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Bjerrum plot

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Bjerrum plot

Example Bjerrum plot: Change in carbonate system of seawater from ocean acidification.

A Bjerrum plot is a graph of the concentrations of the different species of a polyprotic acid in a solution, as functions of the solution's pH,[1] when the solution is at 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+ and OH are also plotted.

Most often, 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 CO
2
; in basic (alkalinic) conditions, the dominant form is CO32−; and in between, the dominant form is HCO3. At every pH, the concentration of carbonic acid is assumed to be negligible compared to the concentration of CO2, 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 CO
2
.[2]

The Bjerrum plots for other polyprotic acids, including silicic, boric, sulphuric and phosphoric acids, can also be constructed.[1]

Bjerrum plot equations for carbonate system

Distribution of DIC (Carbonate) species with pH for 25C and 5,000 ppm salinity (e.g. salt-water swimming pool) - Bjerrum plot

If carbon dioxide, carbonic acid, hydrogen ions, bicarbonate and carbonate are all dissolved in water, and at chemical equilibrium, their equilibrium concentrations are often assumed to be given by:

[\textrm{CO}_2]_{eq} = \frac_{eq} = \frac{K_1K_2}{\textrm{d}t}= -k_1[\textrm{CO}_2] + k_{-1}[\textrm{H}^+][\textrm{HCO}_3^-],
\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-}],
\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-}],
\frac{\textrm{d}[\textrm{CO}_3^{2-}]}{\textrm{d}t}= k_2[\textrm{HCO}_3^-] - k_{-2}[\textrm{H}^+][\textrm{CO}_3^{2-}],

where [ ] denotes concentration, t is time, and k1 and k-1 are appropriate proportionality constants for reaction (1), called respectively the forwards and reverse rate constants for this reaction. (Similarly k2 and k-2 for reaction (2).)

 

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 (1)'s rate constants equals the ratio of its equilibrium concentrations, and this ratio, called K1, is called the equilibrium constant for reaction (1), i.e.

K_1 = \frac{k_1}{k_{-1}} = \frac,         (3)        

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

Similarly, from the fourth equation for the equilibrium constant K2 for reaction (2),

K_2 = \frac{k_2}{k_{-2}} = \frac_{eq}}.           (4)

Rearranging (3) gives

[\textrm{HCO}_3^-]_{eq} = \frac{K_1[\textrm{CO}_2]_{eq}},         (5)

and rearranging (4), then substituting in (5), gives

[\textrm{CO}_3^{2-}]_{eq} = \frac{K_2[\textrm{HCO}_3^-]_{eq}} = \frac{K_1K_2[\textrm{CO}_2]_{eq}}
          = [\textrm{CO}_2]_{eq} \left(1 + \frac{K_1} + \frac{K_1K_2}{[\textrm{H}^+]_{eq}^2}\right)                 substituting in (5) and (6)
          = [\textrm{CO}_2]_{eq} \left(\frac{[\textrm{H}^+]_{eq}^2 + K_1[\textrm{H}^+]_{eq}+K_1K_2}{[\textrm{H}^+]_{eq}^2}\right).

 

Re-arranging this gives the equation for CO
2
:

[\textrm{CO}_2]_{eq} = \frac{[\textrm{H}^+]_{eq}^2}{[\textrm{H}^+]_{eq}^2 + K_1[\textrm{H}^+]_{eq} + K_1K_2} \times \textrm{DIC}.

 

The equations for HCO3 and CO32− are obtained by substituting this into (5) and (6).

See also

References

  1. ^ a b Andersen, C. B. (2002). "Understanding carbonate equilibria by measuring alkalinity in experimental and natural systems". Journal of Geoscience Education 50 (4): 389–403. 
  2. ^ D.A. Wolf-Gladrow (2007). "Total alkalinity: the explicit conservative expression and its application to biogeochemical processes". Marine Chemistry 106 (1). 
  3. ^ 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). [1] Retrieved 30NOV2013.
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