Editing Buffer solution

{{Short description|Aqueous solution of a weak acid and its conjugate base}}
{{Acids and bases}}

A '''buffer solution''' is a solution where the pH does not change significantly on dilution or if an acid or base is added at constant temperature.<ref name="openstax">{{cite book|title= Anatomy and Physiology|author=J. Gordon Betts|publisher=OpenStax|chapter=Inorganic compounds essential to human functioning |date=25 April 2013 |url=https://openstax.org/books/anatomy-and-physiology/pages/2-4-inorganic-compounds-essential-to-human-functioning|access-date=14 May 2023|isbn=978-1-947172-04-3}}</ref> Its pH changes very little when a small amount of [[strong acid]] or [[Base (chemistry)#Strong bases|base]] is added to it. Buffer solutions are used as a means of keeping pH at a nearly constant value in a wide variety of chemical applications. In nature, there are many [[living systems]] that use buffering for pH regulation. For example, the [[bicarbonate buffering system]] is used to regulate the [[pH]] of [[blood]], and bicarbonate also acts as a [[Ocean acidification|buffer in the ocean]].

==Principles of buffering==
[[File:Buffer titration graph.svg|thumb|250px|right|Figure 1. Simulated [[titration]] of an acidified solution of a weak acid ({{math|1=p''K''<sub>a</sub> = 4.7}}) with alkali]]
Buffer solutions resist  pH change because of a [[chemical equilibrium]] between the weak acid HA and its conjugate base A<sup>−</sup>:
{{block indent|em=1.5|text=HA {{eqm}} H<sup>+</sup> + A<sup>−</sup>}}
When some strong acid is added to an equilibrium mixture of the weak acid and its conjugate base, hydrogen ions (H<sup>+</sup>) are added, and the equilibrium is shifted to the left, in accordance with [[Le Chatelier's principle]]. Because of this, the hydrogen ion concentration increases by less than the amount expected for the quantity of strong acid added.
Similarly, if strong alkali is added to the mixture, the hydrogen ion concentration decreases by less than the amount expected for the quantity of alkali added. In Figure 1, the effect is illustrated by the simulated titration of a weak acid with [[Acid dissociation constant|p''K''<sub>a</sub>]]&nbsp;=&nbsp;4.7. The relative concentration of undissociated acid is shown in blue, and of its conjugate base in red. The pH changes relatively slowly in the buffer region, pH&nbsp;=&nbsp;p''K''<sub>a</sub>&nbsp;±&nbsp;1, centered at pH&nbsp;=&nbsp;4.7, where [HA]&nbsp;=&nbsp;[A<sup>−</sup>]. The hydrogen ion concentration decreases by less than the amount expected because most of the added hydroxide ion is consumed in the reaction
{{block indent|em=1.5|text=OH<sup>−</sup> + HA → H<sub>2</sub>O + A<sup>−</sup>}}
and only a little is consumed in the neutralization reaction (which is the reaction that results in an increase in pH)
{{block indent|em=1.5|text=OH<sup>−</sup> + H<sup>+</sup> → H<sub>2</sub>O.}}

Once the acid is more than 95% [[deprotonation|deprotonated]], the pH rises rapidly because most of the added alkali is consumed in the neutralization reaction.

===Buffer capacity===
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:<ref name=Skoog/><ref name=Urbansky/>
<math display="block">\beta = \frac{dC_b}{d(\mathrm{pH})},</math>   
where <math>dC_b</math> is an infinitesimal amount of added base, or
<math display="block">\beta = -\frac{dC_a}{d(\mathrm{pH})},</math> 
where <math>dC_a</math> is an infinitesimal amount of added acid. pH is defined as −log<sub>10</sub>[H<sup>+</sup>], and ''d''(pH) is an infinitesimal change in pH. 

