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[[File:Carbonate system of seawater.svg|thumb|Example Bjerrum plot: Change in carbonate system of seawater from [[ocean acidification]].]]
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A '''Bjerrum plot''' is a graph of the [[concentration]]s of the different species of a [[Acid#Polyprotic acids|polyprotic acid]] in a [[solution]], as functions of the solution's [[pH]],<ref name=Andersen>{{Cite journal |last=Andersen|first=C. B.|coauthors= |year=2002 |title=Understanding carbonate equilibria by measuring alkalinity in experimental and natural systems |url= |archiveurl= |archivedate=  | journal=Journal of Geoscience Education |volume=50 |issue=4 |pages=389–403 |doi= |pmid= |bibcode=}}</ref> when the solution is at [[Chemical equilibrium|equilibrium]]. Due to the many [[orders of magnitude]] spanned by the concentrations, they are commonly plotted on a [[logarithmic scale]]. Sometimes the ratios of the concentrations are plotted rather than the actual concentrations. Occasionally H<sup>+</sup> and OH<sup>−</sup> are also plotted.
 
Normally the carbonate system is plotted, where the polyprotic acid is [[carbonic acid]] (a [[diprotic acid]]), and the different species are [[carbonic acid]], [[carbon dioxide]], [[bicarbonate]], and [[carbonate]]. In acidic conditions, the dominant form is {{CO2}}; in [[Base (chemistry)|basic]] (alkalinic) conditions, the dominant form is CO<sub>3</sub><sup>2−</sup>; and in between, the dominant form is HCO<sub>3</sub><sup>−</sup>. At every pH, the concentration of carbonic acid is assumed to be [[negligible]] compared to the concentration of CO<sub>2</sub>, and so is often omitted from Bjerrum plots. These plots are typically used in ocean chemistry to track the response of an ocean to changes in both pH and of inputs in carbonate and {{chem|CO|2}}.<ref name="wolfgladrow2007">{{cite journal | author=D.A. Wolf-Gladrow| title=Total alkalinity: the explicit conservative expression and its application to biogeochemical processes| journal=Marine Chemistry| year=2007| volume=106| issue=1| url=http://www.soest.hawaii.edu/oceanography/faculty/zeebe_files/Publications/WolfGladrowMarChem07.pdf}}</ref>
 
The Bjerrum plots for other polyprotic acids, including [[Silicic acid|silicic]], [[Boric acid|boric]], [[Sulfuric acid|sulphuric]] and [[Phosphoric acid|phosphoric]] acids, can also be constructed.<ref name=Andersen />
 
==Bjerrum plot equations for carbonate system==
 
If [[carbon dioxide]], [[carbonic acid]], [[hydron (chemistry)|hydrogen ions]], [[bicarbonate]] and [[carbonate]] are all dissolved in [[water]], and at [[chemical equilibrium]], their equilibrium [[concentration]]s are often assumed to be given by:
 
: <math> [\textrm{CO}_2]_{eq} =  \frac{[\textrm{H}^+]_{eq}^2}{[\textrm{H}^+]_{eq}^2 + K_1[\textrm{H}^+]_{eq} + K_1K_2} \times \textrm{DIC},  </math>
: <math> [\textrm{HCO}_3^-]_{eq} =  \frac{K_1[\textrm{H}^+]_{eq}}{[\textrm{H}^+]_{eq}^2 + K_1[\textrm{H}^+]_{eq} + K_1K_2} \times \textrm{DIC},  </math>
: <math> [\textrm{CO}_3^{2-}]_{eq} =  \frac{K_1K_2}{[\textrm{H}^+]_{eq}^2 + K_1[\textrm{H}^+]_{eq} + K_1K_2} \times \textrm{DIC},  </math>
where the subscript 'eq' denotes that these are equilibrium concentrations, ''K<sub>1</sub>'' is the [[equilibrium constant]] for the reaction {{chem|CO|2}} + {{chem|H|2|O}} {{eqm}} H<sup>+</sup> + HCO<sub>3</sub><sup>−</sup> (ie. the first [[acid dissociation constant]] for carbonic acid), ''K<sub>2</sub>'' is the [[equilibrium constant]] for the reaction HCO<sub>3</sub><sup>−</sup> {{eqm}} H<sup>+</sup> + CO<sub>3</sub><sup>2−</sup> (ie. the second [[acid dissociation constant]] for carbonic acid), and DIC is the (unchanging) total [[concentration]] of [[Total inorganic carbon|dissolved inorganic carbon]] in the system, i.e. [{{CO2}}] + [HCO<sub>3</sub><sup>−</sup>] + [CO<sub>3</sub><sup>2−</sup>]. ''K<sub>1</sub>'', ''K<sub>2</sub>'' and DIC each have units of a [[concentration]], eg. [[Mole (unit)|mol]]/[[litre]].
 
