View source for Chemistry:PH

{{Short description|Measure of the level of acidity or basicity of an aqueous solution}}
{{Lowercase title}}
[[File:PH_scale_3.jpg|upright=1|thumb|right|Test tubes containing solutions of pH 1–10 colored with [[Chemistry:Universal indicator|an indicator]]]]
{{Acids and bases}}In [[Chemistry:Chemistry|chemistry]], '''pH''' ({{IPAc-en|p|iː|ˈ|eɪ|tʃ}} {{respelling|pee|AYCH}}), also referred to as ''acidity'' or ''basicity'', historically denotes "[[Physics:Electric potential|potential]] of [[Chemistry:Hydrogen|hydrogen]]" (or "power of hydrogen").<ref>{{cite journal |last1=Jensen |first1=William B. |date=2004 |title=The Symbol for pH |url=http://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/102.%20pH.pdf |url-status=live |journal=Journal of Chemical Education |volume=81 |issue=1 |pages=21 |bibcode=2004JChEd..81...21J |doi=10.1021/ed081p21 |archive-url=https://web.archive.org/web/20191214110759/http://www.che.uc.edu/jensen/w.%20b.%20jensen/reprints/102.%20ph.pdf |archive-date=14 December 2019 |access-date=15 July 2020}}</ref> It is a [[Logarithmic scale|logarithmic scale]] used to specify the [[Chemistry:Acid|acidity]] or [[Chemistry:Base|basicity]] of [[Chemistry:Aqueous solution|aqueous solution]]s. Acidic solutions (solutions with higher concentrations of hydrogen ([[Physics:Hydrogen ion#Cation (positively charged)|{{chem2|H+}}]]) [[Physics:Ion|ions]]) are measured to have lower pH values than basic or [[Chemistry:Alkaline|alkaline]] solutions.

The pH scale is logarithmic and inversely indicates the [[Chemistry:Thermodynamic activity|activity]] of [[Chemistry:Hydronium|hydrogen ions]] in the solution

: <math chem="">\ce{pH} = - \log(a_\ce{H+}) \thickapprox -\log([\ce{H+}])</math>

where [H<sup>+</sup>] is the [[Chemistry:Equilibrium chemistry|equilibrium]] [[Physics:Molar concentration|molar concentration]] (mol/L) of H<sup>+</sup> in the solution. At 25&nbsp;[[Celsius|°C]] (77[[Fahrenheit|°F]]), solutions with a pH less than 7 are acidic, and solutions with a pH greater than 7 are basic. Solutions with a pH of 7 at 25&nbsp;[[Celsius|°C]] are neutral (i.e. have the same concentration of H<sup>+</sup> ions as OH<sup>−</sup> ions, i.e. the same as pure water). The neutral value of the pH depends on the temperature and is lower than 7 if the temperature increases above 25&nbsp;°C. The pH range is commonly given as zero to 14, but a pH value can be less than 0 for very concentrated [[Chemistry:Acid strength|strong acids]] or greater than 14 for very concentrated [[Chemistry:Base#Strong bases|strong bases]].<ref>{{cite journal |last1=Lim |first1=Kieran F. |year=2006 |title=Negative pH Does Exist |journal=Journal of Chemical Education |volume=83 |issue=10 |pages=1465 |bibcode=2006JChEd..83.1465L |doi=10.1021/ed083p1465 |doi-access=}}</ref>

The pH scale is traceable to a set of standard solutions whose pH is established by international agreement.<ref name="covington2">{{cite journal |last1=Covington |first1=A. K. |last2=Bates |first2=R. G. |last3=Durst |first3=R. A. |year=1985 |title=Definitions of pH scales, standard reference values, measurement of pH, and related terminology |url=http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf |url-status=live |journal=Pure Appl. Chem. |volume=57 |issue=3 |pages=531–542 |doi=10.1351/pac198557030531 |s2cid=14182410 |archive-url=https://web.archive.org/web/20070924235637/http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf |archive-date=24 September 2007}}</ref> Primary pH standard values are determined using a [[Chemistry:Galvanic cell|concentration cell with transference]] by measuring the potential difference between a hydrogen electrode and a [[Chemistry:Standard electrode potential (data page)|standard electrode]] such as the [[Physics:Silver chloride electrode|silver chloride electrode]]. The pH of aqueous solutions can be measured with a [[Physics:Glass electrode|glass electrode]] and a [[Chemistry:PH meter|pH meter]] or a color-changing [[Chemistry:PH indicator|indicator]]. Measurements of pH are important in [[Chemistry:Chemistry|chemistry]], [[Earth:Agronomy|agronomy]], medicine, water treatment, and many other applications.

