View source for Biology:Bjerrum plot

{{short description|Graph of polyprotic acid concentration compared to pH}}
[[File:Carbonate system of seawater.svg|thumb|upright=1.35|Example Bjerrum plot: Change in carbonate system of seawater from [[Biology:Ocean acidification|ocean acidification]].]]

A '''Bjerrum plot''' (named after Niels Bjerrum), sometimes also known as a '''Sillén diagram''' (after Lars Gunnar Sillén), or a '''Hägg diagram''' (after Gunnar Hägg)<ref>{{Cite journal |last=Togni |first=Antonio |date=2022-05-25 |title=Teaching Acid-Base Equilibria by Using Log-log Diagrams: Chemical Education |url=https://chimia.ch/chimia/article/view/2022_481 |journal=CHIMIA |volume=76 |issue=5 |pages=481 |doi=10.2533/chimia.2022.481 |issn=2673-2424|hdl=20.500.11850/557050 |hdl-access=free }}</ref> is a [[Chart|graph]] of the [[Chemistry:Concentration|concentration]]s of the different species of a [[Chemistry:Acid#Polyprotic acids|polyprotic acid]] in a [[Chemistry:Solution|solution]], as a function of [[Chemistry:PH|pH]],<ref name=Andersen>{{Cite journal |last=Andersen|first=C. B.|year=2002 |title=Understanding carbonate equilibria by measuring alkalinity in experimental and natural systems | journal=Journal of Geoscience Education |volume=50 |issue=4 |pages=389–403 |doi=10.5408/1089-9995-50.4.389 |bibcode=2002JGeEd..50..389A|s2cid=17094010 }}</ref> when the solution is at [[Chemistry:Chemical equilibrium|equilibrium]]. Due to the many orders of magnitude spanned by the concentrations, they are commonly plotted on a [[Logarithmic scale|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.

Most often, the carbonate system is plotted, where the polyprotic acid is [[Chemistry:Carbonic acid|carbonic acid]] (a diprotic acid), and the different species are dissolved [[Chemistry:Carbon dioxide|carbon dioxide]], [[Chemistry:Carbonic acid|carbonic acid]], [[Chemistry:Bicarbonate|bicarbonate]], and [[Chemistry:Carbonate|carbonate]]. In acidic conditions, the dominant form is {{CO2}}; in [[Chemistry:Base|basic]] (alkaline) conditions, the dominant form is {{chem|CO|3|2−}}; and in between, the dominant form is {{chem|HCO|3|−}}. At every pH, the concentration of carbonic acid is assumed to be negligible compared to the concentration of dissolved {{chem|CO|2}}, and so is often omitted from Bjerrum plots. These plots are very helpful in solution chemistry and natural water chemistry. In the example given here, it illustrates the response of seawater pH and carbonate speciation due to the input of man-made {{chem|CO|2}} emission by the fossil fuel combustion.<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| pages=287–300| doi=10.1016/j.marchem.2007.01.006| bibcode=2007MarCh.106..287W| url=http://www.soest.hawaii.edu/oceanography/faculty/zeebe_files/Publications/WolfGladrowMarChem07.pdf}}</ref>

The Bjerrum plots for other polyprotic acids, including [[Chemistry:Silicic acid|silicic]], [[Chemistry:Boric acid|boric]], [[Chemistry:Sulfuric acid|sulfuric]] and [[Chemistry:Phosphoric acid|phosphoric]] acids, are other commonly used examples.<ref name=Andersen />

==Bjerrum plot equations for carbonate system==
[[File:Carbonate Bjerrum.gif|thumb|upright=1.2|alt=Distribution of DIC (Carbonate) species with pH for 25C and 5,000 ppm salinity (e.g. salt-water swimming pool) - Bjerrum plot]]
If [[Chemistry:Carbon dioxide|carbon dioxide]], [[Chemistry:Carbonic acid|carbonic acid]], [[Chemistry:Hydron|hydrogen ions]], [[Chemistry:Bicarbonate|bicarbonate]] and [[Chemistry:Carbonate|carbonate]] are all dissolved in [[Chemistry:Water|water]], and at [[Chemistry:Chemical equilibrium|chemical equilibrium]], their equilibrium [[Chemistry:Concentration|concentration]]s are often assumed to be given by:

