pKa

# Detailed discussion of pKa vs pKa0 of the buffer

### Dependence of buffer pKa on ionic strength (and hence concentration) of the buffer solution

For a buffer HAz = H+ + Az-1, the buffer dissociation constant Ka is defined according to the “Law of mass action” as:

Ka = [H+][Az-1] / [HAz]

where [H+], [Az-1] and [HAz] are concentrations in moles per liter of the corresponding species.

pKa is defined as follows:

pKa = -log(Ka) = -log ([H+][Az-1] / [HAz] ) = -log [H+] - log ([Az-1] / [HAz]) = pH - log ([Az-1] / [HAz])

The thus defined pKa is equal to the pH of the buffer solution when concentrations of the two buffering species are equal:

pKa =pH - log([ [AZ-1]/[HAZ]), pH = pKa when [AZ-1] = [HAZ].

In reality, things are more complicated. In a solution of nonzero ionic strength, any ion X is surrounded by a number of ions of the opposite charge. This “shielding” effect causes concentration of the ion X appear smaller that the actual concentration [X]. This effect become more pronounced as the ionic strength (or buffer concentration) increases.

Because of this shielding effect, pKa becomes dependent on the solution concentration and "pKa" looses its meaning as a buffer-specific constant. To circumvent this problem the concentration of species of interest are replaced by so called activities. An activity αx of ion X is defined as

αx = γx ∙ [X]
where [X] is concentration of the ion X in moles per liter
and γx is so called “activity coefficient”

This way, for a buffer HAz = H+ + Az-1, the buffer true “thermodynamic” dissociation constant Ka0 is defined as:

Ka0 = (αH+ ∙ αAz-1) / αHAz

pKa0 = pH – log (αAz-1 / αHAz ) = pH – log (γA [Az-1] / γHA [HAz]) = pH - log ([Az-1] / [HAz]) - log (γA / γHA)

pKa0 = pKa - log (γA / γHA)

Unfortunately, it is impossible to precisely calculate activity coefficient values. However, several approximate models were proposed which allows for estimation of activity coefficients in solutions of a given ionic strength.

For ionic strengths up to 0.1M, the activity coefficient γ of a charged species is given by Debye-Hückel equation:

log γi = - A∙zi2 I  / (1+B√ I ), where
A - the Debye–Hückel constant, dependent on dielectric constant of the solvent, and it is accepted that A = 0.509 for water at 25 oC,
B - empirical constant, referred to as an "ion size" parameter, and it is accepted that B = 1.6 for treatment of biological buffers.

Therefore, assuming z is the charge of the proton bearing (acidic) form of the buffer:

pKa0 = pKa - log (γA / γHA) = pKa - log γA + log γHA = pKa + A∙(z-1)2 I  / (1+B√ I ) - A∙z2 I  / (1+B√ I )

pKa0 = pKa + ((z-1)2 - z)2 A√ I  / (1+B√ I ) = pKa - (2z-1)A√ I  / (1+B√ I )

pKa = pKa0 + A(2z+1)√ I  / (1+B√ I )

For ionic strengths 0.1 - 0.5 M, activity coefficients are better approximated by Davies equation:

log γi = - A∙zi2(√ I  / (1+√ I ) - 0.2∙I), and therefore

pKa = pKa0 + A∙(2z+1)∙[ I  / (1+√ I ) - 0.2∙I]

For buffers that are generally appropriate for cation-exchange chromatography, (2z-1) is negative. Therefore, pKa < pKa0, and, in general, the pKa of the buffer decreases with increasing buffer concentration, and therefore the pH of a prepared stock buffer solution is increased on dilution.

Conversely, for buffers that are generally appropriate for anion-exchange chromatography, (2z-1) is positive. Therefore, pKa > pKa0, and, in general, the pKa of the buffer increases with increasing buffer concentration, and therefore the pH of a prepared stock buffer solution is decreased on dilution.

For your convenience, ionic strength at pH=pKa, and absolute values of ΔpKa = |pKa - pKa0| are calculated for different buffers at various concentrations and presented in three tables below.

 The best buffers for cation-exchange (HA = H+ + A-1; 2z-1=-1) and anion-exchange (HA+ = H+ + A; 2z-1=1) chromatography Buffer concentration 10mM 20mM 30mM 40mM 50mM 60mM 70mM 80mM 90mM 100mM 200mM Ionic strength at pH=pKa, I 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.100 ΔpKa = |pKa - pKa0| 0.032 0.044 0.052 0.059 0.064 0.069 0.073 0.077 0.081 0.084 0.107

 Less suitable buffers for cation-exchange (HA-1 = H+ + A-2; 2z-1=-3) and anion-exchange (HA+2 = H+ + A+1; 2z-1=3) chromatography: Barely suitable buffers for cation-exchange chromatography (HA-2 = H+ + A-3; 2z-1=-5) Buffer concentration 10mM 20mM 30mM 40mM 50mM 10mM 20mM 30mM Ionic strength at pH=pKa, I 0.020 0.040 0.060 0.080 0.100 0.045 0.090 0.135 ΔpKa = |pKa - pKa0| 0.176 0.231 0.269 0.297 0.321 0.403 0.516 0.589

### Dependence of buffer pKa on the temperature

The pKa0 (and, of course, the pKa) of a given buffer is temperature dependent in a non-linear and unpredictable way. For small temperature deviations from 25 oC, the temperature effect can be estimated from the van’t Hoff equation:

d(pKa0)/dt = - ΔH0/(2.3RT2), where

R = 8.314 J/(mol K) – universal gas constant,
ΔH0 - molar enthalpy of buffer dissociation reaction, J/mol, tabulated below for many common buffering species,
T – temperature in K (25 oC = 298.25 K).

At 25 oC: d(pKa0)/dt = - ΔH0/(2.3 * 8.314 * 298.152) = - ΔH0/1,700,000

The links below will transfer you to tables of buffers suitable for cation and anion exchange chromatography. For your convenience, these tables contain values of pKa0, d(pKa0)/dt at 298.25 K, and a calculator based on models explained above that allows you to estimate pKa values of each buffer at temperatures form 3oC (cold room) to 37oC, and concentrations from 1mM up to 500mM.

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