Editing PH (section)

== pH calculations ==
When calculating the pH of a solution containing acids or bases, a [[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]] must be solved, and for weak bases, a cubic equation is required. In general, a set of [[non-linear]] [[simultaneous equation]]s 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 ''[[Amphoterism]]''). It [[Self-ionization of water|dissociates]] according to the equilibrium
: {{chem2|2 H2O <-> H3O+ (aq) + OH- (aq)}}
with a [[Acid dissociation constant|dissociation constant]], {{math|''K''<sub>w</sub>}} defined as
: <math chem="">K_\text{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 [[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 acid]]s and [[Strong base|bases]] are compounds that are essentially fully dissociated in water. This means that in an acidic solution, the concentration of hydrogen cations (H<sup>+</sup>) can be considered equal to the concentration of the acid. Similarly, in a basic solution, the concentration of hydroxide ions (OH<sup>−</sup>) 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<sup>+</sup>, and the pOH is defined as the negative logarithm of the concentration of OH<sup>−</sup>. For example, the pH of a 0.01&nbsp;M solution of hydrochloric acid (HCl) is equal to 2 (pH = −log<sub>10</sub>(0.01)), while the pOH of a 0.01&nbsp;M solution of sodium hydroxide (NaOH) is equal to 2 (pOH = −log<sub>10</sub>(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 {{val|5|e=−8|u=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 {{val|5|e=−8|u=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 [[weak acid]] or the conjugate acid of a weak base can be treated using the same formalism.
* Acid HA: <chem>HA <=> H+ + A-</chem>
* Base A: <chem>HA+ <=> H+ + A</chem>
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]] 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 0.01&nbsp;M solution of [[benzoic acid]], p''K''<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>

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 equation]]s 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 [[determination of equilibrium constants]] by [[potentiometric titration]].