Question

Which of the following isotopes is likely to be most stable?
A) \[_{30}^{71}{\text{Zn}}\]
B) \[_{30}^{66}{\text{Zn}}\]
C) \[_{20}^{40}{\text{Ca}}\]
D) none of these

Answer 1

Hint: A nucleus is stable when its neutron to proton ratio is around 1. If the neutron to proton ratio is either high or low, then the nucleus is unstable and undergoes a radioactive decay process. The nuclei having neutron to proton ratio around one are in the zone of stability. The nuclei outside this zone of stability are unstable nuclei.

Complete step by step answer:
In the symbol \[_Z^A{\text{X}}\] for an isotope, A is the mass number and Z is the atomic number. The atomic number is equal to the number of electrons. It is also equal to the number of protons. The mass number is equal to the sum of the number of protons and neutrons. The number of neutrons is equal to the difference between the mass number and the atomic number.
For the nucleus \[_{30}^{71}{\text{Zn}}\] the atomic number is 30 and the mass number is 71. The number of protons is 30 and the number of neutrons is 71 - 30 = 41. The neutron to proton ratio \[\dfrac{n}{p} = \dfrac{{41}}{{30}} = 1.37 > 1\]. The neutron to proton ratio is high. Hence, the nucleus is outside the zone of stability. Hence, the nucleus is unstable.
For the nucleus \[_{30}^{66}{\text{Zn}}\] the atomic number is 30 and the mass number is 66. The number of protons is 30 and the number of neutrons is 66 - 30 = 41. The neutron to proton ratio \[\dfrac{n}{p} = \dfrac{{36}}{{30}} = 1.2 > 1\]. The neutron to proton ratio is high. Hence, the nucleus is outside the zone of stability. Hence, the nucleus is unstable.
For the nucleus \[_{20}^{40}{\text{Ca}}\] the atomic number is 20 and the mass number is 40. The number of protons is 20 and the number of neutrons is 40 - 20 = 20. The neutron to proton ratio \[\dfrac{n}{p} = \dfrac{{20}}{{20}} = 1\] . The neutron to proton ratio is one. Hence, the nucleus is in the zone of stability. Hence, the nucleus is stable.
The isotope \[_{20}^{40}{\text{Ca}}\] is most likely to be stable.

Hence the correct option is (C).

Note: There is another concept of magic number. Magic numbers for protons are 2, 8, 20, 28, 50, 82, 114, 122, 124, and 164. The magic numbers for neutrons are 2, 8, 20, 28, 50, 82, 126.
Nuclei having a magic number of protons and neutrons are stable.