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I missed my classes on ionic equilibrium & I came to know that certain equations were derived for cases such as 'Monobasic acid', 'Two monobasic acids', 'Dibasic acid'etc.

These equations relate H3O+ concentration with the acid's initial concentration, its dissociation constant & water's dissociation constant.

These equations were told to me that, they provide highly accurate answers, instead of the commonly used Henderson–Hasselbalch equation. These equations are of either cubic or bi-quadratic form, taking the H3O+ concentration as the variable.

Can someone explain this? Unfortunately; I couldn't find any sources on the internet (or perhaps I didn't use the proper keywords?) If anyone is able to drop by a link to a webpage that explains these derivations and the use of these equations, it would be greatly helpful. Thank-you.

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Before anything else I must clarify that there is nothing wrong with Henderson–Hasselbalch equation. It is formally exact because it comes from the definition of the $K_a$ as long as you ignore ionic strength which I believe is not what you are interested in at least at this stage. If you say there's anything inaccurate about it it is because people are making assumptions when applying the equation.

Now for the main matter. If you take a look at this note (not what you are looking for I am afraid, but we'll come to that in a moment), near the end of page 4 we have the quadratic form of the equation

$$K_a=\frac{[\ce{H3O+}]^2}{c(\ce{HA})-\ce{H3O+}}$$

based on approximation that $[\ce{H3O+}]\approx[\ce{A-}]$. Now, what you are looking for is the formula without making this assumption. So first examine what's wrong about this assumption.

Mass balance for $\ce A$ states, $$c(\ce{HA})=[\ce{HA}]+[\ce{A-}],$$

charge balance states, $$[\ce{H3O+}]=[\ce{A-}]+[\ce{OH-}],$$

you see that the approximation ignores the second term in RHS, and we do know, $$[\ce{H3O+}][\ce{OH-}]=K_{\rm w},$$

Now it is not difficult to see what we need to put into the the definition of $K_a$. $$K_a=\frac{[\ce{H3O+}][\ce{A-}]}{[\ce{HA}]}=\frac{[\ce{H3O+}]\left([\ce{H3O+}]-\frac{K_{\rm w}}{[\ce{H3O+}]}\right)}{c(\ce{HA})-[\ce{H3O+}]+\frac{K_{\rm w}}{[\ce{H3O+}]}}$$

This is the cubic equation you are looking for, it is not that difficult to derive but quite a pain to solve numerically if you don't have a calculator that solves equation for you (there are various numerical techniques but that's beyond the scope of this answer). The reason that it is not explicitly included in many books is that you don't need to go for such complex form under most occasions and, more importantly, these forms do vary a lot if, say I am talking about $[\ce{H2A}]$ instead of $[\ce{HA}]$.

So the bottom line: instead of pursuing these complex forms (which may seem interesting, but you just can't remember all of them), stick to the charge balance equation and mass balance equation and optionally, if you have been taught, the proton balance equation, and also the definitions of $K_a$'s. With these in mind you can derive the most accurate form of high-order equations for $[\ce{H3O+}]$ whenever you want to, and these balance are not that hard to grisp compared to these exotic high-order equations.

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