==History==
In 1909, the Danish chemist Søren Peter Lauritz Sørensen introduced the concept of pH at the Carlsberg Laboratory,<ref name="Sørensen2">{{cite journal |last1=Sørensen |first1=S. P. L. |year=1909 |title=Über die Messung und die Bedeutung der Wasserstoffionenkonzentration bei enzymatischen Prozessen |url=https://core.ac.uk/download/pdf/14517358.pdf |url-status=live |journal=Biochem. Z. |volume=21 |pages=131–304 |archive-url=https://web.archive.org/web/20210415205740/https://core.ac.uk/download/pdf/14517358.pdf |archive-date=15 April 2021 |access-date=22 March 2021 |quote=Original German: Für die Zahl p schlage ich den Namen Wasserstoffionenexponent und die Schreibweise p<sub>H</sub>• vor. Unter dem Wasserstoffionexponenten (p<sub>H</sub>•) einer Lösungwird dann der Briggsche Logarithmus des reziproken Wertes des auf Wasserstoffionenbezagenen Normalitäts faktors de Lösungverstanden.}} Two other publications appeared in 1909, one in French and one in Danish.</ref> originally using the notation "p<sub>H•</sub>", with H• as a subscript to the lowercase p. The concept was later revised in 1924 to the modern pH to accommodate definitions and measurements in terms of electrochemical cells.<blockquote>For the sign ''p'', I propose the name 'hydrogen ion exponent' and the symbol p<sub>H•</sub>. Then, for the hydrogen ion exponent (p<sub>H•</sub>) of a solution, the negative value of the [[Common logarithm|Briggsian logarithm]] of the related hydrogen ion [[Chemistry:Equivalent concentration|normality factor]] is to be understood.<ref name="Sørensen2" /></blockquote>Sørensen did not explain why he used the letter p, and the exact meaning of the letter is still disputed.<ref>{{Cite journal |last=Francl |first=Michelle |date=August 2010 |title=Urban legends of chemistry |url=https://www.nature.com/articles/nchem.750.epdf |url-status=live |journal=Nature Chemistry |volume=2 |issue=8 |pages=600–601 |bibcode=2010NatCh...2..600F |doi=10.1038/nchem.750 |issn=1755-4330 |pmid=20651711 |archive-url=https://web.archive.org/web/20200806053215/https://www.nature.com/articles/nchem.750.epdf |archive-date=6 August 2020 |access-date=21 July 2019}}</ref> Sørensen described a way of measuring pH using ''potential'' differences, and it represents the negative ''power'' of 10 in the concentration of hydrogen ions. The letter ''p'' could stand for the French ''puissance,'' German ''Potenz,'' or Danish ''potens'', all meaning "power", or it could mean "potential". All of these words start with the letter ''p'' in [[Social:French language|French]], [[Social:German language|German]], and [[Social:Danish language|Danish]], which were the languages in which Sørensen published: Carlsberg Laboratory was French-speaking; German was the dominant language of scientific publishing; Sørensen was Danish. He also used the letter ''q'' in much the same way elsewhere in the paper, and he might have arbitrarily labelled the test solution "p" and the reference solution "q"; these letters are often paired.<ref>{{cite journal |last1=Myers |first1=Rollie J. |year=2010 |title=One-Hundred Years of pH |journal=Journal of Chemical Education |volume=87 |issue=1 |pages=30–32 |bibcode=2010JChEd..87...30M |doi=10.1021/ed800002c}}</ref> Some literature sources suggest that "pH" stands for the [[Social:Latin language|Latin term]] ''pondus hydrogenii'' (quantity of hydrogen) or ''potentia hydrogenii'' (power of hydrogen), although this is not supported by Sørensen's writings.<ref name="Otterson2">{{cite journal |last1=Otterson |first1=David W. |date=2015 |title=Tech Talk: (11) pH Measurement and Control Basics. |url=https://journals.sagepub.com/doi/pdf/10.1177/0020294015600474 |journal=Measurement and Control |volume=48 |issue=10 |pages=309–312 |doi=10.1177/0020294015600474 |s2cid=110716297 |access-date=16 June 2022|doi-access=free }}</ref><ref name="Lian2">{{cite journal |last1=Lian |first1=Ying |last2=Zhang |first2=Wei |last3=Ding |first3=Longjiang |last4=Zhang |first4=Xiaoai |last5=Zhang |first5=Yinglu |last6=Wang |first6=Xu-dong |date=2019 |title=Nanomaterials for Intracellular pH Sensing and Imaging. |url=https://www.sciencedirect.com/science/article/pii/B9780128144978000084 |journal=Novel Nanomaterials for Biomedical, Environmental and Energy Applications. |series=Micro and Nano Technologies |pages=241–273 |doi=10.1016/B978-0-12-814497-8.00008-4 |isbn=9780128144978 |s2cid=104410918 |access-date=16 June 2022}}</ref><ref name="Bradley2">{{cite news |last1=Bradley |first1=David |date=21 February 2018 |title=When it comes to caustic wit and an acid tongue, mind your Ps and Qs. |publisher=Materials Today |url=https://www.materialstoday.com/materials-chemistry/comment/caustic-wit-acid-tongues-mind-your-ps-and-qs/ |access-date=16 June 2022}}</ref>

In modern [[Chemistry:Chemistry|chemistry]], the p stands for "the negative [[Common logarithm|decimal logarithm of]]", and is used in the term p''K''<sub>a</sub> for [[Chemistry:Acid dissociation constant|acid dissociation constants]],<ref name="Jens2">{{cite journal |author=Nørby, Jens |year=2000 |title=The origin and the meaning of the little p in pH |journal=Trends in Biochemical Sciences |volume=25 |issue=1 |pages=36–37 |doi=10.1016/S0968-0004(99)01517-0 |pmid=10637613}}</ref> so pH is "the negative [[Common logarithm|decimal logarithm of]] H<sup>+</sup> ion concentration", while pOH is "the negative decimal logarithm of OH- ion concentration". 

Bacteriologist [[Biography:Alice Catherine Evans|Alice Catherine Evans]], who influenced dairying and food safety, credited William Mansfield Clark and colleagues, including herself, with developing pH measuring methods in the 1910s, which had a wide influence on laboratory and industrial use thereafter. In her memoir, she does not mention how much, or how little, Clark and colleagues knew about Sørensen's work a few years prior.<ref name="Evans-Memoirs2">{{cite web |last1=Evans |first1=Alice C. |year=1963 |title=Memoirs |url=https://history.nih.gov/archives/downloads/aliceevans.pdf |url-status=dead |archive-url=https://web.archive.org/web/20171215000804/https://history.nih.gov/archives/downloads/aliceevans.pdf |archive-date=15 December 2017 |access-date=2018-03-27 |website=NIH Office of History |publisher=National Institutes of Health Office of History}}</ref> She said:<blockquote>In these studies [of bacterial metabolism] Dr. Clark's attention was directed to the effect of acid on the growth of bacteria. He found that it is the intensity of the acid in terms of hydrogen-ion concentration that affects their growth. But existing methods of measuring acidity determined the quantity, not the intensity, of the acid. Next, with his collaborators, Dr. Clark developed accurate methods for measuring hydrogen-ion concentration. These methods replaced the inaccurate titration method of determining the acid content in use in biologic laboratories throughout the world. Also they were found to be applicable in many industrial and other processes in which they came into wide usage.<ref name="Evans-Memoirs2" /></blockquote>The first [[Engineering:Electronics|electronic]] method for measuring pH was invented by [[Biography:Arnold Orville Beckman|Arnold Orville Beckman]], a professor at the [[Organization:California Institute of Technology|California Institute of Technology]] in 1934.<ref>{{cite web |title=Origins: Birth of the pH Meter |url=https://eands.caltech.edu/origins-birth-of-the-ph-meter/ |url-status=dead |archive-url=https://web.archive.org/web/20181106180207/https://eands.caltech.edu/origins-birth-of-the-ph-meter/ |archive-date=6 November 2018 |access-date=11 March 2018 |website=Caltech Engineering & Science Magazine}}</ref> It was in response to a request from the local citrus grower Sunkist, which wanted a better method for quickly testing the pH of lemons they were picking from their nearby orchards.<ref>{{cite web |last1=Tetrault |first1=Sharon |date=June 2002 |title=The Beckmans |url=https://books.google.com/books?id=nf0DAAAAMBAJ&q=ph+caltech+beckman+sunkist&pg=PA96 |url-status=live |archive-url=https://web.archive.org/web/20210415222325/https://books.google.com/books?id=nf0DAAAAMBAJ&q=ph+caltech+beckman+sunkist&pg=PA96 |archive-date=15 April 2021 |access-date=11 March 2018 |website=Orange Coast |publisher=Orange Coast Magazine}}</ref>

== Definition ==
===pH===
The pH of a solution is defined as the decimal [[Logarithm|logarithm]] of the reciprocal of the [[Physics:Hydrogen ion|hydrogen ion]] activity, ''a''<sub>H</sub>+.<ref name="covington22">{{cite journal |last1=Covington |first1=A. K. |last2=Bates |first2=R. G. |last3=Durst |first3=R. A. |year=1985 |title=Definitions of pH scales, standard reference values, measurement of pH, and related terminology |url=http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf |url-status=live |journal=Pure Appl. Chem. |volume=57 |issue=3 |pages=531–542 |doi=10.1351/pac198557030531 |s2cid=14182410 |archive-url=https://web.archive.org/web/20070924235637/http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf |archive-date=24 September 2007}}</ref> Mathematically, pH is expressed as:

