Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Bjerrum plot

From Wikipedia, the free encyclopedia
Graph of polyprotic acid concentration compared to pH
Example Bjerrum plot: Change in carbonate system of seawater fromocean acidification.

ABjerrum plot (named afterNiels Bjerrum), sometimes also known as aSillén diagram (after Lars Gunnar Sillén), or aHägg diagram (afterGunnar Hägg)[1] is agraph of theconcentrations of the different species of apolyprotic acid in asolution, as a function ofpH,[2] when the solution is atequilibrium. Due to the manyorders of magnitude spanned by the concentrations, they are commonly plotted on alogarithmic 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 iscarbonic acid (adiprotic acid), and the different species are dissolvedcarbon dioxide,carbonic acid,bicarbonate, andcarbonate. In acidic conditions, the dominant form is CO2; inbasic (alkaline) conditions, the dominant form isCO2−
3; and in between, the dominant form isHCO−
3. At every pH, the concentration of carbonic acid is assumed to be negligible compared to the concentration of dissolvedCO
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-madeCO
2 emission by the fossil fuel combustion.[3]

The Bjerrum plots for other polyprotic acids, includingsilicic,boric,sulfuric andphosphoric acids, are other commonly used examples.[2]

Bjerrum plot equations for carbonate system

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

Ifcarbon dioxide,carbonic acid,hydrogen ions,bicarbonate andcarbonate are all dissolved inwater, and atchemical equilibrium, their equilibriumconcentrations are often assumed to be given by:

[CO2]eq=[H+]eq2[H+]eq2+K1[H+]eq+K1K2×DIC,[HCO3−]eq=K1[H+]eq[H+]eq2+K1[H+]eq+K1K2×DIC,[CO32−]eq=K1K2[H+]eq2+K1[H+]eq+K1K2×DIC,{\displaystyle {\begin{aligned}[]\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{aligned}}}

where the subscript 'eq' denotes that these are equilibrium concentrations,K1 is theequilibrium constant for the reactionCO
2 +H
2O ⇌ H+ +HCO−
3 (i.e. the firstacid dissociation constant for carbonic acid),K2 is theequilibrium constant for the reactionHCO−
3 ⇌ H+ +CO2−
3 (i.e. the secondacid dissociation constant for carbonic acid), and DIC is the (unchanging) totalconcentration ofdissolved inorganic carbon in the system, i.e. [CO2] + [HCO−
3] + [CO2−
3].K1,K2 and DIC each have units of aconcentration, e.g.mol/L.

A Bjerrum plot is obtained by using these three equations to plot these three species againstpH = −log10 [H+]eq, for givenK1,K2 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 andHCO−
3 intersect at[H+]eq =K1, and the curves forHCO−
3 andCO2−
3 intersect at[H+]eq =K2. Therefore, the values ofK1 andK2 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 ofK1 andK2, and therefore the curves in the Bjerrum plot, vary substantially with temperature and salinity.[4]

Chemical and mathematical derivation of Bjerrum plot equations for carbonate system

[edit]

Suppose that the reactions betweencarbon dioxide,hydrogen ions,bicarbonate andcarbonateions, all dissolved inwater, are as follows:

CO
2 +H
2O ⇌ H+ +HCO−
3		1
HCO−
3 ⇌ H+ +CO2−
3		2

Note that reaction1 is actually the combination of twoelementary reactions:

CO
2 +H
2O ⇌H
2CO
3 ⇌ H+ +HCO−
3

Assuming themass action law applies to these two reactions, that water isabundant, and that the different chemical species are always well-mixed, theirrate equations are

d[CO2]dt=−k1[CO2]+k−1[H+][HCO3−],d[H+]dt=k1[CO2]−k−1[H+][HCO3−]+k2[HCO3−]−k−2[H+][CO32−],d[HCO3−]dt=k1[CO2]−k−1[H+][HCO3−]−k2[HCO3−]+k−2[H+][CO32−],d[CO32−]dt=k2[HCO3−]−k−2[H+][CO32−]{\displaystyle {\begin{aligned}{\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{aligned}}}

where[ ] denotesconcentration,t is time, andK1 andk−1 are appropriateproportionality constants for reaction1, called respectively the forwards and reverserate constants for this reaction. (SimilarlyK2 andk−2 for reaction2.)

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 reaction1's rate constants equals the ratio of its equilibrium concentrations, and this ratio, calledK1, is called theequilibrium constant for reaction1, i.e.

K1=k1k−1=[H+]eq[HCO3−]eq[CO2]eq{\displaystyle K_{1}={\frac {k_{1}}{k_{-1}}}={\frac {[{\textrm {H}}^{+}]_{\text{eq}}[{\textrm {HCO}}_{3}^{-}]_{\text{eq}}}{[{\textrm {CO}}_{2}]_{\text{eq}}}}}3

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

Similarly, from the fourth equation for theequilibrium constantK2 for reaction2,

K2=k2k−2=[H+]eq[CO32−]eq[HCO3−]eq{\displaystyle 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}}}}}4

Rearranging3 gives

[HCO3−]eq=K1[CO2]eq[H+]eq{\displaystyle \left[{\textrm {HCO}}_{3}^{-}\right]_{\text{eq}}={\frac {K_{1}\left[{\textrm {CO}}_{2}\right]_{\text{eq}}}{\left[{\textrm {H}}^{+}\right]_{\text{eq}}}}}    5

and rearranging4, then substituting in5, gives

[CO32−]eq=K2[HCO3−]eq[H+]eq=K1K2[CO2]eq[H+]eq2{\displaystyle \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}}}}    6

The totalconcentration ofdissolved inorganic carbon in the system is given by substituting in5 and6:

DIC=[CO2]+[HCO3−]+[CO32−]=[CO2]eq(1+K1[H+]eq+K1K2[H+]eq2)=[CO2]eq([H+]eq2+K1[H+]eq+K1K2[H+]eq2){\displaystyle {\begin{aligned}{\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_{1}K_{2}}{\left[{\textrm {H}}^{+}\right]_{\text{eq}}^{2}}}\right)\end{aligned}}}

Re-arranging this gives the equation forCO
2:

[CO2]eq=[H+]eq2[H+]eq2+K1[H+]eq+K1K2×DIC{\displaystyle \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}}}

The equations forHCO−
3 andCO2−
3 are obtained by substituting this into5 and6.

See also

[edit]

References

[edit]
  1. ^Togni, Antonio (2022-05-25)."Teaching Acid-Base Equilibria by Using Log-log Diagrams: Chemical Education".CHIMIA.76 (5): 481.doi:10.2533/chimia.2022.481.hdl:20.500.11850/557050.ISSN 2673-2424.
  2. ^abAndersen, C. B. (2002). "Understanding carbonate equilibria by measuring alkalinity in experimental and natural systems".Journal of Geoscience Education.50 (4):389–403.Bibcode:2002JGeEd..50..389A.doi:10.5408/1089-9995-50.4.389.S2CID 17094010.
  3. ^D.A. Wolf-Gladrow (2007)."Total alkalinity: the explicit conservative expression and its application to biogeochemical processes"(PDF).Marine Chemistry.106 (1):287–300.Bibcode:2007MarCh.106..287W.doi:10.1016/j.marchem.2007.01.006.
  4. ^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 30 November 2013.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bjerrum_plot&oldid=1170162690"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2026 Movatter.jp