Editing Buffer solution (section)

== Calculating buffer pH ==

=== Monoprotic acids===

First write down the equilibrium expression
{{block indent|em=1.5|text=HA {{eqm}} A<sup>−</sup> + H<sup>+</sup>}}
This shows that when the acid dissociates, equal amounts of hydrogen ion and anion are produced. The equilibrium concentrations of these three components can be calculated in an [[ICE table]] (ICE standing for "initial, change, equilibrium").
:{| class="wikitable"
|+ ICE table for a monoprotic acid
|-
!
! [HA] !! [A<sup>−</sup>] !! [H<sup>+</sup>]
|-
! I
| ''C''<sub>0</sub> || 0 || ''y''
|-
! C
| −''x'' || ''x'' || ''x''
|-
! E
| ''C''<sub>0</sub> − ''x'' || ''x'' || ''x'' + ''y''
|}
The first row, labelled '''I''', lists the initial conditions: the concentration of acid is ''C''<sub>0</sub>, initially undissociated, so the concentrations of A<sup>−</sup> and H<sup>+</sup> would be zero; ''y'' is the initial concentration of ''added'' strong acid, such as hydrochloric acid. If strong alkali, such as sodium hydroxide, is added, then ''y'' will have a negative sign because alkali removes hydrogen ions from the solution. The second row, labelled '''C''' for "change", specifies the changes that occur when the acid dissociates. The acid concentration decreases by an amount −''x'', and the concentrations of A<sup>−</sup> and H<sup>+</sup> both increase by an amount +''x''. This follows from the equilibrium expression. The third row, labelled '''E''' for "equilibrium", adds together the first two rows and shows the concentrations at equilibrium.

To find ''x'', use the formula for the equilibrium constant in terms of concentrations:
<math chem display="block">K_\text{a} = \frac{[\ce{H+}] [\ce{A-}]}{[\ce{HA}]}.</math>

Substitute the concentrations with the values found in the last row of the ICE table:
<math display="block">K_\text{a} = \frac{x(x + y)}{C_0 - x}.</math>

Simplify to <math display="block">x^2 + (K_\text{a} + y) x - K_\text{a} C_0 = 0.</math>

With specific values for ''C''<sub>0</sub>, ''K''<sub>a</sub> and ''y'', this equation can be solved for ''x''. Assuming that pH&nbsp;=&nbsp;−log<sub>10</sub>[H<sup>+</sup>], the pH can be calculated as pH&nbsp;=&nbsp;−log<sub>10</sub>(''x''&nbsp;+&nbsp;''y'').

===Polyprotic acids===
[[File:Citric acid speciation.svg|thumb|alt=This image plots the relative percentages of the protonation species of citric acid as a function of p H. Citric acid has three ionizable hydrogen atoms and thus three p K A values. Below the lowest p K A, the triply protonated species prevails; between the lowest and middle p K A, the doubly protonated form prevails; between the middle and highest p K A, the singly protonated form prevails; and above the highest p K A, the unprotonated form of citric acid is predominant.| [[Determination of equilibrium constants#speciation calculations|% species formation]] calculated for a 10-millimolar solution of citric acid]]

Polyprotic acids are acids that can lose more than one proton. The constant for dissociation of the first proton may be denoted as ''K''<sub>a1</sub>, and the constants for dissociation of successive protons as ''K''<sub>a2</sub>, etc. [[Citric acid]] is an example of a polyprotic acid H<sub>3</sub>A, as it can lose three protons.
:{| class="wikitable" style="width: 230px; 
|+ Stepwise dissociation constants
|-
! |Equilibrium!!Citric acid
|-
| H<sub>3</sub>A {{eqm}} H<sub>2</sub>A<sup>−</sup> + H<sup>+</sup>||p''K''<sub>a1</sub> = 3.13
|-
| H<sub>2</sub>A<sup>−</sup> {{eqm}} HA<sup>2−</sup> + H<sup>+</sup>|| p''K''<sub>a2</sub> = 4.76
|-
| HA<sup>2−</sup> {{eqm}} A<sup>3−</sup> + H<sup>+</sup>|| p''K''<sub>a3</sub> = 6.40
|}

When the difference between successive p''K''<sub>a</sub> values is less than about 3, there is overlap between the pH range of existence of the species in equilibrium. The smaller the difference, the more the overlap. In the case of citric acid, the overlap is extensive and solutions of citric acid are buffered over the whole range of pH&nbsp;2.5 to 7.5.