: <math chem="">\ce{pH} = - \log_{10}(a_\ce{H+}) = \log_{10}\left(\frac 1 {a_\ce{H+}} \right) </math>

For example, for a solution with a hydrogen ion activity of 5×10<sup>−6</sup> (i.e., the concentration of hydrogen ions in [[Mole (unit)|moles]] per litre), the pH of the solution can be calculated as follows:

: <math chem="">\ce{pH} = - \log_{10}(5\times10^{-6}) = 5.3 </math>

The concept of pH was developed because [[Physics:Ion-selective electrode|ion-selective electrodes]], which are used to measure pH, respond to activity. The electrode potential, ''E'', follows the [[Physics:Nernst equation|Nernst equation]] for the hydrogen ion, which can be expressed as:

: <math chem=""> E = E^0 + \frac{RT}{F} \ln(a_\ce{H+})=E^0 - \frac{2.303 RT}{F} \ce{pH}</math>

where ''E'' is a measured potential, ''E''<sup>0</sup> is the standard electrode potential, ''R'' is the [[Physics:Gas constant|gas constant]], ''T'' is the temperature in [[Kelvin]], ''F'' is the [[Physics:Faraday constant|Faraday constant]]. For {{chem2|H+}}, the number of electrons transferred is one. The electrode potential is proportional to pH when pH is defined in terms of activity.

The precise measurement of pH is presented in International Standard [[Physics:ISO 31-8|ISO 31-8]] as follows:<ref>Quantities and units – Part 8: Physical chemistry and molecular physics, Annex C (normative): pH. [[Organization:International Organization for Standardization|International Organization for Standardization]], 1992.</ref> A [[Chemistry:Galvanic cell|galvanic cell]] is set up to measure the [[Physics:Electromotive force|electromotive force]] (e.m.f.) between a reference electrode and an electrode sensitive to the hydrogen ion activity when they are both immersed in the same aqueous solution. The reference electrode may be a [[Physics:Silver chloride electrode|silver chloride electrode]] or a [[Physics:Saturated calomel electrode|calomel electrode]], and the hydrogen-ion selective electrode is a [[Physics:Standard hydrogen electrode|standard hydrogen electrode]].

: {{math|Reference electrode {{!}} concentrated solution of KCl {{!}}{{!}} test solution {{!}} H<sub>2</sub> {{!}} Pt}}

Firstly, the cell is filled with a solution of known hydrogen ion activity and the electromotive force, ''E''<sub>S</sub>, is measured. Then the electromotive force, ''E''<sub>X</sub>, of the same cell containing the solution of unknown pH is measured.

: <math chem="">\ce{pH(X)} = \ce{pH(S)}+\frac{E_\ce{S} - E_\ce{X} }{z}</math>

The difference between the two measured electromotive force values is proportional to pH. This method of calibration avoids the need to know the [[Chemistry:Standard electrode potential|standard electrode potential]]. The proportionality constant, 1/''z'', is ideally equal to <math>\frac{1}{2.303RT/F}\ </math>, the "Nernstian slope".

In practice, a [[Physics:Glass electrode|glass electrode]] is used instead of the cumbersome hydrogen electrode. A combined glass electrode has an in-built reference electrode. It is calibrated against [[Chemistry:Buffer solution|Buffer solution]]s of known hydrogen ion ({{chem2|H+}}) activity proposed by the International Union of Pure and Applied Chemistry (IUPAC).<ref name="covington3">{{cite journal |last1=Covington |first1=A. K. |last2=Bates |first2=R. G. |last3=Durst |first3=R. A. |year=1985 |title=Definitions of pH scales, standard reference values, measurement of pH, and related terminology |url=http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf |url-status=live |journal=Pure Appl. Chem. |volume=57 |issue=3 |pages=531–542 |doi=10.1351/pac198557030531 |s2cid=14182410 |archive-url=https://web.archive.org/web/20070924235637/http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf |archive-date=24 September 2007}}</ref> Two or more buffer solutions are used in order to accommodate the fact that the "slope" may differ slightly from ideal. To calibrate the electrode, it is first immersed in a standard solution, and the reading on a [[Chemistry:PH meter|pH meter]] is adjusted to be equal to the standard buffer's value. The reading from a second standard buffer solution is then adjusted using the "slope" control to be equal to the pH for that solution. Further details, are given in the IUPAC recommendations.<ref name="covington22" /> When more than two buffer solutions are used the electrode is calibrated by fitting observed pH values to a straight line with respect to standard buffer values. Commercial standard buffer solutions usually come with information on the value at 25&nbsp;°C and a correction factor to be applied for other temperatures.

The pH scale is logarithmic and therefore pH is a [[Physics:Dimensionless quantity|dimensionless quantity]].<ref>{{Cite book |url=https://goldbook.iupac.org/ |title=The IUPAC Compendium of Chemical Terminology: The Gold Book |date=2019 |publisher=International Union of Pure and Applied Chemistry (IUPAC) |editor-last=Gold |editor-first=Victor |edition=4 |location=Research Triangle Park, NC |language=en |doi=10.1351/goldbook.p04525}}</ref>

=== p[H] ===
This was the original definition of Sørensen in 1909,<ref name="Sor2">{{cite web |title=Carlsberg Group Company History Page |url=http://www.carlsberggroup.com/Company/heritage/Research/Pages/pHValue.aspx |url-status=live |archive-url=https://web.archive.org/web/20140118043012/http://www.carlsberggroup.com/Company/heritage/Research/Pages/pHValue.aspx |archive-date=18 January 2014 |access-date=7 May 2013 |publisher=Carlsberggroup.com}}</ref> which was superseded in favor of pH in 1924. [H] is the concentration of hydrogen ions, denoted [{{chem2|H(+)}}] in modern chemistry. More correctly, the [[Chemistry:Thermodynamic activity|thermodynamic activity]] of {{chem2|H(+)}} in dilute solution should be replaced by [{{chem2|H(+)}}]/c<sub>0</sub>, where the standard state concentration c<sub>0</sub> = 1&nbsp;mol/L. This ratio is a pure number whose logarithm can be defined.

It is possible to measure the concentration of hydrogen ions directly using an electrode calibrated in terms of hydrogen ion concentrations. One common method is to [[Chemistry:Titration|titrate]] a solution of known concentration of a strong acid with a solution of known concentration of strong base in the presence of a relatively high concentration of background electrolyte. By knowing the concentrations of the acid and base, the concentration of hydrogen ions can be calculated and the measured potential can be correlated with concentrations. The calibration is usually carried out using a [[Chemistry:Gran plot#Electrode calibration|Gran plot]].<ref>{{cite journal |last=Rossotti |first=F.J.C. |author2=Rossotti, H. |year=1965 |title=Potentiometric titrations solution containing the background electrolyte. |journal=J. Chem. Educ. |volume=42 |doi=10.1021/ed042p375}}</ref> This procedure makes the activity of hydrogen ions equal to the numerical value of concentration.

