Buffer solution

Buffer capacity is a quantitative measure of the resistance to change of pH of a solution containing a buffering agent with respect to a change of acid or alkali concentration. It can be defined as follows:[2][3]

$${\displaystyle \beta ={\frac {dC_{b}}{d(\mathrm {pH} )}},}$$

where ${\displaystyle dC_{b}}$ is an infinitesimal amount of added base, or

$${\displaystyle \beta =-{\frac {dC_{a}}{d(\mathrm {pH} )}},}$$

where ${\displaystyle dC_{a}}$ is an infinitesimal amount of added acid. pH is defined as −log₁₀[H⁺], and d(pH) is an infinitesimal change in pH.

With either definition the buffer capacity for a weak acid HA with dissociation constant K_a can be expressed as[4][5][3]

$${\displaystyle \beta =2.303\left([{\ce {H+}}]+{\frac {T_{{\ce {HA}}}K_{a}[{\ce {H+}}]}{(K_{a}+[{\ce {H+}}])^{2}}}+{\frac {K_{\text{w}}}{[{\ce {H+}}]}}\right),}$$

where [H⁺] is the concentration of hydrogen ions, and ${\displaystyle T_{\text{HA}}}$ is the total concentration of added acid. K_w is the equilibrium constant for self-ionization of water, equal to 1.0×10⁻¹⁴. Note that in solution H⁺ exists as the hydronium ion H₃O⁺, and further aquation of the hydronium ion has negligible effect on the dissociation equilibrium, except at very high acid concentration.

Figure 2. Buffer capacity β for a 0.1 M solution of a weak acid with a pK_a = 7

This equation shows that there are three regions of raised buffer capacity (see figure 2).

• In the central region of the curve (coloured green on the plot), the second term is dominant, and
  $${\displaystyle \beta \approx 2.303{\frac {T_{{\ce {HA}}}K_{a}[{\ce {H+}}]}{(K_{a}+[{\ce {H+}}])^{2}}}.}$$

  
  Buffer capacity rises to a local maximum at pH = pK_a. The height of this peak depends on the value of pK_a. Buffer capacity is negligible when the concentration [HA] of buffering agent is very small and increases with increasing concentration of the buffering agent.[3] Some authors show only this region in graphs of buffer capacity.[2]
  Buffer capacity falls to 33% of the maximum value at pH = pK_a ± 1, to 10% at pH = pK_a ± 1.5 and to 1% at pH = pK_a ± 2. For this reason the most useful range is approximately pK_a ± 1. When choosing a buffer for use at a specific pH, it should have a pK_a value as close as possible to that pH.[2]
• With strongly acidic solutions, pH less than about 2 (coloured red on the plot), the first term in the equation dominates, and buffer capacity rises exponentially with decreasing pH:
  $${\displaystyle \beta \approx 10^{-\mathrm {pH} }.}$$

  
  This results from the fact that the second and third terms become negligible at very low pH. This term is independent of the presence or absence of a buffering agent.
• With strongly alkaline solutions, pH more than about 12 (coloured blue on the plot), the third term in the equation dominates, and buffer capacity rises exponentially with increasing pH:
  $${\displaystyle \beta \approx 10^{\mathrm {pH} -\mathrm {p} K_{\text{w}}}.}$$

  
  This results from the fact that the first and second terms become negligible at very high pH. This term is also independent of the presence or absence of a buffering agent.

Buffering agent	pKa	Useful pH range
Citric acid	3.13, 4.76, 6.40	2.1–7.4
Acetic acid	4.8	3.8–5.8
KH2PO4	7.2	6.2–8.2
CHES	9.3	8.3–10.3
Borate	9.24	8.25–10.25

For buffers in acid regions, the pH may be adjusted to a desired value by adding a strong acid such as hydrochloric acid to the particular buffering agent. For alkaline buffers, a strong base such as sodium hydroxide may be added. Alternatively, a buffer mixture can be made from a mixture of an acid and its conjugate base. For example, an acetate buffer can be made from a mixture of acetic acid and sodium acetate. Similarly, an alkaline buffer can be made from a mixture of the base and its conjugate acid.

By combining substances with pK_a values differing by only two or less and adjusting the pH, a wide range of buffers can be obtained. Citric acid is a useful component of a buffer mixture because it has three pK_a values, separated by less than two. The buffer range can be extended by adding other buffering agents. The following mixtures (McIlvaine's buffer solutions) have a buffer range of pH 3 to 8.[7]

0.2 M Na2HPO4 (mL)	0.1 M citric acid (mL)	pH
20.55	79.45	3.0
38.55	61.45	4.0
51.50	48.50	5.0
63.15	36.85	6.0
82.35	17.65	7.0
97.25	2.75	8.0

A mixture containing citric acid, monopotassium phosphate, boric acid, and diethyl barbituric acid can be made to cover the pH range 2.6 to 12.[8]

Other universal buffers are the Carmody buffer[9] and the Britton–Robinson buffer, developed in 1931.

