Why is Technetium (Element 43) radioactive?

According to the liquid-drop-model of nuclei the most stable isotope would be 100-Tc. But with two odd numbers of protons (43) and neutrons (57) it could only be stable if there weren't other stable nuclei with two even numbers of protons and neutrons, which are 100-Mo (42/58) and 100-Ru (44/56). The second is stable and the first almost ##(7.3 \cdot 10^{18}a)##. (Also a consequence of this model, however, I haven't found a short way to see it.)
Thus the heavy isotopes decay per neutron emission and the lighter per electron capture.

 

Did you check the Wikipedia page? It has a good introduction. There are several factors, mainly the very low energy of nuclei of elements nearby combined with the odd number of protons of technetium.

 

I have taken the German one and wrote a kind of summarization. Unfortunately the English site wasn't as detailed concerning the radioactivity of the isotopes. The good thing is, I learned something about the stability of nuclei myself.

Edit: the 100-Tc was a result from a formula on the liquid-drop-model page plus Wolfram.

 

As far as I can see it, the liquid-drop-model is - as quoted - semi-empirical, i.e. it takes the observations and tries to explain them by patterns. Those patterns represent the more general principle, that nature always tries to find local energy minima, stable points of balance where energy from elsewhere is needed to leave those states. Since all three atomic forces play a role here the overall situation can be pretty complex. The patterns mentioned in the liquid-drop-model are meant to reduce this complexity onto computational principles. And it works astonishingly well as I experienced as I took them to calculate 100-Tc as a preferred state.

 

All the odd isotopes argon. scnr.

It is not that odd for the even-numbered elements - ³⁷Ar would have 19 neutrons, ³⁹Ar would have 21 neutrons. 20 is a magic number (the nuclear equivalent of noble gases - completed shells), so ³⁷Cl and ³⁹K are strongly bound, and the argon isotopes decay to those.

¹³⁹Ce has 81 neutrons, ¹⁴¹ has 83. And 82 is a magic number again.

While the Wikipedia page about technetium is a bit short, the linked article isotopes of technetium has more details.

Magic numbers for reference: 2, 8, 20, 28, 50, 82, and 126

 

Preferred in which way, and where do those numbers come from?

Cerium has the most stable isotope at 82 neutrons, 78, 80 and 84 neutrons also lead to isotopes without observed decay.

 

It's not the 55 neutrons, it's the paired 42 and 44 protons. There can be only one stable isotope with an odd sum of protons and neutrons, in both cases the elements nearby have a slightly lower energy. Same for promethium.

 

None of them is magic, see above. But paired protons and neutrons in general lead to lower energies, and in this case the paired protons win by a small margin.