Which of the following nuclides would you expect to be radioactive: ${ }_{26}^{58} \mathrm{Fe},{ }_{27}^{60} \mathrm{Co},{ }_{41}^{92} \mathrm{Nb},$ mercury- \(202,\) radium \(-226 ?\) Justify your choices.

For each nuclide, calculate the neutron-to-proton (n/p) ratio. The neutron number (N) can be found by subtracting the atomic number Z (number of protons) from the mass number A (sum of protons and neutrons). The calculation of the n/p ratio for each nuclide is as follows: For \({}_{26}^{58} \mathrm{Fe}\), \(N = A - Z = 58 - 26 = 32\). The n/p ratio = \(\frac{32}{26} \approx 1.23\). For \({}_{27}^{60} \mathrm{Co}\), \(N = 60 - 27 = 33\). The n/p ratio = \(\frac{33}{27} \approx 1.22\). For \({}_{41}^{92} \mathrm{Nb}\), \(N = 92 - 41 = 51\). The n/p ratio = \(\frac{51}{41} \approx 1.24\). For mercury-202, \(A=202\) and \(Z=80\), \(N = 202 - 80 = 122\). The n/p ratio = \(\frac{122}{80} \approx 1.53\). For radium-226, \(A=226\) and \(Z=88\), \(N = 226 - 88 = 138\). The n/p ratio = \(\frac{138}{88} \approx 1.57\).

Using the neutron-to-proton ratios calculated in Step 1, along with the concept of magic numbers, analyze each nuclide for stability: For \({}_{26}^{58} \mathrm{Fe}\), the n/p ratio is close to 1, suggesting it may be stable. Neither the proton number (26) nor neutron number (32) corresponds to magic numbers. For \({}_{27}^{60} \mathrm{Co}\), the n/p ratio is close to 1, suggesting it may be stable. The neutron number (33) is not a magic number, but the proton number (27) is close to the magic number 28. For \({}_{41}^{92} \mathrm{Nb}\), the n/p ratio is close to 1, suggesting it may be stable. Neither the proton number (41) nor neutron number (51) corresponds to magic numbers. For mercury-202, the n/p ratio is higher, suggesting it may be unstable (i.e., radioactive). Neither the proton number (80) nor neutron number (122) corresponds to magic numbers. For radium-226, the n/p ratio is higher, suggesting it may be unstable (i.e., radioactive). Neither the proton number (88) nor neutron number (138) corresponds to magic numbers.

From the analysis in Step 2, it is expected that mercury-202 and radium-226 would be radioactive due to their higher neutron-to-proton ratios and lack of magic numbers. The other nuclides (\({_26}^{58} \mathrm{Fe}\), \({_27}^{60} \mathrm{Co}\), and \({_41}^{92} \mathrm{Nb}\)) have neutron-to-proton ratios close to 1 and are not expected to be radioactive.