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.

A while ago, this question was already asked here on this forum, but I wonder if the given explanation is the only reason for the different state of matter. It's absolutely true, that the reactivity of the alkali metals increases from top to bottom, but can this solely explain the phenomenon? @rch provides the solubility of $\ce{NaOH}$ and $\ce{KOH}$ to back up his answer, but I don't think that this is sufficient. You cannot simply change the hydroxide ion by a fatty acid and assume, that there are no substantial amendments in the reaction behavior, can you?

Does anyone can explain this in more detail?