Question

Calculate the binding energy per nucleon for the following isotopes.

(a) ⁹Be

b) ¹⁹⁵Pt

(c) ⁶⁴Zn

(d) ⁷Li

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. 9 Be:

No. of nucleons: 9

No. of protons: 4

Total mass of protons: 4 m_p = 4 * 1.007 = 4.028 u

No. of neutrons: 5

Total mass of neutrons: 5 m_n = 5 * 1.008 = 5.04 u

Mass of the nucleus: 9.012 u

mass defect: (4.028 + 5.04 - 9.012) = 0.056 u

convert mass units from u to MeV/c²

mass defect: 0.056 u = 0.056 * 931.5 MeV/c² = 52.164 MeV/c²

BE: mass defect * c² = 52.164 MeV

BE/nucleon: 52.164/9 = 5.8 MeV

2. 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 nucleus: 194.96 u

mass defect: (78.55 + 117.94 - 194.96) = 1.53 u

convert mass units from u to MeV/c²

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

BE: mass defect * c² = 1425.2 MeV

BE/nucleon: 52.164/9 = 7.31 MeV

3.

64-Zn

No. of nucleons: 64

No. of protons: 30

Total mass of protons: 30 m_p = 30 * 1.007 = 30.21 u

No. of neutrons: 34

Total mass of neutrons: 34 m_n = 34 * 1.008 = 34.272 u

Mass of the nucleus: 63.929 u

mass defect: (30.21 + 34.272 - 63.929) = 0.553 u

convert mass units from u to MeV/c²

mass defect: 0.553 u = 0.553 * 931.5 MeV/c² = 515.12 MeV/c²

BE: mass defect * c² = 515.12 MeV

BE/nucleon: 515.12/64 = 8.05 MeV

4.

7 Li

No. of nucleons: 7

No. of protons: 3

Total mass of protons: 3 m_p = 3 * 1.007 = 3.021 u

No. of neutrons: 4

Total mass of neutrons: 4 m_n = 4 * 1.008 = 4.032 u

Mass of the nucleus: 7.016 u

mass defect: (3.021 + 4.032 - 7.016) = 0.037 u

convert mass units from u to MeV/c²

mass defect: 0.037 u = 0.037 * 931.5 MeV/c² = 34.47 MeV/c²

BE: mass defect * c² = 34.47 MeV

BE/nucleon: 34.47/7 = 4.924 MeV