Question

If \[{M_O}\] is the mass of an oxygen isotope \[_8{O^{17}}\], \[{M_P}\] and \[{M_n}\] are the masses of a proton and a neutron, respectively, the nuclear binding energy of the isotope is
A. \[{M_O}{c^2}\]
B. \[\left( {{M_O} - 17{M_n}} \right){c^2}\]
C. \[\left( {{M_O} - 8{M_p}} \right){c^2}\]
D. \[(8{M_p} + 9{M_n} - {M_O}){c^2}\]

Answer 1

Hint:
The total mass of any nucleus is less than the sum of the masses of its constituent elements. This difference in mass is known as mass defect. This mass defect is connected to an energy by the mass- energy relationship and this energy is called nuclear binding energy.
Formula used:
For any nucleus of atomic mass A, having Z number of protons and (A-Z) number of neutrons, the mass defect is given by:
\[\Delta m = \left[ {Z{m_p} + \left( {A - Z} \right){m_n}} \right] - {m_{nuc}}\]………………… (1)
Where, mnuc is the mass of the nucleus.
The mass energy relationship is given by:
\[E = m{c^2}\]………………… (2)
Nuclear binding energy is given by:
\[{E_b} = \Delta m{c^2}\]…………… (3)

Complete step-by-step answer:
Note that the given nucleus is the nucleus of an isotope of oxygen, 8O17.
The usual notation for writing any atom is zXA.
Comparing them, you can write, the atomic number Z=8 and atomic mass A=17.
You know that neutron number is (A-Z) i.e., (17-8)=9
Hence from the equation 1 you can easily calculate the mass defect
\[\Delta m = \left[ {(8{M_p} + 9{M_n}) - {M_O}} \right]\]
Where, Mp and Mn are the masses of one proton and one neutron respectively and MO is the mass of the nucleus.
Now you can calculate the binding energy using equation 3.
\[{E_b} = \left[ {(8{M_p} + 9{M_n}) - {M_O}} \right]{c^2}\]
Hence the option D is the correct one.
Note:
You must know the two terms atomic number and atomic mass distinctly. The atomic mass gives the total number of nucleons where the atomic number gives the total number of protons. And one more thing, mass of electrons do not contribute much in the total mass of an atom. The mass of the atom is nearly equal to the mass of its nucleus.