0.1 M solution of weak acid has pH=4.0. Calculate pK_a.

If pH=4.0 then [H⁺]=10^-4 - that means that the dissociation fraction is about
10^-4/0.1×100%=0.1% - well below 5%. That in turn means that pH of the solution is described reasonably accurately by the simplified formula 8.13:

All we have to do is rearrange to get the equation for K_a:

Plugging numbers in we get K_a=(10^-4)²/0.1=10^-7, so pK_a=7.

Our pH calculator can help us check this result - and indeed, entering 0.1 as concentration and 7 as pK_a we get pH=4.0 which confirms our calculations.