The glass electrode (and other [[Ion selective electrode])s should be calibrated in a medium similar to the one being investigated. For instance, if one wishes to measure the pH of a seawater sample, the electrode should be calibrated in a solution resembling seawater in its chemical composition.

The difference between p[H] and pH is quite small, and it has been stated that pH = p[H] + 0.04.<ref>{{VogelQuantitative}}, Section 13.23, "Determination of pH"</ref> However, it is common practice to use the term "pH" for both types of measurement.

=== pOH ===
[[File:PHscalenolang.svg|thumb|Relation between pH and pOH. Red represents the acidic region. Blue represents the basic region.]]
pOH is sometimes used as a measure of the concentration of hydroxide ions, {{chem2|OH−}}. pOH values are derived from pH measurements. The concentration of hydroxide ions in water is related to the concentration of hydrogen ions by

: <math chem="">[\ce{OH^-}] = \frac{K_\ce{W}}{[\ce{H^+}]}</math>

where ''K''<sub>W</sub> is the [[Physics:Self-ionization of water|self-ionization]] constant of water. Taking [[Logarithm]]s,

: <math chem="">\ce{pOH} = \ce{p}K_\ce{W} - \ce{pH}.</math>

So, at room temperature, pOH ≈ 14 − pH. However this relationship is not strictly valid in other circumstances, such as in measurements of soil alkalinity.

== Measurement ==

=== pH Indicators ===
{| class="wikitable floatright"
! colspan="3" |Average pH of common solutions
|-
!Substance
!pH range
!Type
|-
|[[Chemistry:Sulfuric acid|Battery acid]]
| style="background-color: #CC0000; text-align: center; color: #ffffff" |< 1
| rowspan="6" style="text-align: center" |[[Chemistry:Acid|Acid]]
|-
|[[Biology:Gastric acid|Gastric acid]]
| style="background-color: #EE0000; text-align: center; color: #ffffff" |1.0 – 1.5
|-
|Orange juice
| style="background-color: #FF6600; text-align: center" |3.3 – 4.2
|-
|[[Chemistry:Vinegar|Vinegar]]
| style="background-color: #FF9900; text-align: center" |4-5
|-
|Black coffee
| style="background-color: #ffff00 ; text-align: center" |5 – 5.03
|-
|Milk
| style="background-color: #99cc33; text-align: center" |6.5 – 6.8
|-
|Pure water at 25&nbsp;°C
| style="background-color: #339933; text-align: center; color: #ffffff" |7
| style="text-align: center" |Neutral
|-
|Sea water
| style="background-color: #19cdff; text-align: center; color: #000000" |7.5 – 8.4
| rowspan="4" style="text-align: center" |[[Chemistry:Base|Base]]
|-
|[[Chemistry:Ammonia|Ammonia]]
| style="background-color: #3333ff; text-align: center; color: #FFFFFF" |11.0 – 11.5
|-
|[[Chemistry:Bleach|Bleach]]
| style="background-color: #330099; text-align: center; color: #FFFFFF" |12.5
|-
|[[Chemistry:Lye|1 M NaOH]]
| style="background-color: #330066; text-align: center; color: #FFFFFF" |14
|}
{{Main|Chemistry:PH indicator}}
pH can be measured using indicators, which change color depending on the pH of the solution they are in. By comparing the color of a test solution to a standard color chart, the pH can be estimated to the nearest whole number. For more precise measurements, the color can be measured using a [[Chemistry:Colorimeter|colorimeter]] or [[Physics:Spectrophotometer|spectrophotometer]]. A [[Chemistry:Universal indicator|Universal indicator]] is a mixture of several indicators that can provide a continuous color change over a range of pH values, typically from about pH 2 to pH 10. Universal indicator paper is made from absorbent paper that has been impregnated with universal indicator.  An alternative method of measuring pH is using an electronic [[Chemistry:PH meter|pH meter]], which directly measures the voltage difference between a pH-sensitive electrode and a reference electrode.

===Non-aqueous solutions===
pH values can be measured in non-aqueous solutions, but they are based on a different scale from aqueous pH values, because the [[Physics:Standard state|standard states]] used for calculating hydrogen ion concentrations (activities) are different. The hydrogen ion activity, ''a<sub>H<sup>+</sup></sub>'', is defined<ref name="GoldBook2">{{GoldBookRef|title=activity (relative activity), ''a''|file=A00115}}</ref><ref name="GreenBook2">{{GreenBookRef2nd|pages=49–50}}</ref> as:

: <math chem="">a_\ce{H+} = \exp\left (\frac{\mu_\ce{H+} - \mu^{\ominus}_\ce{H+}}{RT}\right )</math>

where ''μ''<sub>H<sup>+</sup></sub> is the [[Chemistry:Chemical potential|chemical potential]] of the hydrogen ion, <math chem="">\mu^{\ominus}_\ce{H+}</math> is its chemical potential in the chosen standard state, ''R'' is the [[Physics:Gas constant|gas constant]] and ''T'' is the [[Physics:Thermodynamic temperature|thermodynamic temperature]]. Therefore, pH values on the different scales cannot be compared directly because of differences in the solvated proton ions, such as lyonium ions, which require an intersolvent scale which involves the transfer activity coefficient of [[Physics:Lyonium ion|hydronium/lyonium ion]].

pH is an example of an [[Chemistry:Acidity function|acidity function]], but there are others that can be defined. For example, the [[Chemistry:Hammett acidity function|Hammett acidity function]], ''H''<sub>0</sub>, has been developed in connection with [[Chemistry:Superacid|Superacid]]s.

===Unified absolute pH scale===
In 2010, a new approach to measuring pH was proposed, called the "unified absolute pH scale". This approach allows for a common reference standard to be used across different solutions, regardless of their pH range. The unified absolute pH scale is based on the absolute chemical potential of the proton, as defined by the [[Chemistry:Lewis acids and bases|Lewis acid–base]] theory. This scale is applicable to liquids, gases, and even solids.<ref name="Krossing2">{{Cite journal |last1=Himmel |first1=Daniel |last2=Goll |first2=Sascha K. |last3=Leito |first3=Ivo |last4=Krossing |first4=Ingo |date=2010-08-16 |title=A Unified pH Scale for All Phases |journal=Angewandte Chemie International Edition |volume=49 |issue=38 |pages=6885–6888 |doi=10.1002/anie.201000252 |issn=1433-7851 |pmid=20715223}}</ref> The advantages of the unified absolute pH scale include consistency, accuracy, and applicability to a wide range of sample types. It is precise and versatile because it serves as a common reference standard for pH measurements. However, implementation efforts, compatibility with existing data, complexity, and potential costs are some challenges.