For effective range see Buffer capacity, above. Also see Good's buffers for the historic design principles and favourable properties of these buffer substances in biochemical applications.

Common name (chemical name)	Structure	pKa, 25 °C	Temp. effect, dpH/dT (K−1)[10]	Mol. weight
TAPS, ([tris(hydroxymethyl)methylamino]propanesulfonic acid)		8.43	−0.018	243.3
Bicine, (2-(bis(2-hydroxyethyl)amino)acetic acid)		8.35	−0.018	163.2
Tris, (tris(hydroxymethyl)aminomethane, or 2-amino-2-(hydroxymethyl)propane-1,3-diol)		8.07[lower-alpha 1]	−0.028	121.14
Tricine, (N-[tris(hydroxymethyl)methyl]glycine)		8.05	−0.021	179.2
TAPSO, (3-[N-tris(hydroxymethyl)methylamino]-2-hydroxypropanesulfonic acid)		7.635		259.3
HEPES, (4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid)		7.48	−0.014	238.3
TES, (2-[[1,3-dihydroxy-2-(hydroxymethyl)propan-2-yl]amino]ethanesulfonic acid)		7.40	−0.020	229.20
MOPS, (3-(N-morpholino)propanesulfonic acid)		7.20	−0.015	209.3
PIPES, (piperazine-N,N′-bis(2-ethanesulfonic acid))		6.76	−0.008	302.4
Cacodylate, (dimethylarsenic acid)		6.27		138.0
MES, (2-(N-morpholino)ethanesulfonic acid)		6.15	−0.011	195.2

First write down the equilibrium expression

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").

ICE table for a monoprotic acid

	[HA]	[A−]	[H+]
I	C0	0	y
C	−x	x	x
E	C0 − x	x	x + y

The first row, labelled I, lists the initial conditions: the concentration of acid is C₀, initially undissociated, so the concentrations of A⁻ and H⁺ 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⁻ and H⁺ 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:

$${\displaystyle K_{\text{a}}={\frac {[{\ce {H+}}][{\ce {A-}}]}{[{\ce {HA}}]}}.}$$

Substitute the concentrations with the values found in the last row of the ICE table:

$${\displaystyle K_{\text{a}}={\frac {x(x+y)}{C_{0}-x}}.}$$

Simplify to

$${\displaystyle x^{2}+(K_{\text{a}}+y)x-K_{\text{a}}C_{0}=0.}$$

With specific values for C₀, K_a and y, this equation can be solved for x. Assuming that pH = −log₁₀[H⁺], the pH can be calculated as pH = −log₁₀(x + y).

% 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_a1, and the constants for dissociation of successive protons as K_a2, etc. Citric acid is an example of a polyprotic acid H₃A, as it can lose three protons.

Stepwise dissociation constants

Equilibrium	Citric acid
H3A ⇌ H2A− + H+	pKa1 = 3.13
H2A− ⇌ HA2− + H+	pKa2 = 4.76
HA2− ⇌ A3− + H+	pKa3 = 6.40

When the difference between successive pK_a 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 2.5 to 7.5.

Calculation of the pH with a polyprotic acid requires a speciation calculation to be performed. In the case of citric acid, this entails the solution of the two equations of mass balance:

$${\displaystyle {\begin{aligned}C_{{\ce {A}}}&=[{\ce {A^3-}}]+\beta _{1}[{\ce {A^3-}}][{\ce {H+}}]+\beta _{2}[{\ce {A^3-}}][{\ce {H+}}]^{2}+\beta _{3}[{\ce {A^3-}}][{\ce {H+}}]^{3},\\C_{{\ce {H}}}&=[{\ce {H+}}]+\beta _{1}[{\ce {A^3-}}][{\ce {H+}}]+2\beta _{2}[{\ce {A^3-}}][{\ce {H+}}]^{2}+3\beta _{3}[{\ce {A^3-}}][{\ce {H+}}]^{3}-K_{\text{w}}[{\ce {H+}}]^{-1}.\end{aligned}}}$$

C_A is the analytical concentration of the acid, C_H is the analytical concentration of added hydrogen ions, β_q are the cumulative association constants. K_w is the constant for self-ionization of water. There are two non-linear simultaneous equations in two unknown quantities [A³⁻] and [H⁺]. Many computer programs are available to do this calculation. The speciation diagram for citric acid was produced with the program HySS.[11]

N.B. The numbering of cumulative, overall constants is the reverse of the numbering of the stepwise, dissociation constants.

Relationship between cumulative association constant (β) values and stepwise dissociation constant (K) values for a tribasic acid.

Equilibrium	Relationship
A3− + H+ ⇌ AH2+	Log β1= pka3
A3− + 2H+ ⇌ AH2+	Log β2 =pka2 + pka3
A3− + 3H+⇌ AH3	Log β3 = pka1 + pka2 + pka3

Cumulative association constants are used in general-purpose computer programs such as the one used to obtain the speciation diagram above.