: <math>\begin{align}[]
       \left[\textrm{CO}_2\right]_\text{eq} &= \frac{\left[\textrm{H}^+\right]_\text{eq}^2}{\left[\textrm{H}^+\right]_\text{eq}^2 + K_1\left[\textrm{H}^+\right]_\text{eq} + K_1 K_2} \times \textrm{DIC}, \\[3pt]
    \left[\textrm{HCO}_3^-\right]_\text{eq} &= \frac{K_1\left[\textrm{H}^+\right]_\text{eq}}{\left[\textrm{H}^+\right]_\text{eq}^2 + K_1\left[\textrm{H}^+\right]_\text{eq} + K_1 K_2} \times \textrm{DIC}, \\[3pt]
  \left[\textrm{CO}_3^{2-}\right]_\text{eq} &= \frac{K_1 K_2}{\left[\textrm{H}^+\right]_\text{eq}^2 + K_1\left[\textrm{H}^+\right]_\text{eq} + K_1 K_2} \times \textrm{DIC},
\end{align}</math>

where the subscript 'eq' denotes that these are equilibrium concentrations, ''K''<sub>1</sub> is the [[Chemistry:Equilibrium constant|equilibrium constant]] for the reaction {{chem|CO|2}} + {{chem|H|2|O}} {{eqm}} H<sup>+</sup> + {{chem|HCO|3|−}} (i.e. the first [[Chemistry:Acid dissociation constant|acid dissociation constant]] for carbonic acid), ''K''<sub>2</sub> is the [[Chemistry:Equilibrium constant|equilibrium constant]] for the reaction {{chem|HCO|3|−}} {{eqm}} H<sup>+</sup> + {{chem|CO|3|2−}} (i.e. the second [[Chemistry:Acid dissociation constant|acid dissociation constant]] for carbonic acid), and DIC is the (unchanging) total [[Chemistry:Concentration|concentration]] of [[Chemistry:Total inorganic carbon|dissolved inorganic carbon]] in the system, i.e. [{{CO2}}] + [{{chem|HCO|3|−}}] + [{{chem|CO|3|2−}}]. ''K''<sub>1</sub>, ''K''<sub>2</sub> and DIC each have units of a [[Chemistry:Concentration|concentration]], e.g. [[Mole (unit)|mol]]/[[Litre|L]].

A Bjerrum plot is obtained by using these three equations to plot these three species against {{nowrap|pH {{=}} −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 {{chem|HCO|3|−}} intersect at {{nowrap|[H<sup>+</sup>]<sub>eq</sub> {{=}} ''K''<sub>1</sub>}}, and the curves for {{chem|HCO|3|−}} and {{chem|CO|3|2−}} intersect at {{nowrap|[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. An example with linear Y axis is shown in the accompanying graph. The values of ''K''<sub>1</sub> and ''K''<sub>2</sub>, and therefore the curves in the Bjerrum plot, vary substantially with temperature and salinity.<ref>Mook W (2000) Chemistry of carbonic acid in water. In 'Environmental Isotopes in the Hydrological Cycle: Principles and Applications' pp. 143-165. (INEA / UNESCO: Paris). [http://www-naweb.iaea.org/napc/ih/documents/global_cycle/vol%20I/cht_i_09.pdf] Retrieved 30 November 2013.</ref>