=== Extremes of pH measurements ===
The measurement of pH can become difficult at extremely acidic or alkaline conditions, such as below pH 2.5 (ca. 0.003&nbsp;[[Mole (unit)|mol]]/dm<sup>3</sup> acid) or above pH 10.5 (above ca. 0.0003&nbsp;mol/dm<sup>3</sup> alkaline). This is due to the breakdown of the [[Physics:Nernst equation|Nernst equation]] in such conditions when using a glass electrode. There are several factors that contribute to this problem. Firstly, [[Chemistry:Liquid junction potential|liquid junction potential]]s may not be independent of pH.<ref name="Feldman2">{{cite journal |author=Feldman, Isaac |year=1956 |title=Use and Abuse of pH measurements |journal=Analytical Chemistry |volume=28 |issue=12 |pages=1859–1866 |doi=10.1021/ac60120a014}}</ref> Secondly, the high [[Chemistry:Ionic strength|ionic strength]] of concentrated solutions can affect the electrode potentials. At high pH the glass electrode may be affected by "alkaline error", because the electrode becomes sensitive to the concentration of cations such as {{chem2|Na+}} and {{chem2|K+}} in the solution.<ref>{{VogelQuantitative}}, Section 13.19 The glass electrode</ref> To overcome these problems, specially constructed electrodes are available.

Runoff from mines or mine tailings can produce some extremely low pH values.<ref>{{cite journal |last1=Nordstrom |first1=D. Kirk |last2=Alpers |first2=Charles N. |date=March 1999 |title=Negative pH, efflorescent mineralogy, and consequences for environmental restoration at the Iron Mountain Superfund site, California |url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1495&context=usgsstaffpub |url-status=live |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=96 |issue=7 |pages=3455–62 |bibcode=1999PNAS...96.3455N |doi=10.1073/pnas.96.7.3455 |pmc=34288 |pmid=10097057 |archive-url=https://web.archive.org/web/20170923012227/http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1495&context=usgsstaffpub |archive-date=23 September 2017 |access-date=4 November 2018 |doi-access=free}}</ref>

== Applications ==
Pure water has a pH of 7 at 25°C, meaning it is neutral. When an [[Chemistry:Acid|acid]] is dissolved in water, the pH will be less than 7, while a [[Chemistry:Base|base]], or [[Chemistry:Alkali|alkali]], will have a pH greater than 7. A strong acid, such as [[Chemistry:Hydrochloric acid|hydrochloric acid]], at concentration 1&nbsp;mol dm<sup>−3</sup> has a pH of 0, while a strong alkali like [[Chemistry:Sodium hydroxide|sodium hydroxide]], at the same concentration, has a pH of 14. Since pH is a logarithmic scale, a difference of one in pH is equivalent to a tenfold difference in hydrogen ion concentration.

Neutrality is not exactly 7 at 25°C, but 7 serves as a good approximation in most cases. Neutrality occurs when the concentration of hydrogen ions ([{{chem2|H+}}]) equals the concentration of hydroxide ions ([{{chem2|OH−}}]), or when their activities are equal. Since [[Physics:Self-ionization of water|self-ionization of water]] holds the product of these concentration [{{chem2|H+}}] × [{{chem2|OH−}}] = K<sub>w</sub>, it can be seen that at neutrality [{{chem2|H+}}] = [{{chem2|OH−}}] = {{radic|K<sub>w</sub>}}, or pH = pK<sub>w</sub>/2. pK<sub>w</sub> is approximately 14 but depends on ionic strength and temperature, and so the pH of neutrality does also. Pure water and a solution of [[Chemistry:Sodium chloride|NaCl]] in pure water are both neutral, since [[Physics:Self-ionization of water|dissociation of water]] produces equal numbers of both ions. However the pH of the neutral NaCl solution will be slightly different from that of neutral pure water because the hydrogen and hydroxide ions' activity is dependent on [[Chemistry:Ionic strength|ionic strength]], so K<sub>w</sub> varies with ionic strength.

When pure water is exposed to air, it becomes mildly acidic. This is because water absorbs [[Chemistry:Carbon dioxide|carbon dioxide]] from the air, which is then slowly converted into [[Chemistry:Bicarbonate|bicarbonate]] and hydrogen ions (essentially creating [[Chemistry:Carbonic acid|carbonic acid]]).

: {{chem|CO|2|+ H|2|O {{eqm}} HCO|3|-|+ H|+}}

=== pH in soil ===

[[File:Soil_pH_effect_on_nutrient_availability.svg|thumb|Nutritional elements availability within soil varies with pH. Light blue color represents the ideal range for most plants.]]
The United States Department of Agriculture [[Organization:Natural Resources Conservation Service|Natural Resources Conservation Service]], formerly Soil Conservation Service classifies [[Biology:Soil pH|soil pH]] ranges as follows:<ref>{{cite web |author=Soil Survey Division Staff |title=Soil survey manual.1993. Chapter 3, selected chemical properties. |url=http://soils.usda.gov/technical/manual/contents/chapter3.html |url-status=dead |archive-url=https://web.archive.org/web/20110514151830/http://soils.usda.gov/technical/manual/contents/chapter3.html |archive-date=14 May 2011 |access-date=2011-03-12 |publisher=Soil Conservation Service. U.S. Department of Agriculture Handbook 18}}</ref>
{| class="wikitable" style="align: center;"
! scope="col" |Denomination
! scope="col" |pH range
|-
|Ultra acidic
|< 3.5
|-
|Extremely acidic
|3.5–4.4
|-
|Very strongly acidic
|4.5–5.0
|-
|Strongly acidic
|5.1–5.5
|-
|Moderately acidic
|5.6–6.0
|-
|Slightly acidic
|6.1–6.5
|-
|Neutral
|6.6–7.3
|-
|Slightly alkaline
|7.4–7.8
|-
|Moderately alkaline
|7.9–8.4
|-
|Strongly alkaline
|8.5–9.0
|-
|Very strongly alkaline
|9.0–10.5
|-
|Hyper alkaline
|> 10.5
|}
In Europe, topsoil pH is influenced by soil parent material, erosional effects, climate and vegetation. A recent map<ref>{{Cite journal |last1=Ballabio |first1=Cristiano |last2=Lugato |first2=Emanuele |last3=Fernández-Ugalde |first3=Oihane |last4=Orgiazzi |first4=Alberto |last5=Jones |first5=Arwyn |last6=Borrelli |first6=Pasquale |last7=Montanarella |first7=Luca |last8=Panagos |first8=Panos |date=2019 |title=Mapping LUCAS topsoil chemical properties at European scale using Gaussian process regression |journal=Geoderma |language=en |volume=355 |pages=113912 |bibcode=2019Geode.355k3912B |doi=10.1016/j.geoderma.2019.113912 |pmc=6743211 |pmid=31798185 |doi-access=free}}</ref> of topsoil pH in Europe shows the alkaline soils in Mediterranean, Hungary, East Romania, North France. Scandinavian countries, Portugal, Poland and North Germany have more acid soils.