A Bjerrum plot is obtained by using these three equations to plot these three species against pH = –{{nowrap|log<sub>10</sub>[H<sup>+</sup>]<sub>eq</sub>}}, for given ''K<sub>1</sub>'', ''K<sub>2</sub>'' and DIC. The fractions in these equations give the three species' relative proportions, and so if DIC is unknown, or the actual concentrations are unimportant, these proportions may be plotted instead.
 
These three equations show that the curves for {{CO2}} and HCO<sub>3</sub><sup>−</sup> intersect at [H<sup>+</sup>]<sub>eq</sub> = ''K<sub>1</sub>'', and the curves for HCO<sub>3</sub><sup>−</sup> and CO<sub>3</sub><sup>2−</sup> intersect at [H<sup>+</sup>]<sub>eq</sub> = ''K<sub>2</sub>''. Therefore, the values of ''K<sub>1</sub>'' and ''K<sub>2</sub>'' that were used to create a given Bjerrum plot can easily be found from that plot, by reading off the concentrations at these points of intersection.
 
==Chemical and mathematical derivation of Bjerrum plot equations for carbonate system==
 
Suppose that the reactions between [[carbon dioxide]], [[hydron (chemistry)|hydrogen ions]], [[bicarbonate]] and [[carbonate]] [[ions]], all dissolved in [[water]], are as follows:
: {{chem|CO|2}} + {{chem|H|2|O}} {{eqm}} H<sup>+</sup> + HCO<sub>3</sub><sup>−</sup> {{space}} {{space}} {{space}} {{space}} {{space}} {{space}} {{space}} (1)
: {{space}} {{space}} {{space}} HCO<sub>3</sub><sup>−</sup> {{eqm}} H<sup>+</sup> + CO<sub>3</sub><sup>2−</sup>. {{space}} {{space}} {{space}} {{space}} {{space}} {{space}} {{space}} (2)
 
(Note that reaction (1) is actually the combination of two [[elementary reaction]]s: {{chem|CO|2}} + {{chem|H|2|O}} {{eqm}} {{chem|H|2|CO|3}} {{eqm}} H<sup>+</sup> + HCO<sub>3</sub><sup>−</sup>.)
 
Assuming the [[mass action law]] applies to these two reactions, that water is [[Abundance (chemistry)|abundant]], and that the different chemical species are always well-mixed, their [[rate equation]]s are:
: <math>\frac{\textrm{d}[\textrm{CO}_2]}{\textrm{d}t}= -k_1[\textrm{CO}_2] + k_{-1}[\textrm{H}^+][\textrm{HCO}_3^-], </math>
: <math>\frac{\textrm{d}[\textrm{H}^+]}{\textrm{d}t}= k_1[\textrm{CO}_2] - k_{-1}[\textrm{H}^+][\textrm{HCO}_3^-] + k_2[\textrm{HCO}_3^-] - k_{-2}[\textrm{H}^+][\textrm{CO}_3^{2-}], </math>
: <math>\frac{\textrm{d}[\textrm{HCO}_3^-]}{\textrm{d}t}= k_1[\textrm{CO}_2] - k_{-1}[\textrm{H}^+][\textrm{HCO}_3^-] - k_2[\textrm{HCO}_3^-] + k_{-2}[\textrm{H}^+][\textrm{CO}_3^{2-}], </math>
: <math>\frac{\textrm{d}[\textrm{CO}_3^{2-}]}{\textrm{d}t}= k_2[\textrm{HCO}_3^-] - k_{-2}[\textrm{H}^+][\textrm{CO}_3^{2-}], </math>
where [{{space}}] denotes [[concentration]], ''t'' is time, and ''k<sub>1</sub>'' and ''k<sub>-1</sub>'' are appropriate [[Proportionality (mathematics)|proportionality]] constants for reaction (1), called respectively the forwards and reverse [[Reaction rate constant|rate constants]] for this reaction. (Similarly ''k<sub>2</sub>'' and ''k<sub>-2</sub>'' for reaction (2).)
 
