Soap is made by a saponification reaction, where a fat reacts with hydroxide ions to form a surfactant and glycerol.

To make a solid soap $\ce{NaOH}$ is used, while $\ce{KOH}$ is used for liquid soaps.

I don't understand why the alkali metal has such a great impact on the state of matter. Usually the argumentation is based on intramolecular interactions, such as Van-der-Waals forces or hydrogen bonds, but if the same fat is used once with $\ce{NaOH}$ and once with $\ce{KOH}$ the resulting surfactants are basically the same, so the interactions shouldn't differ too much.

The only reason I could think of is the size of the alkali metal. Potassium has an atomic radius of 231 pm which is quite a bit more than the radius of sodium, being 186 pm. But why the atomic radius should have an impact on the state of matter of the soap is still unclear to me. Maybe I'm also completely wrong with this assumption.

Does anyone can explain this?