Ncert Exercises & Solutions

13.4 Obtain the binding energy of the nuclei \({ }_{26}^{56} \mathrm{Fe}\) and \({ }_{83}^{209} \mathrm{Bi}\) in units of MeV from the following data:

m (\({ }_{26}^{56} \mathrm{Fe}\)) = 55.934939 u and m (\({ }_{83}^{209} \mathrm{Bi}\)) = 208.980388 u

Hint: The binding energy can be obtained by finding the mass defect in the nucleus.
Step 1: Find the mass defect in nucleus \({ }_{26}^{56} \mathrm{Fe}\).

Atomic mass of

{}_{26}^{56}{Fe}, m_{1} = 55.934939 u

{}_{26}^{56}{Fe}

nucleus has 26 protons and (56 - 26)=30 neutrons

Hence, the mass defect of the nucleus,

∆ m = 26 \times m_{H} + 30 \times m_{n} - m_{1}

where,

Mass of a proton,

m_{H} = 1.007825 u

Mass of neutron,

m_{n} = 1.008665 u

∴ ∆ m = 26 \times 1.007825 + 30 \times 1.008665 - 55.934939 = 26.20345 + 30.25995 - 55.934939 = 0.528461 u

But 1 u = 931.5 MeV / c^{2}

∴ ∆ m = 0.528461 \times 931.5 MeV / c^{2}

Step 2: Find the binding energy of nucleus \({ }_{26}^{56} \mathrm{Fe}\).

The binding energy of this nucleus is given as:

E_{b1} = ∆ {mc}^{2}

where, c = speed of light

∴ E_{b1} = 0.52861 \times 931.5 (\frac{MeV}{c^{2}}) c^{2} = 492 .400215 MeV

Average binding energy per nucleon

=\(\frac{492.400215}{56}=8.792 ~MeV\)
Step 3: Find the mass defect in nucleus \({ }_{83}^{209} \mathrm{Bi}\).
Atomic mass of

{}_{83}^{209}{Bi}, m_{2} = 208.980388 u

{}_{83}^{209}{Bi}

nucleus has 83 protons and

(209 - 83)

126 neutrons.

Hence, the mass defect of this nucleus is given as:

∆ m' = 83 \times m_{H} + 126 \times m_{n} - m_{2}

where,

Mass of a proton,

m_{H} = 1.007825 u

Mass of neutron,

m_{n} = 1.008665 u

∆ m' = 83 \times 1.007825 + 126 \times 1.008665 - 208.980388 = 83.649475 + 127.091790 - 208.980399 = 1.760877 u

But

1 u = 931.5 MeV / c^{2}

Step 4: Find the binding energy of \({ }_{83}^{209} \mathrm{Bi}\) nucleus.
Hence, the binding energy of this nucleus is given by,

E_{b 2} = ∆ m' c^{2}

= 1.760877 \times 931.5 (\frac{MeV}{c^{2}}) c^{2} = 1640.26 MeV

Average binding energy per nucleon

= \frac{1640.26}{209} = 7.848 MeV

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