===pH in plants===
[[File:Lemon_-_whole_and_split.jpg|thumb|Lemon juice tastes sour because it contains 5% to 6% citric acid and has a pH of 2.2 (high acidity).]]
Plants contain pH-dependent pigments that can be used as [[Chemistry:PH indicator|pH indicators]], such as those found in [[Biology:Hibiscus|hibiscus]], red cabbage ([[Chemistry:Anthocyanin|anthocyanin]]), and grapes ([[Red wine|red wine]]). [[Biology:Citrus|Citrus]] fruits have acidic juice primarily due to the presence of [[Chemistry:Citric acid|citric acid]], while other [[Chemistry:Carboxylic acid|carboxylic acids]] can be found in various living systems. The [[Chemistry:Protonation|protonation]] state of [[Chemistry:Phosphate|phosphate]] derivatives, including [[Chemistry:Adenosine triphosphate|ATP]], is pH-dependent. [[Biology:Hemoglobin|Hemoglobin]], an oxygen-transport enzyme, is also affected by pH in a phenomenon known as the [[Medicine:Root effect|Root effect]].

=== pH in the ocean ===

The pH of [[Earth:Seawater|seawater]] plays an important role in the ocean's [[Earth:Carbon cycle#Ocean|carbon cycle]]. There is evidence of ongoing [[Biology:Ocean acidification|ocean acidification]] (meaning a drop in pH value): Between 1950 and 2020, the average pH of the ocean surface fell from approximately 8.15 to 8.05.<ref>{{Cite journal |last1=Terhaar |first1=Jens |last2=Frölicher |first2=Thomas L. |last3=Joos |first3=Fortunat |date=2023 |title=Ocean acidification in emission-driven temperature stabilization scenarios: the role of TCRE and non-{{CO2}} greenhouse gases |journal=Environmental Research Letters |language=en |volume=18 |issue=2 |pages=024033 |bibcode=2023ERL....18b4033T |doi=10.1088/1748-9326/acaf91 |issn=1748-9326 |s2cid=255431338|doi-access=free }}</ref> Carbon dioxide emissions from human activities are the primary cause of ocean acidification, with [[Earth:Carbon dioxide in Earth's atmosphere|atmospheric carbon dioxide (CO<sub>2</sub>) levels]] exceeding 410 ppm (in 2020). CO<sub>2</sub> from the [[Astronomy:Atmosphere|atmosphere]] is absorbed by the oceans. This produces [[Chemistry:Carbonic acid|carbonic acid]] (H<sub>2</sub>CO<sub>3</sub>) which dissociates into a bicarbonate ion ({{Chem|HCO|3|-|}}) and a [[Physics:Hydrogen ion|hydrogen ion]] (H<sup>+</sup>). The presence of free hydrogen ions (H<sup>+</sup>) lowers the pH of the ocean. 

==== Three pH scales in oceanography ====
The measurement of pH in seawater is complicated by the [[Chemistry:Chemical property|chemical properties]] of seawater, and three distinct pH scales exist in [[Chemistry:Chemical oceanography|chemical oceanography]].<ref name="zeebe2">Zeebe, R. E. and Wolf-Gladrow, D. (2001) ''CO<sub>2</sub> in seawater: equilibrium, kinetics, isotopes'', Elsevier Science B.V., Amsterdam, Netherlands {{ISBN|0-444-50946-1}}</ref> In practical terms, the three seawater pH scales differ in their pH values up to 0.10, differences that are much larger than the accuracy of pH measurements typically required, in particular, in relation to the ocean's [[Chemistry:Total inorganic carbon|carbonate system]].<ref name="zeebe2" /> Since it omits consideration of sulfate and fluoride ions, the ''free scale'' is significantly different from both the total and seawater scales. Because of the relative unimportance of the fluoride ion, the total and seawater scales differ only very slightly.

As part of its [[Philosophy:Operational definition|operational definition]] of the pH scale, the IUPAC defines a series of [[Chemistry:Buffer solution|Buffer solution]]s across a range of pH values (often denoted with National Bureau of Standards (NBS) or [[Organization:National Institute of Standards and Technology|National Institute of Standards and Technology]] (NIST) designation). These solutions have a relatively low [[Chemistry:Ionic strength|ionic strength]] (≈0.1) compared to that of seawater (≈0.7), and, as a consequence, are not recommended for use in characterizing the pH of seawater, since the ionic strength differences cause changes in [[Chemistry:Standard electrode potential|electrode potential]]. To resolve this problem, an alternative series of buffers based on [[Earth:Artificial seawater|artificial seawater]] was developed.<ref>{{cite journal |author=Hansson, I. |year=1973 |title=A new set of pH-scales and standard buffers for seawater |journal=Deep-Sea Research |volume=20 |issue=5 |pages=479–491 |bibcode=1973DSRA...20..479H |doi=10.1016/0011-7471(73)90101-0}}</ref> This new series resolves the problem of ionic strength differences between samples and the buffers, and the new pH scale is referred to as the ''total scale'', often denoted as ''pH<sub>T</sub>''. The total scale was defined using a medium containing [[Chemistry:Sulfate|sulfate]] ions. These ions experience [[Chemistry:Protonation|protonation]], {{chem2|H+}} + {{chem|SO|4|2-|↔ HSO|4|-}}, such that the total scale includes the effect of both [[Physics:Proton|protons]] (free hydrogen ions) and hydrogen sulfate ions:

: [{{chem2|H+}}]<sub>T</sub> = [{{chem2|H+}}]<sub>F</sub> + [{{chem|HSO|4|-}}]

An alternative scale, the ''free scale'', often denoted ''pH<sub>F</sub>'', omits this consideration and focuses solely on [{{chem2|H+}}]<sub>F</sub>, in principle making it a simpler representation of hydrogen ion concentration. Only [{{chem2|H+}}]<sub>T</sub> can be determined,<ref>{{cite journal |author=Dickson, A. G. |year=1984 |title=pH scales and proton-transfer reactions in saline media such as sea water |journal=Geochim. Cosmochim. Acta |volume=48 |issue=11 |pages=2299–2308 |bibcode=1984GeCoA..48.2299D |doi=10.1016/0016-7037(84)90225-4}}</ref> therefore [{{chem2|H+}}]<sub>F</sub> must be estimated using the [{{chem|SO|4|2-}}] and the stability constant of {{chem|HSO|4|-}}, {{nowrap|K{{su|b=S|p=*}}}}:

: [{{chem2|H+}}]<sub>F</sub> = [{{chem2|H+}}]<sub>T</sub> − [{{chem|HSO|4|-}}] = [{{chem2|H+}}]<sub>T</sub> ( 1 + [{{chem|SO|4|2-}}] / K{{su|b=S|p=*}} )<sup>−1</sup>

However, it is difficult to estimate K{{su|b=S|p=*}} in seawater, limiting the utility of the otherwise more straightforward free scale.

