# 10.4: The Equations

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- 10929

The whole purpose of Stewart's model is to discover what determines the \(\ce{[H^{+}]}\) (and thus pH) in aqueous solutions such as body fluids. Lets look at two simple systems to gain some experience in deciding what determines the \(\ce{[H^{+}]}\) in these systems.

Example \(\PageIndex{1}\): Pure Water

Consider first a solution of pure water and ask the question here: What determines the [H^{+}]?

**Solution**

We can determine a formula for this as follows:

Water dissociates into H^{+} and OH- to a very small degree:

\[\ce{H2O <=> H^{+} + OH^{-}}\nonumber \]

The dissociation equilibrium equation for this reaction is:

\[\ce{[H^{+}] \times [OH^{-}] = K_{w} \times [H_{2}O]} \nonumber \]

where \(K_w\) is the **dissociation constant** for water.

The value for Kw is temperature dependent. The term [H_{2}O] is very large (55.5M at 37°C) and the values of \(\ce{[H^{+}]}\) and [OH^{-}] are both very small: that is water dissociates to such a very small extent that the value of [H_{2}O] is essentially constant. The terms Kw and [H_{2}O] can be combined into a new constant K'w.

Kw which is called the **ion product for water**. Thus:

\[ K_{w} = \ce{[H^{+}]} \times \ce{[OH^{-}]} \nonumber \]

Electrical neutrality must also be present in the solution. As H^{+} and OH- are the only ions present:

\[[H^{+}] =[OH^{-}] \nonumber \]

These 2 simultaneous equations have two unknowns so a solution for \(\ce{[H^{+}]}\) is possible:

\[[H^{+}] = (K_{w})^{\frac {1} {2}} \nonumber \]

This is the simplest system possible but illustrates the point that analysis of a system results in several equations that can be solved for [H^{+}].

## Overview of Basic PrinciplesThe basic principles used in analyzing all systems and determining the equation for \(\ce{[H^{+}]}\) are simple: - Electroneutrality must be conserved
- Mass must be conserved
- All dissociation equilibriums must be met
The result is a set of simultaneous equations which may be solved. No matter how complex the solution, all these 3 conditions must be met. |

Example \(\PageIndex{2}\): A Solution of Sodium chloride

Now consider a slightly more complicated system: an aqueous solution containing only Na^{+} and Cl^{-}. This example shows how the SID term arises. What determines the \(\ce{[H^{+}]}\) in this solution?

**Solution**

We can write the following equations for this system:

Water Dissociation Equilibrium:

\[K_{w} = [H^{+}] \times [OH^{-}] \nonumber \]

Electrical Neutrality:

\[[Na^{+}] + [H^{+}] = [Cl^{-}] + [OH^{-}] \nonumber \]

Solving for [H^{+}]:

\[ [Na^{+}] - [Cl^{-}] =[OH^{-}] -[H^{+}] \nonumber \]

\[[OH^{-}] = \frac {K_{w}} {H^{+}} \nonumber \]

Combining these:

\[[H^{+}]^{2} + [H^{+}]([Na^{+}] - [Cl^{-}])-K_{w} = 0 \nonumber \]

Now \([Na^{+}] - [Cl^{-}] = SID\) for the solution in this example, so:

\[ [H^{+}]^{2} + (SID \cdot [H^{+}]) - K_{w} =0 \nonumber \]

Solving this quadratic equation, the 2 solutions are:

\[[H^{+}] = \frac {-SID} {2} + \sqrt {K_{w} + \frac {SID^{2}} {4}} \nonumber \]

and

\[[H^{+}] = \frac {-SID} {2} - \sqrt {K_{w} + \frac {SID^{2}} {4}} \nonumber \]

For solutions containing Na^{+} and Cl- in water, the \(\ce{[H^{+}]}\) is determined by the SID **alone** (as this is the only variable on the right hand side of the equation)! This simple example illustrates how the SID term is useful as a independent variable which arises out of the equations used to analyse the chemical systems in body fluids.

## The Equation Set for Body Fluids

The preceding two examples outline the approach that can be taken with any aqueous solution. Even though body fluids are much more complex, Stewart was able to find the equations which describe the system and solve them for [H^{+}].

Body fluids are aqueous solutions which contain strong ions (inorganic and organic) and weak ions (the volatile CO_{2}/HCO_{3}^{- }system and various non-volatile weak acids HA). The independent variables which determine the \(\ce{[H^{+}]}\) in all body fluids are the pCO_{2}, SID and [A_{Tot}].

All the other variables (eg [H^{+}], [OH^{-}], [HCO_{3}^{-}], [A^{-}] ) are dependent on the values of the 3 independent variables. There are six simultaneous equations necessary to describe this system (see table below)

*A full discussion and derivation of these equations is not presented here: the interested reader is referred to Peter Stewart's book "How to Understand Acid-Base" (1981)*

## The Six Simultaneous Equations used by Stewart |
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1. Water Dissociation Equilibrium |

\[[H^{+}] \times [OH^{-}] = K_{w}\nonumber \] |

2. Electrical Neutrality Equation |

\[[SID] + [H^{+}] = [HCO_{3}] + [A^{-}] + [CO_{3}^{2-}] + [OH^{-}] \nonumber \] |

3. Weak Acid Dissociation Equilibrium |

\[ [H^{+}] \times [A^{-}] = KA \times [HA] \nonumber \] |

4. Conservation of Mass for "A" |

\[[A_{Tot}]=[HA] + [A^{-}] \nonumber \] [A |

5. Bicarbonate Ion Formation Equilibrium |

\[ [H^{+}] \times [HCO_{3}^{-}] = KC \times pCO_{2} \nonumber \] |

6. Carbonate Ion Formation Equilibrium |

\[ [H^{+}] \times [CO_{3}^{2-}] = K3 \times [HCO_{3}^{-}\nonumber \] |

Equation 5 is the basis of the familiar Henderson-Hasselbalch equation. It is interesting to note that the traditional approach to acid-base physiology uses the Henderson-Hasselbalch equation alone and ignores all the other equations!

The three basic constraints that lead to these six equations are chemical or physical laws that must be obeyed by the system:

- Electrical neutrality must be present in the solution
- Conservation of mass must occur
- All dissociation equilibria must be satisfied simultaneously