==Chemical and mathematical derivation of Bjerrum plot equations for carbonate system==
Suppose that the reactions between [[Chemistry:Carbon dioxide|carbon dioxide]], [[Chemistry:Hydron|hydrogen ions]], [[Chemistry:Bicarbonate|bicarbonate]] and [[Chemistry:Carbonate|carbonate]] ions, all dissolved in [[Chemistry:Water|water]], are as follows:
{{NumBlk|:|{{chem|CO|2}} + {{chem|H|2|O}} {{eqm}} H<sup>+</sup> + {{chem|HCO|3|−}}|{{EquationRef|1}}}}
{{NumBlk|:|{{chem|HCO|3|−}} {{eqm}} H<sup>+</sup> + {{chem|CO|3|2−}}|{{EquationRef|2}}}}

Note that reaction {{EquationNote|1}} is actually the combination of two [[Physics:Elementary reaction|elementary reaction]]s:
: {{chem|CO|2}} + {{chem|H|2|O}} {{eqm}} {{chem|H|2|CO|3}} {{eqm}} H<sup>+</sup> + {{chem|HCO|3|−}}

Assuming the mass action law applies to these two reactions, that water is [[Chemistry:Abundance|abundant]], and that the different chemical species are always well-mixed, their [[Chemistry:Rate equation|rate equation]]s are
: <math>\begin{align}
       \frac{\textrm{d}\left[\textrm{CO}_2\right]}{\textrm{d}t} &= -k_1\left[\textrm{CO}_2\right] + k_{-1}\left[\textrm{H}^+\right]\left[\textrm{HCO}_3^-\right], \\
        \frac{\textrm{d}\left[\textrm{H}^+\right]}{\textrm{d}t} &= k_1\left[\textrm{CO}_2\right] - k_{-1}\left[\textrm{H}^+\right]\left[\textrm{HCO}_3^-\right] + k_2\left[\textrm{HCO}_3^-\right] - k_{-2}\left[\textrm{H}^+\right]\left[\textrm{CO}_3^{2-}\right], \\
    \frac{\textrm{d}\left[\textrm{HCO}_3^-\right]}{\textrm{d}t} &= k_1\left[\textrm{CO}_2\right] - k_{-1}\left[\textrm{H}^+\right]\left[\textrm{HCO}_3^-\right] - k_2\left[\textrm{HCO}_3^-\right] + k_{-2}\left[\textrm{H}^+\right]\left[\textrm{CO}_3^{2-}\right], \\
  \frac{\textrm{d}\left[\textrm{CO}_3^{2-}\right]}{\textrm{d}t} &= k_2\left[\textrm{HCO}_3^-\right] - k_{-2}\left[\textrm{H}^+\right]\left[\textrm{CO}_3^{2-}\right]
\end{align}</math>

where {{nowrap|[&emsp;]}} denotes [[Chemistry:Concentration|concentration]], ''t'' is time, and ''K''<sub>1</sub> and ''k''<sub>−1</sub> are appropriate [[Proportionality (mathematics)|proportionality]] constants for reaction {{EquationNote|1}}, called respectively the forwards and reverse [[Chemistry:Reaction rate constant|rate constants]] for this reaction. (Similarly ''K''<sub>2</sub> and ''k''<sub>−2</sub> for reaction {{EquationNote|2}}.)

{{em|At any 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 {{EquationNote|1}}'s rate constants equals the ratio of its equilibrium concentrations, and this ratio, called ''K''<sub>1</sub>, is called the [[Chemistry:Equilibrium constant|equilibrium constant]] for reaction {{EquationNote|1}}, i.e.
{{NumBlk|:|<math>K_1 = \frac{k_1}{k_{-1}} = \frac{[\textrm{H}^+]_\text{eq}[\textrm{HCO}_3^-]_\text{eq}}{[\textrm{CO}_2]_\text{eq}}</math>|{{EquationRef|3}}}}

where the subscript 'eq' denotes that these are equilibrium concentrations.

