Question

Problem 1. Using the atomic mass data, calculate the average binding energy per nucleon (in MeV units) for the following nuclei:    a) ⁴He.      b) ²³⁵U. Atomic mass data: M(⁴He) = 4.002602 amu, M(²³⁵U) = 235.04393 amu, M(¹H) = 1.00797 amu, m_n = 1.008665 amu.

Answer 1

Answer -Given ,

mass of ⁴He =  4.002602 amu

mass of ²³⁵U = 235.04393 amu

Mass of ¹H or proton = 1.00797 amu

Mass of neutron = 1.008665 amu

Average Binding Energy per nucleon of ⁴He and ²³⁵U= ?

For ⁴He,

We know that,

Nuclear Binding Energy = mc²

Also, m = Nm_n + Zm_1H - m_atom

wherem = difference in mass

N = no. of neutrons

Z = no of protons

m = mass of atom

We know that, ⁴He have 2 protons and 2 neutrons

So, m = 2 * 1.008665 u + 2 * 1.00797 u - 4.002602 u

m = 2.01733 + 2.01594 - 4.002602 u

m = 0.030668 u

Put this in nuclear binding Energy Formula,

Nuclear Binding Energy = mc²

Nuclear Binding Energy = 0.030668 u * c²

Also, 1u = 931.494 MeV/c²

So, Nuclear Binding Energy = 0.030668 *931.494 MeV

Nuclear Binding Energy = 28.57 MeV

Now,

binding energy per nucleon = Nuclear Binding Energy / number of Nucleon

In ⁴He, there are 4 Nucleon i.e. 2 protons and 2 neutrons

So,  binding energy per nucleon = 28.57 MeV /4 = 7.14 MeV

Binding energy per nucleon of ⁴He is 7.14 MeV

For ²³⁵U ,

We know that,

Nuclear Binding Energy = mc²

Also, m = Nm_n + Zm_1H - m_atom

wherem = difference in mass

N = no. of neutrons

Z = no of protons

m = mass of atom

We know that, ²³⁵He have 92 protons and 143 neutrons

So, m = 143 * 1.008665 u + 92 * 1.00797 u - 235.04393 u

m = 144.24 + 92.73 - 235.04393 u

m = 1.926 u

Put this in nuclear binding Energy Formula,

Nuclear Binding Energy = mc²

Nuclear Binding Energy = 1.926 u * c²

Also, 1u = 931.494 MeV/c²

So, Nuclear Binding Energy = 1.926 * 931.494 MeV

Nuclear Binding Energy = 1794.06 MeV

Now,

binding energy per nucleon = Nuclear Binding Energy / number of Nucleon

In ⁴He, there are 235 Nucleon i.e. 92 protons and 143 neutrons

So,  binding energy per nucleon = 1794.05 MeV /235 = 7.63 MeV

Binding energy per nucleon of ²³⁵U is 7.63 MeV

Answer 2

Answer -Given ,

mass of ⁴He =  4.002602 amu

mass of ²³⁵U = 235.04393 amu

Mass of ¹H or proton = 1.00797 amu

Mass of neutron = 1.008665 amu

Average Binding Energy per nucleon of ⁴He and ²³⁵U= ?

For ⁴He,

We know that,

Nuclear Binding Energy = mc²

Also, m = Nm_n + Zm_1H - m_atom

wherem = difference in mass

N = no. of neutrons

Z = no of protons

m = mass of atom

We know that, ⁴He have 2 protons and 2 neutrons

So, m = 2 * 1.008665 u + 2 * 1.00797 u - 4.002602 u

m = 2.01733 + 2.01594 - 4.002602 u

m = 0.030668 u

Put this in nuclear binding Energy Formula,

Nuclear Binding Energy = mc²

Nuclear Binding Energy = 0.030668 u * c²

Also, 1u = 931.494 MeV/c²

So, Nuclear Binding Energy = 0.030668 *931.494 MeV

Nuclear Binding Energy = 28.57 MeV

Now,

binding energy per nucleon = Nuclear Binding Energy / number of Nucleon

In ⁴He, there are 4 Nucleon i.e. 2 protons and 2 neutrons

So,  binding energy per nucleon = 28.57 MeV /4 = 7.14 MeV

Binding energy per nucleon of ⁴He is 7.14 MeV

For ²³⁵U ,

We know that,

Nuclear Binding Energy = mc²

Also, m = Nm_n + Zm_1H - m_atom

wherem = difference in mass

N = no. of neutrons

Z = no of protons

m = mass of atom

We know that, ²³⁵He have 92 protons and 143 neutrons

So, m = 143 * 1.008665 u + 92 * 1.00797 u - 235.04393 u

m = 144.24 + 92.73 - 235.04393 u

m = 1.926 u

Put this in nuclear binding Energy Formula,

Nuclear Binding Energy = mc²

Nuclear Binding Energy = 1.926 u * c²

Also, 1u = 931.494 MeV/c²

So, Nuclear Binding Energy = 1.926 * 931.494 MeV

Nuclear Binding Energy = 1794.06 MeV

Now,

binding energy per nucleon = Nuclear Binding Energy / number of Nucleon

In ⁴He, there are 235 Nucleon i.e. 92 protons and 143 neutrons

So,  binding energy per nucleon = 1794.05 MeV /235 = 7.63 MeV

Binding energy per nucleon of ²³⁵U is 7.63 MeV