{\alpha})}Q, respectively.
  • #1
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Homework Statement


In [itex]\alpha[/itex] decay a nucleus X at rest decays to a daughter nucleus Y and an [itex]\alpha[/itex] particle. Conservation of momentum and kinetic energy gives:
[itex]M_{\alpha}v_{\alpha}+M_{Y}v_{Y}=0[/itex]

[itex]\frac{1}{2}M_{\alpha}v_{\alpha}^{2}+\frac{1}{2}M_{Y}v_{Y}^{2}=Q[/itex]

Where the Q value is the available energy found through [itex]Q=(M(X)-M(Y)-M(\alpha))c^{2}[/itex]

Show the kinetic energy of the two decay products are given by

[itex]E^{\alpha}_{k}=\frac{M_{Y}}{(M_{Y}+M_{\alpha})}Q[/itex]

[itex]E^{Y}_{k}=\frac{M_{\alpha}}{(M_{Y}+M_{\alpha})}Q[/itex]


Homework Equations


[itex]\frac{1}{2}M_{\alpha}v_{\alpha}^{2}+\frac{1}{2}M_{Y}v_{Y}^{2}=Q[/itex]

[itex]M_{\alpha}v_{\alpha}+M_{Y}v_{Y}=0[/itex]

[itex]Q=(M(X)-M(Y)-M(\alpha))c^{2}[/itex]

[itex]E^{\alpha}_{k}=\frac{M_{Y}}{(M_{Y}+M_{\alpha})}Q[/itex]

[itex]E^{Y}_{k}=\frac{M_{\alpha}}{(M_{Y}+M_{\alpha})}Q[/itex]

The Attempt at a Solution


I have tried rearranging the energy equation to get [itex]E^{\alpha}_{k}=Q-\frac{1}{2}M_{Y}v_{Y}^{2}[/itex]
since
[itex]E^{\alpha}_{k}=\frac{1}{2}M_{\alpha}v_{\alpha}^{2}[/itex]

Then rearranging the conservation of momentum equation and substituting in for various variables but I can't get anything that looks like the required expressions. I know this is a fairly simple algebra exercise but I just can't figure out what to do so any advice or suggestions would be appreciated.
 
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  • #2
I managed to figure this out. I rearranged the expression for Q to get [itex]E^{\alpha}_{K}[/itex]like in my initial attempt but then I took Q out as a common factor which was the key. It was then just a matter of doing about 3 pages of algebra on what was inside the brackets.
 

1. What is the definition of kinetic energy in the context of alpha decay?

The kinetic energy in alpha decay refers to the energy possessed by the daughter nucleus and alpha particle as they move away from each other after the decay process. It is the result of the conversion of nuclear potential energy into kinetic energy.

2. How is the kinetic energy of the daughter nucleus and alpha particle determined in alpha decay?

The kinetic energy of the daughter nucleus and alpha particle can be determined by using the equation KE = (1/2)mv2, where m is the mass and v is the velocity of the particle. This equation is based on the conservation of energy principle.

3. How does the mass of the daughter nucleus and alpha particle affect their kinetic energy in alpha decay?

The mass of the particles does not have a direct impact on their kinetic energy. However, the mass difference between the parent nucleus and the daughter nucleus does affect the amount of energy released in the form of kinetic energy during alpha decay.

4. Is the kinetic energy of the daughter nucleus and alpha particle always the same in alpha decay?

No, the kinetic energy of the particles can vary depending on the specific isotope undergoing alpha decay. The amount of energy released in the form of kinetic energy is unique to each isotope and is determined by the mass difference between the parent and daughter nuclei.

5. Can the kinetic energy of the daughter nucleus and alpha particle be converted into other forms of energy?

Yes, the kinetic energy of the particles can be converted into other forms of energy, such as heat or light, through various interactions with their surroundings. However, the total amount of energy remains constant due to the law of conservation of energy.

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