Similarly, from the fourth equation for the [[Chemistry:Equilibrium constant|equilibrium constant]] ''K''<sub>2</sub> for reaction {{EquationNote|2}},
{{NumBlk|:|<math>K_2 = \frac{k_2}{k_{-2}} = \frac{\left[\textrm{H}^+\right]_\text{eq}\left[\textrm{CO}_3^{2-}\right]_\text{eq}}{\left[\textrm{HCO}_3^-\right]_\text{eq}}</math>|{{EquationRef|4}}}}

Rearranging {{EquationNote|3}} gives
{{NumBlk|:|<math>\left[\textrm{HCO}_3^-\right]_\text{eq} = \frac{K_1\left[\textrm{CO}_2\right]_\text{eq}}{\left[\textrm{H}^+\right]_\text{eq}}</math> {{space}} {{space}} {{space}} {{space}}|{{EquationRef|5}}}}

and rearranging {{EquationNote|4}}, then substituting in {{EquationNote|5}}, gives
{{NumBlk|:|<math>\left[\textrm{CO}_3^{2-}\right]_\text{eq} = \frac{K_2\left[\textrm{HCO}_3^-\right]_\text{eq}}{\left[\textrm{H}^+\right]_\text{eq}} = \frac{K_1 K_2\left[\textrm{CO}_2\right]_\text{eq}}{\left[\textrm{H}^+\right]_\text{eq}^2}</math> {{space}} {{space}} {{space}} {{space}}|{{EquationRef|6}}}}

The total [[Chemistry:Concentration|concentration]] of [[Chemistry:Total inorganic carbon|dissolved inorganic carbon]] in the system is given by substituting in {{EquationNote|5}} and {{EquationNote|6}}:
: <math>\begin{align}
  \textrm{DIC}
    &= \left[\textrm{CO}_2\right] + \left[\textrm{HCO}_3^-\right] + \left[\textrm{CO}_3^{2-}\right] \\
    &= \left[\textrm{CO}_2\right]_\text{eq} \left(1 + \frac{K_1}{\left[\textrm{H}^+\right]_\text{eq}} + \frac{K_1 K_2}{\left[\textrm{H}^+\right]_\text{eq}^2}\right) \\ 
    &= \left[\textrm{CO}_2\right]_\text{eq} \left(\frac{\left[\textrm{H}^+\right]_\text{eq}^2 + K_1\left[\textrm{H}^+\right]_\text{eq} + K_1K_2}{\left[\textrm{H}^+\right]_\text{eq}^2}\right)
\end{align}</math>

Re-arranging this gives the equation for {{chem|CO|2}}:
: <math>\left[\textrm{CO}_2\right]_\text{eq} = \frac{\left[\textrm{H}^+\right]_\text{eq}^2}{\left[\textrm{H}^+\right]_\text{eq}^2 + K_1\left[\textrm{H}^+\right]_\text{eq} + K_1 K_2} \times \textrm{DIC}</math>

The equations for {{chem|HCO|3|−}} and {{chem|CO|3|2−}} are obtained by substituting this into {{EquationNote|5}} and {{EquationNote|6}}.

== See also ==
* [[Chemistry:Charlot equation|Charlot equation]]
* [[Chemistry:Gran plot|Gran plot]] (also known as Gran titration or the Gran method)
* [[Henderson–Hasselbalch equation]]
* [[Chemistry:Hill equation (biochemistry)|Hill equation (biochemistry)]]
* [[Chemistry:Ion speciation|Ion speciation]]
* [[Earth:Fresh water|Fresh water]]
* [[Earth:Seawater|Seawater]]
* [[Earth:Thermohaline circulation|Thermohaline circulation]]

== References ==
{{reflist}}

[[Category:Acid–base chemistry]]
[[Category:Aquatic ecology]]
[[Category:Chemical oceanography]]
[[Category:Geochemistry]]
[[Category:Oceanography]]
[[Category:Soil chemistry]]
[[Category:Thermodynamics]]
[[Category:Water chemistry]]

{{Sourceattribution|Bjerrum plot}}