The following theory was developed for an experiment using solutions of 4-NP in a series of buffers at different pH values. I think it would also work for a situation where one titrated the chromophore with acid or base, as long as one made corrections for the change in volume during the titration, but perhaps someone else can weigh in on that.

The observed absorbance of any weak acid such as 4-NP at any pH is the sum of the absorbances of the neutral species (HB) and the anionic species (B-). As you will see in this lab, the observed absorbance depends upon pH because the two species have different spectra. Each form has its own molar absorptivity, which is pH-independent. We use Beer's law to write:

A_{obsd} = A_{HB} + A_{B-} = ε_{HB}[HB] + ε_{B-}[B-] (1)

By conservation of mass, we can write:

[B_{t}] = [HB] + [B^{-}] (2)

It is convenient to define a pH-dependent quantity, the **apparent** or observed molar absorptivity:

ε_{H} = (ε_{HB}[HB] + ε_{B-}[B-])/[B_{t}] (3)

The observed absorbance is a function of the apparent molar absorptivity and the total concentration of conjugate acid and conjugate base:

A_{obsd} = ε_{H}[B_{t}] (4)

The relative concentration of each species depends upon the [H+] and the acidity constant, K_{a}. Therefore we can write:

ε_{H} = (ε_{HB}[H^{+}] + ε_{B-}K_{a})/(K_{a} + [H^{+}]) (5)

An alternate form of equation (5) is

ε_{H} = (ε_{HB} + ε_{B-}•10^{(pH-pKa)})/(10^{(pH-pKa)} + 1) (6)

One may fit equation (5) or (6) using nonlinear regression, which will estimate values of K_{a}, ε_{HB} and ε_{B-}.

With respect to 4-NP and equation 6, if one measures absorbance values at 400 nm and plots ε_{H} vs. pH, one will see a sigmoidal curve that rises with increasing pH. The asymptotes are ε_{HB} and ε_{B-}, and the inflection point is pK_{a}. If one measures absorbance values at 317 nm and plots ε_{H}, then one will still get a sigmoidal curve, but it will fall with rising pH.

If one measures absorbance values at the isosbestic point (348 nm) one will get an uninformative straight line. Putting it another way, suppose that I have a solution of a known total concentration of 4-NP at an unknown pH. If I measure absorbance values at 348 nm, then I don't get any information on pH. If I measure absorbance values at 400 nm or 317 nm, then I do get information on pH. I'll try to write something more this evening if I have time.