This Demonstration is based on  by Limpanuparb et al.
Snapshot 1: calculation for an acid solution diluted to a very low concentration, using a quartic equation
Snapshot 2: calculation for a buffer solution using a cubic equation, with input parameters that the Henderson–Hasselbalch equation neglects
Snapshot 3: calculation for a solution of pure water (no acid or base) at a high temperature, showing the effect of self-ionization of water
The quartic equation used for calculation [2, 3] is
The cubic equation used for calculation  is
Definitions of input variables:
is the acid dissociation constant;
(assume 0 for strong acid).
is the molar concentration of acid.
is the volume of acid in mL.
is the molar concentration of conjugate base.
is the base dissociation constant;
, (assume 0 for strong base).
is the molar concentration of base.
is the volume of base in mL.
is the molar concentration of conjugate acid.
Definitions of output variables:
is the molar concentration of hydronium ion.
is the molar concentration of hydroxide ion.
pH and pOH are the negative of the common logarithm of
Fraction of dissociation
is the ratio of equilibrium concentration of dissociated acid/base to initial concentration of the acid/base.
 T. Limpanuparb et al., unpublished work, submitted to Journal of Chemical Education
 P. Glaister, "A Unified Titration Formula," Journal of Chemical Education
(1), 1999 132. doi:10.1021/ed076p132
 R. de Levie, "Explicit Expressions of the General Form of the Titration Curve in Terms of Concentration: Writing a Single Closed-Form Expression for the Titration Curve for a Variety of Titrations without Using Approximations or Segmentation," Journal of Chemical Education
(3), 1993 209. doi:10.1021/ed070p209
 H. L. Pardue, I. N. Odeh and T. M. Tesfai, "Unified Approximations: A New Approach for Monoprotic Weak Acid-Base Equilibria," Journal of Chemical Education
(9), 2004 1367. doi:10.1021/ed081p1367