Calculation of the pH with a polyprotic acid requires a [[Determination of equilibrium constants#Speciation calculations|speciation calculation]] to be performed. In the case of citric acid, this entails the solution of the two equations of mass balance:

<math chem display="block">\begin{align}
 C_\ce{A} &= [\ce{A^3-}]+ \beta_1 [\ce{A^3-}][\ce{H+}] + \beta_2 [\ce{A^3-}][\ce{H+}]^2 + \beta_3 [\ce{A^3-}][\ce{H+}]^3, \\
 C_\ce{H} &= [\ce{H+}] + \beta_1 [\ce{A^3-}][\ce{H+}] + 2\beta_2 [\ce{A^3-}][\ce{H+}]^2 + 3\beta_3 [\ce{A^3-}][\ce{H+}]^3 - K_\text{w}[\ce{H+}]^{-1}.
\end{align}</math>

''C''<sub>A</sub> is the analytical concentration of the acid, ''C''<sub>H</sub> is the analytical concentration of added hydrogen ions, ''β<sub>q</sub>'' are the [[equilibrium constant#Cumulative and stepwise formation constants|cumulative association constants]]. ''K''<sub>w</sub> is the constant for [[self-ionization of water]]. There are two [[non-linear]] [[simultaneous equation]]s in two unknown quantities [A<sup>3−</sup>] and [H<sup>+</sup>]. Many computer programs are available to do this calculation. The speciation diagram for citric acid was produced with the program HySS.<ref>{{cite journal | last1 = Alderighi | first1 = L. | last2 = Gans | first2 = P. | last3 = Ienco | first3 = A. | last4 = Peters | first4 = D. | last5 = Sabatini | first5 = A. | last6 = Vacca | first6 = A. | year = 1999 | title = Hyperquad simulation and speciation (HySS): a utility program for the investigation of equilibria involving soluble and partially soluble species | journal = Coordination Chemistry Reviews | volume = 184 | issue = 1 | pages = 311–318 | doi = 10.1016/S0010-8545(98)00260-4 | url = http://www.hyperquad.co.uk/hyss.htm | url-status = live | archive-url = https://web.archive.org/web/20070704083413/http://www.hyperquad.co.uk/hyss.htm | archive-date = 2007-07-04 | url-access = subscription }}</ref>

N.B. The numbering of cumulative, overall constants is the reverse of the numbering of the stepwise, dissociation constants.
:{| class="wikitable"
|+ Relationship between cumulative association constant (β) values and stepwise dissociation constant (K) values for a tribasic acid.
! Equilibrium!! Relationship
|-
| A<sup>3−</sup> + H<sup>+</sup> {{eqm}} AH<sup>2+</sup>||Log β<sub>1</sub>= pk<sub>a3</sub>
|-
| A<sup>3−</sup> + 2H<sup>+</sup> {{eqm}} AH<sub>2</sub><sup>+</sup>||Log β<sub>2</sub> =pk<sub>a2</sub> + pk<sub>a3</sub>
|-
| A<sup>3−</sup> + 3H<sup>+</sup>{{eqm}} AH<sub>3</sub>||Log β<sub>3</sub> = pk<sub>a1</sub> + pk<sub>a2</sub> + pk<sub>a3</sub>
|}
Cumulative association constants are used in general-purpose computer programs such as the one used to obtain the speciation diagram above.