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: : \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} where the subscript 'eq' denotes that these are equilibrium concentrations, K1 is the equilibrium constant for the reaction + H+ + (i.e. the first acid dissociation constant for carbonic acid), K2 is the equilibrium constant for the reaction H+ + (i.e. the second acid dissociation constant for carbonic acid), and DIC is the (unchanging) total concentration of dissolved inorganic carbon in the system, i.e. [] + [] + []. K1, K2 and DIC each have units of a concentration, e.g. mol/L. A Bjerrum plot is obtained by using these three equations to plot these three species against , for given K1, 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 and intersect at , and the curves for and intersect at . Therefore, the values of K1 and K2 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 K1 and K2, and therefore the curves in the Bjerrum plot, vary substantially with temperature and salinity. ==Chemical and mathematical derivation of Bjerrum plot equations for carbonate system==

Suppose that the reactions between carbon dioxide, hydrogen ions, bicarbonate and carbonate ions, all dissolved in water, are as follows: Note that reaction is actually the combination of two elementary reactions: : + H+ + Assuming the mass action law applies to these two reactions, that water is abundant, and that the different chemical species are always well-mixed, their rate equations are : \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} where denotes concentration, t is time, and K1 and k−1 are appropriate proportionality constants for reaction , called respectively the forwards and reverse rate constants for this reaction. (Similarly K2 and k−2 for reaction .) , 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 's rate constants equals the ratio of its equilibrium concentrations, and this ratio, called K1, is called the equilibrium constant for reaction , i.e. {{NumBlk|:|K_1 = \frac{k_1}{k_{-1}} = \frac{[\textrm{H}^+]_\text{eq}[\textrm{HCO}_3^-]_\text{eq}}{[\textrm{CO}_2]_\text{eq}}|}} where the subscript 'eq' denotes that these are equilibrium concentrations. Similarly, from the fourth equation for the equilibrium constant K2 for reaction , {{NumBlk|:|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}}|}} Rearranging gives {{NumBlk|:|\left[\textrm{HCO}_3^-\right]_\text{eq} = \frac{K_1\left[\textrm{CO}_2\right]_\text{eq}}{\left[\textrm{H}^+\right]_\text{eq}} |}} and rearranging , then substituting in , gives {{NumBlk|:|\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} |}} The total concentration of dissolved inorganic carbon in the system is given by substituting in and : : \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} Re-arranging this gives the equation for : : \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 for and are obtained by substituting this into and . == See also ==