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10.5: The Solutions

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    The set of six simultaneous equations derived by Stewart (see previous section) include:

    • the 3 independent variables (pCO2, SID and [ATot])
    • the 6 dependent variables ( [HA], [A-], [HCO3-], [CO2-3], [OH-], [H+] )

    These equations can be solved mathematically to express the value of any one of the dependent variables in terms of the 3 independent variables (and the various equilibrium constants). The values of the equilibrium constants have been experimentally determined under a range of conditions and can be obtained from various reference sources.

    To focus only on the solution of the six equations for [H+], one derives a formula of the following form:

    \[ ax^{4} + bx^{3} + cx^{2} +dx + e =0 \]

    Mathematicians call this type of equation a 4th order polynomial or a quartic equation. The unknown value is x and a,b,c,d and e are constants. (The actual value of these "constants" can change - eg with change in temperature - but are a fixed value under a given set of conditions. If, for example, the temperature changes, then different values of the constants have to be used.) The actual equation for [H+] that Stewart derived is listed below.

    Equation used to Solve for [H+]

    \( a \cdot [H^{+}]^{4} + b \cdot [H^{+}]^{3} + c \cdot [H^{+}]^{2} + d \cdot [H^{+}] + e = 0 \)


    • \(a = 1\)
    • \(b = [SID] + KA\)
    • \( c = KA \times ([SID]-[A_{Tot}]) - K_{w} - KC \times pCO_{2} \)
    • \( d = -(KA \times (K_{w} + KC \times pCO_{2}) - K3 \times KC \times CO_{2}) \)
    • \(e = -(KA \times K3 \times KC \times pCO_{2}) \)

    (see Stewart's book for values of the constants in this equation)

    A daunting equation and there is a general formula for solving such quartic equations, but this is very complex. Solution is however fast and easy on an appropriately programmed computer, and can be done online, if you know the values for the constants. Some of the constants are temperature dependent, so not actually "constant". A similar type of equation can be produced for any of the 6 dependent variables. The point here is not to become involved in complicated mathematics but to show that it is possible to solve the equation and determine the hydrogen ion concentration (ie [H+] ) in the solution using only the values of the three independent variables and various equilibrium constants. What is not stated explicitly are the error bounds related to the calculation. These become important (i.e. they are wide) as there are so many variables and constants involved, and such indirect calculation of pH can never be as accurate as actual pH measurement.

    This page titled 10.5: The Solutions is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Kerry Brandis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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