Another scale, known as the ''seawater scale'', often denoted ''pH<sub>SWS</sub>'', takes account of a further protonation relationship between hydrogen ions and [[Chemistry:Fluoride|fluoride]] ions, {{chem2|H+}} + {{chem2|F-}} ⇌ HF. Resulting in the following expression for [{{chem2|H+}}]<sub>SWS</sub>:

: [{{chem2|H+}}]<sub>SWS</sub> = [{{chem2|H+}}]<sub>F</sub> + [{{chem|HSO|4|-}}] + [HF]

However, the advantage of considering this additional complexity is dependent upon the abundance of fluoride in the medium. In seawater, for instance, sulfate ions occur at much greater concentrations (>400 times) than those of fluoride. As a consequence, for most practical purposes, the difference between the total and seawater scales is very small.

The following three equations summarize the three scales of pH:

: pH<sub>F</sub> = −log [{{chem2|H+}}]<sub>F</sub>
: pH<sub>T</sub> = −log([{{chem2|H+}}]<sub>F</sub> + [{{chem|HSO|4|-}}]) = −log[{{chem2|H+}}]<sub>T</sub>
: pH<sub>SWS</sub> = −log({{chem2|H+}}]<sub>F</sub> + [{{chem|HSO|4|-}}] + [HF]) = −log[v]<sub>SWS</sub>

===pH of various body fluids===
:{| class="wikitable"
|+pH of various body fluids<ref name=Boron2012>{{cite book |last1=Boron |first1=Walter, F. |url= https://books.google.com/books?id=54mxMgO5H_YC&dq=pH%20in%20living%20systems&pg=PA652 |title=Medical Physiology: A Cellular And Molecular Approach |last2=Boulpaep |first2=Emile L. |date=13 January 2012 |publisher=Elsevier Health Sciences, Saunders |isbn=9781455711819 |pages=652–671 |oclc=1017876653 |access-date=8 May 2022 |archive-date=8 May 2022 |archive-url=https://web.archive.org/web/20220508051939/https://www.google.co.in/books/edition/_/54mxMgO5H_YC?hl=en&gbpv=1&dq=pH+in+living+systems&pg=PA652 |url-status=live |edition=2nd }}</ref>
|-
! Compartment
! pH
|-
| [[Biology:Gastric acid|Gastric acid]] || 1.5–3.5<ref>{{cite book |last1=Marieb |first1=Elaine N. |url=https://books.google.com/books?id=BxDfnQEACAAJ |title=Human anatomy & physiology |last2=Mitchell |first2=Susan J. |date=30 June 2011 |publisher=Benjamin Cummings |isbn=9780321735287 |location=San Francisco |access-date=8 May 2022 |archive-date=8 May 2022 |archive-url=https://web.archive.org/web/20220508051937/https://www.google.co.in/books/edition/_/BxDfnQEACAAJ?hl=en |url-status=live }}</ref>
|-
| [[Biology:Lysosome|Lysosome]]s || 4.5<ref name=Boron2012/>
|-
| [[Biology:Human skin|Human skin]] || 4.7<ref>{{Cite journal|last1=Lambers|first1=H.|last2=Piessens|first2=S.|last3=Bloem|first3=A.|last4=Pronk|first4=H.|last5=Finkel|first5=P.|date=2006-10-01|url=https://onlinelibrary.wiley.com/doi/10.1111/j.1467-2494.2006.00344.x|url-status=live|title=Natural skin surface pH is on average below 5, which is beneficial for its resident flora|journal=International Journal of Cosmetic Science| volume= 28 |issue= 5|pages=359–370|doi=10.1111/j.1467-2494.2006.00344.x|issn=1468-2494|pmid=18489300|s2cid=25191984|access-date=8 May 2022|archive-date=21 March 2022|archive-url= https://web.archive.org/web/20220321033318/https://onlinelibrary.wiley.com/doi/10.1111/j.1467-2494.2006.00344.x}}</ref>
|-
| Granules of [[Biology:Chromaffin cell|chromaffin cell]]s || 5.5
|-
| [[Biology:Urine|Urine]] || 6.0
|-
| [[Biology:Cytosol|Cytosol]] || 7.2
|-
| [[Biology:Blood|Blood]] (natural pH) || 7.34–7.45<ref name=Boron2012/>
|-
| [[Biology:Cerebrospinal fluid|Cerebrospinal fluid]] (CSF) || 7.5
|-
| [[Biology:Mitochondrial matrix|Mitochondrial matrix]] || 7.5
|-
| Pancreas secretions || 8.1
|}

In living organisms, the pH of various [[Biology:Body fluid|Body fluid]]s, cellular compartments, and organs is tightly regulated to maintain a state of acid-base balance known as [[Chemistry:Acid–base homeostasis|acid–base homeostasis]]. Acidosis, defined by blood pH below 7.35, is the most common disorder of acid–base homeostasis and occurs when there is an excess of acid in the body. In contrast, alkalosis  is characterized by excessively high blood pH.

Blood pH is usually slightly basic, with a pH of 7.365, referred to as physiological pH in biology and medicine. Plaque formation in teeth can create a local acidic environment that results in [[Medicine:Tooth decay|tooth decay]] through demineralization. [[Biology:Enzyme|Enzyme]]s and other [[Biology:Protein|Protein]]s have an optimal pH range for function and can become inactivated or [[Biology:Denaturation (biochemistry)|denatured]] outside this range.

== pH calculations ==
When calculating the pH of a solution containing acids and/or bases, a [[Chemistry:Determination of equilibrium constants#Speciation calculations|chemical speciation calculation]] is used to determine the concentration of all chemical species present in the solution. The complexity of the procedure depends on the nature of the solution. Strong acids and bases are compounds that are almost completely dissociated in water, which simplifies the calculation. However, for weak acids, a [[Quadratic equation|quadratic equation]] must be solved, and for weak bases, a cubic equation is required. In general, a set of non-linear simultaneous equations must be solved.

Water itself is a weak acid and a weak base, so its dissociation must be taken into account at high pH and low solute concentration (see [[Chemistry:Amphoterism|amphoterism]]). It [[Physics:Self-ionization of water|dissociates]] according to the equilibrium

: {{chem2|2 H2O <-> H3O+ (aq) + OH- (aq)}}

with a [[Chemistry:Acid dissociation constant|dissociation constant]], ''{{mvar|K<sub>w</sub>}}'' defined as

: <math chem="">K_w = \ce{[H+][OH^{-}]} </math>

where [H<sup>+</sup>] stands for the concentration of the aqueous hydronium ion and [OH<sup>−</sup>] represents the concentration of the [[Chemistry:Hydroxide ion|hydroxide ion]]. This equilibrium needs to be taken into account at high pH and when the solute concentration is extremely low.

===Strong acids and bases===
Strong acids and bases are compounds that are essentially fully dissociated in water. This means that in an acidic solution, the concentration of hydrogen ions (H+) can be considered equal to the concentration of the acid. Similarly, in a basic solution, the concentration of hydroxide ions (OH-) can be considered equal to the concentration of the base. The pH of a solution is defined as the negative logarithm of the concentration of H+, and the pOH is defined as the negative logarithm of the concentration of OH-. For example, the pH of a 0.01M solution of hydrochloric acid (HCl) is equal to 2 (pH = −log10(0.01)), while the pOH of a 0.01M solution of sodium hydroxide (NaOH) is equal to 2 (pOH = −log10(0.01)), which corresponds to a pH of about 12.

