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Professor Lavelle said in class that strong bases have weak conjugate acids and weak bases have strong conjugate bases. I don't understand why and I also don't understand why stronger bases have conjugate acids with larger pKa values.
Strong acids/bases have to have very weak conjugate bases/acids by definition because otherwise they wouldn't be strong! Strong acids/bases are not in equilibrium in solution; they dissociate completely (there is only the forward reaction). Their conjugate bases/acids are so weak that they will not protonate or deprotonate water molecules and will stay in solution as they are, not affecting [H3O+] and [OH-]. An example of this is Cl-, which doesn't react with water. If their conjugate bases/acids were stronger, they would react with water and the reverse reaction forming the strong acid/base would occur, which completely contradicts the definition of a strong acid/base. Also, similar to pH, small pKa values indicate a stronger acid while large pKa values indicate a weaker acid.
Strong acids include H3O+, HCl and HNO3. For example, in water, a strong acid like hydrochloric acid readily donates a proton to a water molecule and the equilibrium position lies so far to the right that we usually assume that the HCl molecule completely dissociates in water. This means that the reverse reaction does not occur to any appreciable extent, that is Cl- does not accept a proton from H3O+. So Cl- must be a weak base, it has very little tendency to accept a proton. The conjugate base of a strong acid is a weak base.
Stronger bases have conjugate acids that are weak. Since weak acids have large pKa values, the conjugate acid has a large pKa. This is because in the same way that the lower the pH the more acidic a substance is, the lower the pKa the more acidic it is as well.
Try thinking about this on a molecular level. When a strong acid is deprotonated, the resulting conjugate base is remarkably stable. This could be due to inductive effects (shifting of negative charges around atoms with large EN differences) or resonant structures (decreased average negative charge because of shifting bonds). Having a stable conjugate base means the base is unreactive- the negative charge is no longer concentrated in one area, so it does not need to accept a proton. This means strong acids have a large Ka and a small Kb, or a small pKa and a large pKb. Hope that makes sense.