Question

How much energy is required to remove one proton from the nucleus of

{}_{8}^{16}O

? A rough answer to this question is obtained by giving the binding energy per nucleon. A better answer is obtained when we write a reaction that removes a proton from the nucleus. In this case

{}_{8}^{16}O \to {}_{1}^{1}p + {}_{7}^{15}N

. Calculate the energy required for this reaction to take place, known as the proton separation energy. Compare the two energy values. (The atomic mass of oxygen is

15.994 u

; that of nitrogen is

15.000 u

.)

Accepted Answer

Solution:

Giving binding energy per nucleon:

Mass defect $=$ Total mass of nucleons $-$ Mass of nucleus

$δ = Z m_{p} + (A - Z) m_{n} - M_{n u c l e u s}$

Where,

$δ$ is a mass defect

$Z$ is atomic number

$A$ is atomic mass number

$m_{p}$ is mass of proton $i . e . 1.007276 u$

$m_{n}$ is mass of neutron $i . e . 1.008665 u$

$M_{n u c l e u s}$ is mass of nucleus

For Oxygen,

$M_{n u c l e u s} = 15.994 u - 8 \times 0.0005486 u = 15.989 u$

$δ = (8 \times 1.007276 u) + (16 - 8) \times 1.008665 u - 15.989 u = 0.138 u$

Binding energy of by Einstein theory can be given as,

$b i n d i n g e n e r g y = δ c^{2} = 0.138 \times 931.5 MeV = 128.5 MeV$

The Binding energy per nucleon is then,

$\frac{128.5}{16} = 8 MeV$

Hence,

$E = - 8 MeV$

Proton separation energy:

${}_{8}^{16}O \to {}_{1}^{1}p + {}_{7}^{15}N$

Mass difference $=$ Mass of reactant $-$ Mass of product

$∆ m = 15.994 u - (15.000 + 1.007276) u = - 0.0133 u$

Mass difference is negative hence energy being absorbed,

$E = ∆ m \times 931.5 MeV = - (0.0133 \times 931.5) MeV = - 12.37 MeV$

Comparison:

Energy required by proton separation energy is greater than the energy by binding energy per nucleon.

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