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Which of the following has the magic number of neutrons?
(a) $_{ 13 }{ Al^{ 17 } }$
(b) $_{ 26 }{ Fe^{ 56 } }$
(c) $_{ 83 }{ Bi^{ 209 } }$
(d) $_{ 92 }{ U^{ 38 } }$

Answer Verified Verified
Hint: In order to identify the stability of an isotope, we use nuclear stability which depends upon the neutron to proton ratio and the number of nucleons in the nucleus. The number of neutrons can be easily determined by subtracting the atomic number from the mass number. The magic number of neutrons is the number of neutrons that are present in stable isotopes (non-radioactive).

Complete step by step answer:
In order to identify the stability of an isotope, we use nuclear stability which depends upon the neutron to proton ratio and the number of nucleons in the nucleus.
In order to determine the number of neutrons present in an atom, we need to subtract its atomic number from its mass number (the mass number is the sum of the total number of protons and neutrons in an atom).
For lighter elements (these elements will have an atomic number less than 20) the neutron to proton ratio is 1:1 i.e. the number of neutrons and the number of protons should be equal.
In heavy elements (they have their atomic number between 20 and 83); the neutron to proton ratio is around 1.5:1 for stability.
The magic number of neutrons is the number of neutrons that are present in stable isotopes (non-radioactive). These magic numbers are: 2, 8, 20, 28, 50, 82, 126, and 184.
Let us calculate the number of neutrons for each atom:
 (a) $_{ 13 }{ Al^{ 17 } }$ = 27 - 13 = 14
(b) (b) $_{ 26 }{ Fe^{ 56 } }$ = 56 - 26 = 30
(c) $_{ 83 }{ Bi^{ 209 } }$ = 209 - 83 = 126
(d) $_{ 92 }{ U^{ 38 } }$ = 238 - 92 = 146
Hence the correct answer is (c) $_{ 83 }{ Bi^{ 209 } }$.
Note: Just like there is a magic number of neutrons, there is also a magic number of protons. The magic numbers of protons are: 2, 8, 20, 28, 50, 82, and 114. Isotopes that have either of these magic numbers (protons or neutrons) are stable. Some heavier isotopes have both the magic number of protons as well as neutrons (called double magic numbers). There are also many isotopes that do not have a magic number of nucleons but they are still stable.