With either definition the buffer capacity for a weak acid HA with dissociation constant ''K''<sub>a</sub> can be expressed as<ref>{{cite book |last1=Butler |first1=J. N. |title=Ionic Equilibrium: Solubility and pH calculations |date=1998 |publisher=Wiley |pages=133–136 |isbn= 978-0-471-58526-8}}</ref><ref name=Hulanicki/><ref name=Urbansky>{{cite journal |last1=Urbansky |first1=Edward T. |last2=Schock |first2=Michael R.|title=Understanding, Deriving and Computing Buffer Capacity |journal=Journal of Chemical Education |date=2000 |volume=77 |issue=12 |pages=1640–1644 |doi=10.1021/ed077p1640 |bibcode=2000JChEd..77.1640U }}</ref>
<math chem display="block">\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),</math>
where [H<sup>+</sup>] is the concentration of hydrogen ions, and <math>T_\text{HA}</math> is the total concentration of added acid. ''K''<sub>w</sub> is the equilibrium constant for [[self-ionization of water]], equal to 1.0{{e|−14}}. Note that in solution H<sup>+</sup> exists as the [[hydronium]] ion H<sub>3</sub>O<sup>+</sup>, and further [[aquation]] of the hydronium ion has negligible effect on the dissociation equilibrium, except at very high acid concentration. 

[[File:Buffer Capacity 2.png|thumb|250px|Figure 2. Buffer capacity ''β'' for a 0.1&nbsp;M solution of a weak acid with a p''K''<sub>a</sub>&nbsp;=&nbsp;7]]
This equation shows that there are three regions of raised buffer capacity (see figure 2).
* In the central region of the curve (colored green on the plot), the second term is dominant, and <math chem display="block">\beta \approx 2.303 \frac{T_\ce{HA} K_a[\ce{H+}]}{(K_a + [\ce{H+}])^2}.</math> Buffer capacity rises to a local maximum at pH&nbsp;=&nbsp;''pK''<sub>a</sub>. The height of this peak depends on the value of pK<sub>a</sub>. Buffer capacity is negligible when the concentration [HA] of buffering agent is very small and increases with increasing concentration of the buffering agent.<ref name=Urbansky/> Some authors show only this region in graphs of buffer capacity.<ref name=Skoog/>{{pb}} Buffer capacity falls to 33% of the maximum value at pH = p''K''<sub>a</sub> ± 1, to 10% at pH = p''K''<sub>a</sub> ± 1.5 and to 1% at pH = p''K''<sub>a</sub> ± 2. For this reason the most useful range is approximately p''K''<sub>a</sub>&nbsp;±&nbsp;1. When choosing a buffer for use at a specific pH, it should have a p''K''<sub>a</sub> value as close as possible to that pH.<ref name=Skoog>{{cite book |last1=Skoog |first1=Douglas A. |last2=West |first2=Donald M. |last3=Holler |first3=F. James |last4=Crouch |first4=Stanley R. |title=Fundamentals of Analytical Chemistry |date=2014 |publisher=Brooks/Cole |isbn=978-0-495-55828-6 |pages=226 |edition=9th}}</ref> 
* 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: <math display="block">\beta \approx 10^{-\mathrm{pH}}.</math> 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: <math display="block">\beta \approx 10^{\mathrm{pH} - \mathrm{p}K_\text{w}}.</math> 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.

==Applications of buffers==
The pH of a solution containing a buffering agent can only vary within a narrow range, regardless of what else may be present in the solution. In biological systems this is an essential condition for [[enzyme]]s to function correctly. For example, in [[Blood|human blood]] a mixture of [[carbonic acid]] (H{{su|b=2}}CO{{su|b=3}}) and [[bicarbonate]] (HCO{{su|b=3|p=−}}) is present in the [[blood plasma|plasma]] fraction;  this constitutes the major mechanism for maintaining the pH of blood between 7.35 and 7.45. Outside this narrow range (7.40 ± 0.05 pH unit), [[acidosis]] and [[alkalosis]] metabolic conditions rapidly develop, ultimately leading to death if the correct buffering capacity is not rapidly restored. 

