I know the pKa of H2SO4 is −3.
The pKa is the pH under which the acid is halfway deprotonised. The H2SO4 is a strong acid which should be able to dissociate completely if enough water is added.
The molar density of pure H2SO4 is 18,68 mol/dm³. If we had 1 litre of H2SO4 (18,67 moles) and added 37,36 moles of water, the whole H2SO4 should dissociate. If we add only 18,68 moles of water, half of the dissociable protons shall separate, leaving 18,68 moles of H3O+, 9,34 moles of SO4¯ and 9,34 unchanged H2SO4.
18,67 moles of water should be 0,336 litres. So our final solution should be 1,336 dm³.
c(H3O+) = 18,67/1,336 = 13,97 mol/dm³
c(SO4¯) = c(H2SO4) = 9,34/1,336 = 6,99 mol/dm³
Shouldn't the pKa equal −log(13,97•6,99/6,99) = −log(13,97) = −1,145
Why are my assumptions wrong?