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Difficulty: MediumMass Defect and Binding Energy

The mass of a helium nucleus 24He^{4}_{2}\text{He} is 4.0015 u4.0015\text{ u}. If the mass of a proton is 1.0073 u1.0073\text{ u} and the mass of a neutron is 1.0087 u1.0087\text{ u}, what is the binding energy per nucleon of the helium nucleus? (Take 1 u=931 MeV1\text{ u} = 931\text{ MeV})

  1. 7.10 MeV7.10\text{ MeV}Answer
  2. B
    28.40 MeV28.40\text{ MeV}
  3. C
    14.20 MeV14.20\text{ MeV}
  4. D
    0.0305 MeV0.0305\text{ MeV}

Answer

The binding energy per nucleon of the helium nucleus is 7.10 MeV7.10\text{ MeV}.
The correct response accurately calculates the mass defect of the helium-4 nucleus (0.0305 u0.0305\text{ u}), converts it to total binding energy (28.40 MeV28.40\text{ MeV}), and divides by the mass number A=4A = 4 to obtain 7.10 MeV7.10\text{ MeV} per nucleon.

Step-by-Step Solution

1
Calculate the total mass of the individual constituent nucleons (2 protons and 2 neutrons).
Total constituent mass = 2(1.0073 u)+2(1.0087 u)=2.0146 u+2.0174 u=4.0320 u2(1.0073\text{ u}) + 2(1.0087\text{ u}) = 2.0146\text{ u} + 2.0174\text{ u} = 4.0320\text{ u}.
Helium-4 consists of Z=2Z = 2 protons and N=A−Z=4−2=2N = A - Z = 4 - 2 = 2 neutrons.
2
Calculate the mass defect (Δm\Delta m).
Δm=4.0320 u−4.0015 u=0.0305 u\Delta m = 4.0320\text{ u} - 4.0015\text{ u} = 0.0305\text{ u}.
Mass defect is the difference between the total mass of individual nucleons and the actual nuclear mass.
3
Convert mass defect into total binding energy (EbE_b).
Eb=0.0305 u×931 MeV/u=28.3955 MeVE_b = 0.0305\text{ u} \times 931\text{ MeV/u} = 28.3955\text{ MeV}.
Applying mass-energy equivalence using the factor 1 u=931 MeV1\text{ u} = 931\text{ MeV}.
4
Divide total binding energy by the total number of nucleons (A=4A = 4).
\text{Binding Energy per nucleon} = \frac{28.3955\text{ MeV}}{4} = 7.098875\text{ MeV} \approx 7.10\text{ MeV}.
Binding energy per nucleon measures average stability per nuclear particle.

Key Concept

Binding Energy per Nucleon and Mass Defect
Estimated Time:1m 30s
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Topics

Mass Defect and Binding Energy

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