When the actual measured mass of an atom is compared to the sum of the component

Question

When the actual measured mass of an atom is compared to the sum of the component subatomic particles (electrons, protons, and neutrons), the mass of the isotope is always smaller. This mass loss or defect has been converted to energy in the formation of the atom. The energy equivalent of the missing mass is known as the total nuclear binding energy and can be calculated from the mass defect. Binding energies are often expressed in electronvolts (eV). The mass defect can be related to this energy term as 1 amu = 9.31 times 10^8 eV where a mass change of 1amu corresponds to an energy release of 9.31 times 10^8 eV. Often in such calculations the total mass of an electron (0.0005485 amu) and a proton (1.0072765 amu) is expressed in terms of the mass of a jH atom, or 1.0078250 amu.

Explanation / Answer

For given isotope of gold 19779Au

Mass number (A) = 197

Atomic number (Z) = Proton number (P) = Electron number (E) = 79

Hence Neutron number (N) = A – P = 197 – 79 = 118

We have mass of proton (Mp) = 1.0072765 amu.

Mass of Neutron (Mn) = 1.0078250 amu.

Mass of Electron (Me) = 0.0005485 amu.

Using these masses and there numbers in Gold isotope let us find out exact mass of this isotope that formed from given number of Neutrons, protons, electrons.

Expected mass of Au-197 = P x Mp + N x Mn + E x Me

Expected mass of Au-197 = 79 x 1.00727565 + 118 x 1.0078250 + 79 x 0.0005485

Expected mass of Au-197 = 198.5414579 amu

But given that the observed mass of Au-197 = 196.967

From expected mass and observed mass we can find out the mass defect (m).

m = Expected mass – Observed mass

m = 198.5414579 – 196.967

m = 1.57445785 amu.

This is the mass which converted into the energy according to Einstein mass-energy relation (E = mc2). This energy is called as total nuclear binding energy(E)

We are given with that,

1 amu 9.31 x 108 eV total nuclear binding energy.

For Au-197 isotope total nuclear binding energy hence is

E = m (in amu unit) x 9.31 x 108 eV.

E = 1.57445785 x 9.31 x 108 eV.

E = 1.465820258 x 109 eV

E = 1.47 x 109 eV ……………(rounded to 3 significant figures)

Hence the total nuclear binding energy of Au-197 isomer is 1.47 x 109 eV.

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