What makes zinc so desperate to give up its electrons? Silver has one extra electron, and zinc has two, but why is zinc more reactive?
The reason is that Zinc has a filled D-shell, which means that it is almost unique in having completely filled inner shells, with two electrons in the outer shell. All the n=1, n=2 and n=3 orbitals completely filled in Zn, and exactly 2 electrons very loosly bound in the n=4 orbitals.
The most comparable element is Magnesium, which has completely filled n=1 and n=2 orbitals and 2 loose electrons in an n=3 orbit. Magnesium is also very reactive.
Atomic orbitals are oscillating near zero near the origin for higher values of n, so they are more loosly bound as you go up in n, the electron is further away from the nucleus most of the time. Magnesium's configuration is the most unstable configuration that two outer electrons can have. They are highly screened, seeing a core-charge of 2, and the exclusion principle is placing them in a far-away position relative to the core.
Why doesn't Cadmium have similar reactivity? This is because although the n=1,n=2,n=3 shells are filled, and the n=4 S,P,D shells are filled, the n=4 spin-3 F states are not filled until the Lanthanoids much further on up the table. So the n=4 inner F shell states don't Pauli exclude the n=5 S states, so the electrons can reduce their energy and stability by quantum mechanically spreading into those states.
Zinc is not to be compared with silver, by the way, but Cadmium and Mercury, which lie below it, and with Magnesium, which is similar chemically for the reasons explained above.