{{space}}
 
'''At any [[Chemical equilibrium|equilibrium]]''', the concentrations are unchanging, hence the left hand sides of these equations are zero. Then, from the first of these four equations, the ratio of reaction (1)'s rate constants equals the ratio of its equilibrium concentrations, and this ratio, called ''K<sub>1</sub>'', is called the [[equilibrium constant]] for reaction (1), i.e.
: <math> K_1 = \frac{k_1}{k_{-1}} = \frac{[\textrm{H}^+]_{eq}[\textrm{HCO}_3^-]_{eq}}{[\textrm{CO}_2]_{eq}}, </math> {{space}} {{space}} {{space}} {{space}} (3) {{space}} {{space}} {{space}} {{space}}
where the subscript 'eq' denotes that these are equilibrium concentrations.
 
Similarly, from the fourth equation for the [[equilibrium constant]] ''K<sub>2</sub>'' for reaction (2),
: <math> K_2 = \frac{k_2}{k_{-2}} = \frac{[\textrm{H}^+]_{eq}[\textrm{CO}_3^{2-}]_{eq}}{[\textrm{HCO}_3^-]_{eq}}. </math> {{space}} {{space}} {{space}} {{space}} {{space}} (4)
 
Rearranging (3) gives
: <math> [\textrm{HCO}_3^-]_{eq} =  \frac{K_1[\textrm{CO}_2]_{eq}}{[\textrm{H}^+]_{eq}}, </math> {{space}} {{space}} {{space}} {{space}} (5)
and rearranging (4), then substituting in (5), gives
: <math> [\textrm{CO}_3^{2-}]_{eq} =  \frac{K_2[\textrm{HCO}_3^-]_{eq}}{[\textrm{H}^+]_{eq}}
=  \frac{K_1K_2[\textrm{CO}_2]_{eq}}{[\textrm{H}^+]_{eq}^2}. </math> {{space}} {{space}} {{space}} {{space}} (6)
 
{{space}}
 
The total [[concentration]] of [[Total inorganic carbon|dissolved inorganic carbon]] in the system is given by
: <math> \textrm{DIC} =  [\textrm{CO}_2] + [\textrm{HCO}_3^-] + [\textrm{CO}_3^{2-}] </math>
: {{space}} {{space}} {{space}} {{space}} {{space}}  <math>  =  [\textrm{CO}_2]_{eq} \left(1 + \frac{K_1}{[\textrm{H}^+]_{eq}} + \frac{K_1K_2}{[\textrm{H}^+]_{eq}^2}\right) </math> {{space}} {{space}} {{space}} {{space}} {{space}} {{space}} {{space}} {{space}}  substituting in (5) and (6)
: {{space}} {{space}} {{space}} {{space}} {{space}} <math>  = 
[\textrm{CO}_2]_{eq} \left(\frac{[\textrm{H}^+]_{eq}^2 + K_1[\textrm{H}^+]_{eq}+K_1K_2}{[\textrm{H}^+]_{eq}^2}\right). </math>
 
{{space}}
 
This gives the equation for [{{chem|CO|2}}]. The equations for [HCO<sub>3</sub><sup>−</sup>] and [CO<sub>3</sub><sup>2−</sup>] are obtained by substituting this into (5) and (6).
 
== See also ==
* [[Fresh water]]
* [[Seawater]]
* [[Thermohaline circulation]]
 
== References ==
{{reflist}}
 
[[Category:Chemical oceanography]]
[[Category:Aquatic ecology]]
[[Category:Oceanography]]
[[Category:Thermodynamics]]

Latest revision as of 19:30, 23 December 2014

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