Has nothing to do with half-life and everything to do with particle size. Liming agents (like dolomite or calcitic limestone) applied to soil outdoors will last or provide Ca\Mg and neutralize acidity for 6 months to a year or longer. Soils with a high cation exchange capacity require more lime, but it can be applied less frequently. People usually apply lime pellets\aggregate to their lawn once or twice a year; in the fall and in early spring.
Applying lime outside on your lawn, it will take some time to work. When we lime a lawn we have to spread pellets on the surface of the lawn. With an existing lawn we really can't till it into the soil at all. So the best time to lime a lawn is just before a heavy rain. The pellets are made of the lime along with a binder and once they get wet they will disintegrate. The rain will wash the lime into the soil and outside this will take some time.
Eggshells are made primarily of calcium carbonate (CaCO3), which is what more than 90% of the shell is. Oyster shell has a very similar\identical composition. Other forms of garden lime are calcite or dolomite or dolomitic limestone, which are from ground up sedimentary rock\mineral deposits that contain CaCO3, CaMg(CO3)2, and potentially other carbonates. Even when you mix it into soil you can think of it as a precipitate or mineral deposits. But, we mix it directly and thoroughly into the potting soil.
Carbonates react will react in the soil in a couple of ways (and this isn't after a certain amount of time). They react with (neutralize) acids, releasing calcium in the case of CaCO3, as well as carbon dioxide (gas) and water. Carbon dioxide dissolved in water or carbonic acid will also react with some of the CaCO3 to form calcium bicarbonate, which is soluble (Ca2+ ions will dissociate). So, basically acids will dissolve the calcium carbonate and it will dissociate into the soil solution over time. The mineral cations such as calcium (Ca2+) will also be adsorbed by soil particles which hold onto them loosely (CEC).
Particle size matters because the finer the particles are, the more dispersed they are, more surface area is exposed and the reactions will occur more rapidly.