Pedro, you are mistaken when you say that a strong acid has an unstable conjugate base. It’s actually the opposite, the more stable the conjugate base of an acid, the stronger the acid. Why you may ask? Because if it is so willing to give away a hydrogen, it is because it can withstand the negative charge! A deprotonated acid, is an anion, and this anion is better at deprotonating an acid stron than its conjugate acid. Also, in acid-base reactions, the equilibrium will always favour the weaker acid-base pair! Chateliers principle in play. On the other hand, nucleophilicity involves kinetics and also the willingness of a compound or atom to give its lone pair of electrons to a election deficient atom. Nucleophilicity of an atom follows the opposite trend of electronegativity!

A.I

The sentence says, “stronger” nucleophile, not strong nucleophile. The conjugate base of HCl is Cl- which is a better nucleophile than HCl (but not itself a strong base or nucleophile). Thank you for elaborating on the points above.

When you have a conjugate base of a weak acid, the base is necessarily a strong base. That is to say it has a greater tendency to capture protons than the conjugate base of a strong acid (ie.: weaker base).

Judging from the above paragraph alone, one might be tempted to think that a stronger base would make for a stronger nucleophile. However, the strength of a conjugate base is measured by its unstability (the more unstable the conjugate base, the stronger the acid). So when you have a weak acid and therefore a strong conjugate base, you have a very unstable species with an enormous tendency for removing protons in order to neutralize its negative charge. Nucleophily though, is a slightly different thing. It is not as much an ability to neutralize the base itself and its negative charge as an urge to neutralize positive charges by donating electrons.

So, when you have a carbocation, for example, you need a species that attacks the positive charge created by the electron deficit generated by the leaving of the exiting group. If you had a very strong base, it would probably be more likely to remove an acidic proton from one of the beta carbons than to bond to the positively charged centre.