Problem 1.

Given the following masses, find the mass defect and the binding energy of a helium-4 nucleus.

b. Binding energy = (Δm)c

(4.53X10

Problem 2.

a. Given the following masses, 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?

Solution:

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:

Conservation of momentum gives us:

m

and

v

The energy released is in the form of kinetic energy: (1/2)m

Substituting eqn. 1 and simplifying:

v

Problem 3.

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

Solution:

The energy is converted to Joules: (0.217x10

The energy is converted to wavelength:

λ = hc/ΔE = (6.63x10

Problem 4.

The radioactivity of milk in one sample was 1750 Bq/L due to iodine-131 with half-life 8.04 days. For comparison, find the activity of milk due to potassium. Assume that one liter of milk contains 1.60 g of potassium, of which 0.0117% is the isotope 40K with half-life 1.28X10

### Solution:

R = [(ln 2)/T][N/A]

where N = Avogadro number [/kmol], A[kg/kmol] is the mass number, ln 2 = 0.693, and T is the half-life in seconds.

One kilogram of a pure radioactive isotope with half-life T[sec] has activity R[Bq/kg]

The unit of activity is Becquerel (1 Bq = 1 decay/sec)

T = (1.28 x 10

R = [0.693/4.039 x 10

R = [2.583 ×10

Problem 5.

(a) Use the single-particle shell model to predict the ground-state angular momenta (spin) values and parities for the following nuclei. Explain your reasoning in each case.

7Li, 33S, 44Ca, 69Ga, 80Zr, 123Sb, 175Lu.

(b) For the nuclei listed above, predict the magnetic dipole moments from the shell model.

Problem 6.

(a) The following nuclei have the spin quantum number quoted, but not the

parity. By comparing the measured dipole moments with the predictions of the

shell model, determine the parity of these states.

J <μ>

43Ca 7/2 -1.3μN

67Zn 5/2 +0.9μN

153Eu 5/2 +1.7μN

139La 7/2 +2.8μN

(b) Compare your findings in 1(b) and 2(a) with the experimental values.

Comment on any differences.

For solutions to the problems 5 & 6 click here.

Given the following masses, 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} |

Solution:

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

mass defect = 6.6950X10^{-27}kg^{-27}- 6.6447X10^{-27}= 0.0503X10^{-27}kgb. 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 MeVProblem 2.

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

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

U-238 | 238.0508 |

Th-234 | 234.0436 |

He-4 | 4.0026 |

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

Solution:

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} | - | ? |

m

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

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/sProblem 3.

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

Solution:

The energy is converted to Joules: (0.217x10

^{6}eV) (1.602x10^{-19}J/eV) = 3.48x10^{-14}JThe 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}mProblem 4.

The radioactivity of milk in one sample was 1750 Bq/L due to iodine-131 with half-life 8.04 days. For comparison, find the activity of milk due to potassium. Assume that one liter of milk contains 1.60 g of potassium, of which 0.0117% is the isotope 40K with half-life 1.28X10

^{9}yr.### Solution:

R = [(ln 2)/T][N/A]where N = Avogadro number [/kmol], A[kg/kmol] is the mass number, ln 2 = 0.693, and T is the half-life in seconds.

One kilogram of a pure radioactive isotope with half-life T[sec] has activity R[Bq/kg]

The unit of activity is Becquerel (1 Bq = 1 decay/sec)

T = (1.28 x 10

^{9}yr)( 31,557,600 s/y) = 4.039 x 10^{16}sR = [0.693/4.039 x 10

^{16}][ 6.0221415×10^{26}/ 40] = 2.583 ×10^{8}Bq/kgR = [2.583 ×10

^{8}Bq/kg][0.00160 * 0.0117/100 kg/L] = 48.35 Bq/LProblem 5.

(a) Use the single-particle shell model to predict the ground-state angular momenta (spin) values and parities for the following nuclei. Explain your reasoning in each case.

7Li, 33S, 44Ca, 69Ga, 80Zr, 123Sb, 175Lu.

(b) For the nuclei listed above, predict the magnetic dipole moments from the shell model.

Problem 6.

(a) The following nuclei have the spin quantum number quoted, but not the

parity. By comparing the measured dipole moments with the predictions of the

shell model, determine the parity of these states.

J <μ>

43Ca 7/2 -1.3μN

67Zn 5/2 +0.9μN

153Eu 5/2 +1.7μN

139La 7/2 +2.8μN

(b) Compare your findings in 1(b) and 2(a) with the experimental values.

Comment on any differences.

For solutions to the problems 5 & 6 click here.