Question

Calculate the binding energy per nucleon for the following isotopes.

(a) ⁵⁹Co

______ MeV/nucleon

(b) ¹¹B
______MeV/nucleon

(c) ¹²C
______MeV/nucleon

(d) ⁷Li
_____ MeV/nucleon

Answer 1

Follow the steps.

1. calc the mass of the individual hydrogen atoms and neutrons used in the atom

2. take the difference between this and the mass given
3. convert to energy using E =mc2

a) 59Co
initial mass = 27 * 1.007825 + 32 * 1.008665 = 59.4885 amu

given mass = 59

difference = 0.4885 amu = 0.4885*931.494 MeV / amu = 455.0348 MeV

for nucleon = 455.0348/59 = 7.7124 MeV

b) 11B
initial mass = 5 * 1.007825 + 6 * 1.008665 = 11.091115 amu
binding energy per nucleon = 7.715734 MeV

c) 12C
initial mass = 6 * 1.007825 + 6 * 1.008665 = 12.0 amu
binding energy per nucleon = 1.0 MeV

d) 7Li
initial mass = 3 * 1.007825 + 4 * 1.008665 =7.058135 amu
binding energy per nucleon = 7.73605767 MeV

Answer 2

Nuclear Binding Energy

The energy required to break down a nucleus into its component nucleons is called the nuclear binding energy.

⁶³Cu + Energy 29 p⁺ + 34 n^o

Nuclear binding energies are usually expressed in terms of kJ/mole of nuclei or MeV's/nucleon. Calculation of the nuclear binding energy involves the following three steps:

• Determining the Mass Defect
• Conversion of Mass Defect into Energy
• Expressing Nuclear Binding Energy as Energy per Mole of Atoms, or as Energy per Nucleon

Determining the Mass Defect

The difference between the mass of a nucleus and the sum of the masses of the nucleons of which it is composed is called the mass defect.  Three things need to be known in order to calculate the mass defect:

• the actual mass of the nucleus,
• the composition of the nucleus (number of protons and of neutrons),
• the masses of a proton and of a neutron.

To calculate the mass defect:

• add up the masses of each proton and of each neutron that make up the nucleus,
• subtract the actual mass of the nucleus from the combined mass of the components to obtain the mass defect.

Example: Find the mass defect of a copper-63 nucleus if the actual mass of a copper-63 nucleus is 62.91367 amu.

Copper has 29 protons and copper-63 also has (63 - 29) 34 neutrons.
The mass of a proton is 1.00728 amu and a neutron is 1.00867 amu.
The combined mass is calculated:

29 protons(1.00728 amu/proton) + 34 neutrons(1.00867 amu/neutron)
or
63.50590 amu

Dm = 63.50590 amu - 62.91367 amu = 0.59223 amu

• Find the composition of the copper-63 nucleus and determine the combined mass of its components.
• Calculate the mass defect.

Conversion of Mass Defect into Energy

To convert the mass defect into energy:

• Convert the mass defect into kilograms (1 amu = 1.6606 x 10^-27 kg)
• Convert the mass defect into its energy equivalent using Einstein's equation.

Example: Determine the binding energy of the copper-63 atom.

• Convert the mass defect (calculated in the previous example) into kg.

(0.59223 amu/nucleus)(1.6606 x 10^-27 kg/amu) = 9.8346 x 10^-28 kg/nucleus

• Convert this mass into energy using DE = Dmc², where c = 2.9979 x 10⁸ m/s.

E = (9.8346 x 10^-28 kg/nucleus)(2.9979 x 10⁸ m/s)² = 8.8387 x 10^-11 J/nucleus

Expressing Nuclear Binding Energy as Energy per Mole of Atoms, or as Energy per Nucleon

The energy calculated in the previous example is the nuclear binding energy. However, nuclear binding energy is often expressed as kJ/mol of nuclei or as MeV/nucleon.

• To convert the energy to kJ/mol of nuclei we will simply employ the conversion factors for converting joules into kilojoules (1 kJ = 1000 J) and for converting individual particles into moles of particles (Avogadro's Number).

(8.8387 x 10^-11 J/nucleus)(1 kJ/1000 J)(6.022 x 10²³ nuclei/mol) = 5.3227 x 10¹⁰ kJ/mol of nuclei

• To convert the binding energy to MeV (megaelectron volts) per nucleon we will employ the conversion factor for converting joules into MeV (1 MeV = 1.602 x 10^-13 J) and the number of nucleons (protons and neutrons) which make up the nucleus.

(8.8387 x 10^-11 J/nucleus)[1 MeV/(1.602 x 10^-13 J)](1 nucleus/63 nucleons) = 8.758 MeV/nucleon