If the pH value of a solution rises or falls too much, the effectiveness of an enzyme decreases in a process, known as [[denaturation (biochemistry)|denaturation]], which is usually irreversible.<ref name="Scorpio 2000">{{cite book |title=Fundamentals of Acids, Bases, Buffers & Their Application to Biochemical Systems |last=Scorpio |first=R. |year=2000 |publisher=Kendall/Hunt Publishing Company |isbn=978-0-7872-7374-3}}</ref> The majority of biological samples that are used in research are kept in a buffer solution, often [[phosphate buffered saline]] (PBS) at pH&nbsp;7.4.

In industry, buffering agents are used in [[fermentation (biochemistry)|fermentation]] processes and in setting the correct conditions for dyes used in colouring fabrics. They are also used in chemical analysis<ref name=Hulanicki>{{cite book |last= Hulanicki |first= A. |title= Reactions of acids and bases in analytical chemistry |publisher= Horwood |year= 1987 |isbn=978-0-85312-330-9 |translator-first= Mary R.|translator-last= Masson}}</ref> and calibration of [[pH&nbsp;meter]]s.

===Simple buffering agents===
:{| class="wikitable"
! Buffering agent !! p''K''<sub>a</sub> !! Useful pH range
|-
| [[Citric acid]] || 3.13, 4.76, 6.40 || 2.1–7.4
|-
| [[Acetic acid]] || 4.8 || 3.8–5.8
|-
| [[potassium dihydrogenphosphate|KH<sub>2</sub>PO<sub>4</sub>]] || 7.2 || 6.2–8.2
|-
| [[N-Cyclohexyl-2-aminoethanesulfonic acid|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.

==="Universal" buffer mixtures===
By combining substances with p''K''<sub>a</sub> 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 p''K''<sub>a</sub> 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&nbsp;3 to 8.<ref>{{cite journal |last=McIlvaine |first=T. C. |year=1921 |title=A buffer solution for colorimetric comparaison |journal=J. Biol. Chem. |volume=49 |pages=183–186 |url=http://www.jbc.org/content/49/1/183.full.pdf |issue=1 |doi=10.1016/S0021-9258(18)86000-8 |url-status=live |archive-url=https://web.archive.org/web/20150226111238/http://www.jbc.org/content/49/1/183.full.pdf |archive-date=2015-02-26|doi-access=free }}</ref>
:{| class="wikitable"
! 0.2&nbsp;M [[Disodium phosphate|Na<sub>2</sub>HPO<sub>4</sub>]] (mL)
! 0.1&nbsp;M [[citric acid]] (mL)
! pH
|-
| 20.55
| 79.45
| style="background:#ff0000; color:white" | 3.0
|-
| 38.55
| 61.45
| style="background:#ff7777; color:white" |4.0
|-
| 51.50
| 48.50
| style="background:#ff7700;" | 5.0
|-
| 63.15
| 36.85
| style="background:#ffff00;" |6.0
|-
| 82.35
| 17.65
| style="background:#007777; color:white" | 7.0
|-
| 97.25
| 2.75
|style="background:#0077ff; color:white" | 8.0
|}

A mixture containing [[citric acid]], [[monopotassium phosphate]], [[boric acid]], and [[Barbital|diethyl barbituric acid]] can be made to cover the pH range 2.6 to 12.<ref>{{cite book |title=Vogel's textbook of quantitative chemical analysis |last1=Mendham |first1=J. |last2=Denny|first2= R. C. |last3=Barnes |first3=J. D. |last4=Thomas|first4= M. |edition=5th|year=2000 |publisher=Pearson Education |location=Harlow |isbn=978-0-582-22628-9|chapter= Appendix 5}}</ref>

Other universal buffers are the Carmody buffer<ref name=carmody>{{cite journal |last=Carmody |first=Walter R. |title=Easily prepared wide range buffer series |journal=J. Chem. Educ. |year=1961 |volume=38 |issue=11 |pages=559–560 |doi=10.1021/ed038p559 |bibcode = 1961JChEd..38..559C }}</ref> and the [[Britton–Robinson buffer]], developed in 1931.