However, self-ionization of water must also be considered when concentrations of a strong acid or base is very low or high. For instance, a 5×10<sup>−8</sup>M solution of HCl would be expected to have a pH of 7.3 based on the above procedure, which is incorrect as it is acidic and should have a pH of less than 7. In such cases, the system can be treated as a mixture of the acid or base and water, which is an amphoteric substance. By accounting for the self-ionization of water, the true pH of the solution can be calculated. For example, a 5×10<sup>−8</sup>M solution of HCl would have a pH of 6.89 when treated as a mixture of HCl and water. The self-ionization equilibrium of solutions of sodium hydroxide at higher concentrations must also be considered.<ref>{{cite web |last=Maloney |first=Chris |title=pH calculation of a very small concentration of a strong acid. |url=http://sinophibe.blogspot.com/2011/03/ph-calculation-of-very-small.html |url-status=live |archive-url=https://web.archive.org/web/20110708062942/http://sinophibe.blogspot.com/2011/03/ph-calculation-of-very-small.html |archive-date=8 July 2011 |access-date=13 March 2011}}</ref>

===Weak acids and bases===
A [[Chemistry:Weak acid|weak acid]] or the conjugate acid of a weak base can be treated using the same formalism.

* Acid HA: {{chem2|HA <-> H+ + A-}}
* Base A: {{chem2|HA+ <-> H+ + A}}

First, an acid dissociation constant is defined as follows. Electrical charges are omitted from subsequent equations for the sake of generality

: <math chem="">K_a = \frac \ce{[H] [A]}\ce{[HA]}</math>

and its value is assumed to have been determined by experiment. This being so, there are three unknown concentrations, [HA], [H<sup>+</sup>] and [A<sup>−</sup>] to determine by calculation. Two additional equations are needed. One way to provide them is to apply the law of mass conservation in terms of the two "reagents" H and A.

: <math chem="">C_\ce{A} = \ce{[A]}  +  \ce{[HA]}</math>
: <math chem="">C_\ce{H} = \ce{[H]}  +  \ce{[HA]}</math>

C stands for analytical concentration. In some texts, one mass balance equation is replaced by an equation of charge balance. This is satisfactory for simple cases like this one, but is more difficult to apply to more complicated cases as those below. Together with the equation defining K<sub>a</sub>, there are now three equations in three unknowns. When an acid is dissolved in water C<sub>A</sub> = C<sub>H</sub> = C<sub>a</sub>, the concentration of the acid, so [A] = [H]. After some further algebraic manipulation an equation in the hydrogen ion concentration may be obtained.

: <math chem="">[\ce H]^2 + K_a[\ce H] - K_a C_a = 0</math>

Solution of this [[Quadratic equation|quadratic equation]] gives the hydrogen ion concentration and hence p[H] or, more loosely, pH. This procedure is illustrated in an ICE table which can also be used to calculate the pH when some additional (strong) acid or alkaline has been added to the system, that is, when C<sub>A</sub> ≠ C<sub>H</sub>.

For example, what is the pH of a 0.01M solution of [[Chemistry:Benzoic acid|benzoic acid]], pK<sub>a</sub> = 4.19?

* Step 1: <math>K_a = 10^{-4.19} = 6.46\times10^{-5}</math>
* Step 2: Set up the quadratic equation. <math chem="">[\ce{H}]^2 + 6.46\times 10^{-5}[\ce{H}] - 6.46\times 10^{-7} = 0 </math>
* Step 3: Solve the quadratic equation. <math chem="">[\ce{H+}] = 7.74\times 10^{-4};\quad \mathrm{pH} = 3.11</math>

For alkaline solutions, an additional term is added to the mass-balance equation for hydrogen. Since the addition of hydroxide reduces the hydrogen ion concentration, and the hydroxide ion concentration is constrained by the self-ionization equilibrium to be equal to <math chem="">\frac{K_w}\ce{[H+]}</math>, the resulting equation is:

: <math chem="">C_\ce{H} = \frac{[\ce H] + [\ce{HA}] -K_w}\ce{[H]}</math>

===General method===
Some systems, such as with polyprotic acids, are amenable to spreadsheet calculations.<ref>{{cite book |last1=Billo |first1=E.J. |title=EXCEL for Chemists |publisher=Wiley-VCH |year=2011 |isbn=978-0-470-38123-6 |edition=3rd}}</ref> With three or more reagents or when many complexes are formed with general formulae such as A<sub>p</sub>B<sub>q</sub>H<sub>r</sub>, the following general method can be used to calculate the pH of a solution. For example, with three reagents, each equilibrium is characterized by an equilibrium constant, β.

: <math chem="">[\ce{A}_p\ce{B}_q\ce{H}_r] =\beta_{pqr}[\ce A]^{p}[\ce B]^{q}[\ce H]^{r}</math>

Next, write down the mass-balance equations for each reagent:

: <math chem="">\begin{align}
C_\ce{A} &= [\ce A] + \Sigma p \beta_{pqr}[\ce A]^p[\ce B]^q[\ce H]^{r} \\
C_\ce{B} &= [\ce B] + \Sigma q \beta_{pqr}[\ce A]^p[\ce B]^q[\ce H]^r \\
C_\ce{H} &= [\ce H] + \Sigma r \beta_{pqr}[\ce A]^p[\ce B]^q[\ce H]^r - K_w[\ce H]^{-1}
\end{align}</math>

Note that there are no approximations involved in these equations, except that each stability constant is defined as a quotient of concentrations, not activities. Much more complicated expressions are required if activities are to be used.

There are three simultaneous equations in the three unknowns, [A], [B] and [H]. Because the equations are non-linear and their concentrations may range over many powers of 10, the solution of these equations is not straightforward. However, many computer programs are available which can be used to perform these calculations. There may be more than three reagents. The calculation of hydrogen ion concentrations, using this approach, is a key element in the [[Chemistry:Determination of equilibrium constants|determination of equilibrium constants]] by [[Chemistry:Potentiometric titration|potentiometric titration]].

== See also ==
* [[Chemistry:PH indicator|pH indicator]]
* Arterial blood gas
* [[Chemistry:Chemical equilibrium|Chemical equilibrium]]
* [[Chemistry:PCO2|p{{CO2}}]]
* p''K''<sub>a</sub>

==References==
{{Reflist}}

==External links==
{{Wikidata property | P9440 }}
{{Wastewater}}

{{DEFAULTSORT:pH}}
[[Category:Acid–base chemistry]]
[[Category:Equilibrium chemistry]]
[[Category:Units of measurement]]
[[Category:Logarithmic scales of measurement]]
[[Category:General chemistry]]

{{Sourceattribution|PH}}