Question

Calculate the binding energy per nucleon for the following isotopes.

(a) ¹²C
______MeV/nucleon

(b) ¹¹B
________MeV/nucleon

(c) ⁵⁵Mn
________MeV/nucleon

(d) ¹⁹⁵Pt
_______MeV/nucleon

Answer 1

Binding Energy per nucleon = Binding energy / total no. of nucleons

Binding energy BE = mass defect * c²

mass defect = (difference in sum of masses of no of protons and neutrons present and mass of nucleus)

We use masses in atomic mass units (u)

Proton mass : m_p: 1.007 u

Neutron mass: m_n : 1.008 u

1 u = 1.66 * 10^-27 kg = 931.5 MeV/c²

First we need to calculate mass defects for the given nuclei, then calculate BE using above formula and finally find BE per nucleon. (To find mass of the given nucleus you need to consult book/internet).

1. 12 C

No. of nucleons: 12

No. of protons: 6

Total mass of protons: 6 m_p = 6 * 1.007 = 6.042 u

No. of neutrons: 6

Total mass of neutrons: 6 m_n = 6 * 1.008 = 6.048 u

Mass of the nucleus: Mass of 12 C atom - Mass of 6 electrons

Mass of 1 electron: 0.000548u

Mass of 6 electrons = 0.0033 u

Mass of 12 C nucleus: 12- 0.0033=11.996 u

mass defect: (6.042 + 6.048 - 11.996) = 0.094 u

convert mass units from u to MeV/c²

mass defect: 0.094 u = 0.094 * 931.5 MeV/c² = 87.561 MeV/c²

BE: mass defect * c² = 7.3 MeV

BE/nucleon: 83.835/12 = 6.986 MeV/nucleon

2.

11 B

No. of nucleons: 11

No. of protons: 5

Total mass of protons: 5 m_p = 5 * 1.007 = 5.035 u

No. of neutrons: 6

Total mass of neutrons: 6 m_n = 6 * 1.008 = 6.048 u

Mass of the nucleus: Mass of 11 B atom - Mass of 5 electrons

Mass of 1 electron: 0.000548u

Mass of 5 electrons = 0.00274

Mass of 11 B nucleus: 11.009 - 0.0027=11.0063 u

Mass of the nucleus: 11.0063 u

mass defect: (5.035 + 6.048 - 11.0063) = 0.0767 u

convert mass units from u to MeV/c²

mass defect: 0.0767 u = 0.0767 * 931.5 MeV/c² = 71.446 MeV/c²

BE: mass defect * c² = 71.446 MeV

BE/nucleon: 71.446/11 = 6.495 MeV/nucleon

3.

55 Mn

No. of nucleons: 55

No. of protons: 25

Total mass of protons: 25 m_p = 25 * 1.007 = 25.175 u

No. of neutrons: 30

Total mass of neutrons: 30 m_n = 30 * 1.008 = 30.24 u

Mass of the atom: 54.939 u

Mass of the nucleus: Mass of 55 Mn atom - Mass of 25 electrons

Mass of 1 electron: 0.000548u

Mass of 25 electrons = 0.0137 u

Mass of nucleus: 54.939 - 0.0137=54.925 u

mass defect: (25.175 + 30.24 - 54.925) = 0.4897 u

convert mass units from u to MeV/c²

mass defect: 0.4897 u = 0.476 * 931.5 MeV/c² = 443.39 MeV/c²

BE: mass defect * c² = 456.16 MeV

BE/nucleon: 456.16/55 = 8.294 MeV/nucleon

4.

195-Pt

No. of nucleons: 195

No. of protons: 78

Total mass of protons: 78 m_p = 78 * 1.007 = 78.55 u

No. of neutrons: 117

Total mass of neutrons: 117 m_n = 117 * 1.008 = 117.94 u

Mass of the atom: 194.96 u

Mass of the nucleus: Mass of atom - Mass of 78 electrons

Mass of 1 electron: 0.000548u

Mass of 78 electrons = 0.0427 u

Mass of nucleus: 194.96 - 0.0427=194.917 u

mass defect: (78.55 + 117.94 - 194.917) = 1.573 u

convert mass units from u to MeV/c²

mass defect: 1.573 u = 1.53 * 931.5 MeV/c² = 1425.2 MeV/c²

BE: mass defect * c² = 1465.011 MeV

BE/nucleon: 1465.011/195 = 7.513 MeV/Nucleon