Acid-Base Physiology
10.5 Quantitative Acid-Base Analysis: The Solutions
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:
ax4 + bx3 + cx2 + dx + e = 0
Mathematicians call this type of equation a "4th order polynomial". 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.
A daunting equation but solution is fast and easy on an appropriately programmed computer. 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.
[
For the curious, such 4th order polynomial equations can
be solved on-line. To use
this on-line resource to solve for pH you will of course first need to know the values for the constants.]