Question

In Biochemistry- How do you know if you need to use the quadratic equation for weak acid problems? (Or, how do you know that the “x is small assumption” is correct?) Why does this work?

Answer 1

A weak acid is the acid that gets partially ionised in solution to give hydrogen ions and the acid in the dissociated form.

  

Initial concentration(M) C 0 0

Change -x +x +x

Equilibrium concentration(M)C-x x x

Acid dissociation constant (Ka)=[H⁺][A⁺]/[HA]

=(x)(x)/(C-x)

Here x gives the amount of A^- formed from the partial ionisation of HA. If the value of the acid dissociation constant is very small(<<<1, like <10^-4), this means the concentration of A^- and H⁺ (i.e. x) is very small as compared to the initial concentration of HA (i.e. C). So x is very small as compared to C and can be ignored in the denominator as compared to C i.e. C-x≈C

So Ka≈(x)(x)/C

The assumption that "x is small' works because the extent of dissociation of a very weak acid is very small, and thus x actually very small as compared to C.

But if the value of Ka is not very small (of the order of 10^-1-10^-3), we don't ignore x as compared to C and instead solve the quadratic equation.

Ka(C-x)=(x)(x)

HA(aq) = H+(aq) + A- (aq)