===Common buffer compounds used in biology===
For effective range see [[#buffer capacity|Buffer capacity]], above. Also see [[Good's buffers]] for the historic design principles and favourable properties of these buffer substances in biochemical applications. 
{| class="wikitable"
|-
! Common name (chemical name)
! Structure
! [[Acid dissociation constant|p''K''<sub>a</sub>]], <br/>25&nbsp;°C
! {{abbr|Temp.|Temperature}} effect, <br/>{{sfrac|dpH|d''T''}} (K<sup>−1</sup>)<ref>{{Cite web |title=Buffer Reference Center |url=http://www.sigmaaldrich.com/life-science/core-bioreagents/biological-buffers/learning-center/buffer-reference-center.html |publisher=Sigma-Aldrich |access-date=2009-04-17 |url-status=live |archive-url=https://web.archive.org/web/20090417003507/http://www.sigmaaldrich.com/life-science/core-bioreagents/biological-buffers/learning-center/buffer-reference-center.html |archive-date=2009-04-17 }}</ref>
! [[Molecular weight|Mol. <br/>weight]] 
|-
| [[TAPS (buffer)|TAPS]], <br/>([tris(hydroxymethyl)methylamino]propanesulfonic acid)
| [[file:TAPS.svg|200px]] || 8.43 || −0.018 || 243.3
|-
| [[Bicine]], <br/>(2-(bis(2-hydroxyethyl)amino)acetic acid)
| [[file:Bicine.png|150px]] || 8.35 || −0.018 || 163.2
|-
| [[Tris]], <br/>(tris(hydroxymethyl)aminomethane, or <br/>2-amino-2-(hydroxymethyl)propane-1,3-diol)
| [[file:tris.png|100px]] || 8.07{{efn|text=Tris is a base, the p''K''<sub>a</sub> = 8.07 refers to its conjugate acid.}} || −0.028 || 121.14
|-
| [[Tricine]], <br/>(N-[tris(hydroxymethyl)methyl]glycine)
||[[file:Tricine.svg|150px]] || 8.05 || −0.021 || 179.2
|-
| [[TAPSO (buffer)|TAPSO]], <br/>(3-[N-tris(hydroxymethyl)methylamino]-2-hydroxypropanesulfonic acid)
||[[file:TAPSO.svg|200px]] || 7.635 ||  || 259.3
|-
| [[HEPES]], <br/>(4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid)
| [[file: HEPES.png|200px]] || 7.48 || −0.014 || 238.3
|-
| [[TES (buffer)|TES]], <br/><nowiki>(2-[[1,3-dihydroxy-2-(hydroxymethyl)propan-2-yl]amino]ethanesulfonic acid)</nowiki>
| [[file:TES free acid.svg|200px]] || 7.40 || −0.020 || 229.20
|-
| [[MOPS]], <br/>(3-(N-morpholino)propanesulfonic acid)
| [[file:Mops is.svg|150px]] || 7.20 || −0.015 || 209.3
|-
| [[PIPES]], <br/>(piperazine-N,N′-bis(2-ethanesulfonic acid))
| [[file:PIPES.svg|200px]] || 6.76 || −0.008 || 302.4
|-
| [[Cacodylate]], <br/>(dimethylarsenic acid)
| [[file:Cacodylic acid.svg|100px]] || 6.27 ||  || 138.0
|-
| [[MES (buffer)|MES]], <br/>(2-(N-morpholino)ethanesulfonic acid)
| [[file:MES.svg|150px]] || 6.15 || −0.011 || 195.2
|}
{{noteslist}}

== 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.

==See also==
*[[Henderson–Hasselbalch equation]]
*[[Good's buffers]]
*[[Common-ion effect]]
*[[Metal ion buffer]]
*[[Mineral redox buffer]]

==References==
{{reflist}}

==External links==
{{Cite web|title=Biological buffers |url=http://www.reachdevices.com/Protein/BiologicalBuffers.html |publisher=REACH Devices}}

{{Chemical equilibria}}

[[Category:Acid–base chemistry]]
[[Category:Acid–base physiology]]
[[Category:Equilibrium chemistry]]
[[Category:Buffer solutions| ]]