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Solutions are printed below.

1. Given the masses in this table, find the mass defect and the binding energy of a helium-4 nucleus.

particle | mass (kg) |
---|---|

helium-4 nucleus | 6.6447X10^{-27} |

proton | 1.6726X10^{-27} |

neutron | 1.6749X10^{-27} |

2. a. Given the masses in this table, find the energy released when U-238 transmutes to Th-234 by alpha decay.

b. Ignoring relativistic effects, what is the speed of the alpha particle as a result of the alpha decay?

particle | mass (amu) |
---|---|

U-238 | 238.0508 |

Th-234 | 234.0436 |

He-4 | 4.0026 |

3.What is the wavelength of a 0.217 MeV photon emitted during gamma decay?

1.

a. sum of masses of individual nucleons = 2(1.6726X10^{-27}) + 2(1.6749X10^{-27}) = 6.6950X10^{-27} kg

mass defect = 6.6950X10^{-27} - 6.6447X10^{-27} = 0.0503X10^{-27} kg

b. binding energy = (Δm)c^{2} = (0.0503X10^{-27})(3.00X10^{8}) = 4.53X10^{-12} J

(4.53X10^{-12} J)/(1.60X10^{-19}) = 2.83X10^{7} eV = 28.3 MeV

2.

a. mass defect: 238.0508 - (234.0436 + 4.0026) = 0.0046 u

energy equivalent: (0.0046 u)(931.5 MeV/u) = 4.3 MeV

b. The values for part a are converted to mks units:

quantity | symbol | conversion factor | value |
---|---|---|---|

energy released | E_{k} |
1.602x10^{-19} J/eV |
6.88886x10^{-13} J |

mass of Th-234 | m_{Th} |
1.66053886x10^{-27 }kg/amu |
3.8864x10^{-25} kg |

mass of He-4 | m_{He} |
1.66053886x10^{-27 }kg/amu |
6.6465x10^{-27} kg |

speed of Th-234 | v_{Th} |
- | ? |

speed of He-4 | v_{He} |
- | ? |

Conservation of momentum gives us:

m_{Th}v_{Th}+ m_{He}v_{He} = 0

and

v_{Th}+ = -(m_{He}/m_{Th})v_{He} [eqn. 1]

The energy released is in the form of kinetic energy:

(1/2)m_{Th}v_{Th}^{2} + (1/2)m_{He}v_{He}^{2} = E_{k}

Substituting eqn. 1 and simplifying:

v_{He} = √{[2E_{k}] / [(m_{He})(1 + m_{He}/m_{Th})]} = 1.4x10^{7} m/s

3. The energy is converted to Joules:

(0.217x10^{6} eV) (1.602x10^{-19} J/eV) = 3.48x10^{-14} J

The energy is converted to wavelength:

λ = hc/ΔE = (6.63x10^{-34} J*s)(3.00x10^{8} m/s)/(3.48x10^{-14} J) = 5.72x10^{-12} m

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