# Chemical Equilibrium Calculations in Analytical Chemistry

When is a chemical reaction in equilibrium?

There is a very simple test to check if a reaction is in **equilibrium**. If the reaction will be allowed:

- to go in the forward direction up to the point that nothing more appears to happen and all of the
**concentrations**of the species present remain constant and - then the same reaction is allowed to proceed in the reverse direction until nothing more appears to happen and all of the
**concentrations of the species**present remain constant then the**chemical reaction**is in**equilibrium**.

How an equilibrium problem can be solved and the concentrations of the species present be determined?

Solving such a problem consists of a number of steps:

- The nature of all the species present in the solution must be established
- The
**equilibrium constants**relating the concentrations of the various species must be found - Other relations between the unknown concentrations must be found so that there are as many equations as unknowns. This requires the use of
**mass**and**charge balance equations**. - The above described system of n simultaneous equations in n unknown concentrations can be solved for all the unknown concentrations. The exact solution can be tedious and therefore several approximations are made.

What is a mass balance equation?

A **mass balance equation** states that the **number of atoms** of a certain species **must remain constant throughout a chemical reaction**.

Consider, for example, a **saturated solution** of Na_{2}SO_{4} in **distilled water**. If S is the **molar solubility**, the number of **moles** of Na_{2}SO_{4} that dissolve per liter of solution, then:

S = [Na^{+}]

is a **mass balance** on the sodium in solution. It says that the sodium that dissolves is present in the solution only as sodium ions.

A corresponding **mass balance on sulfate** would be:

S = [SO_{4}^{-2}] + [HSO_{4}^{-}]

since the sulfate group normally exists in solution as sulfate ion or as hydrogen sulfate ion.

Let us see as an example a **saturated solution** of **mercuric chloride HgCl _{2}** in water. A

**mass balance**on mercury in such a solution is:

S = [Hg^{+2}] + [HgCl^{+}] + [HgCl_{2}] + [HgCl_{3}^{-}] + [HgCl_{4}^{-2}] + [HgOH^{+}] + [Hg(OH)_{2}]

since the above seven species exist when Hg and Cl are present in a solution. The above **mass balance equation** says that Ηg exists in a solution where Cl is present as the above seven species.

A **mass balance on chlorine** in HgCl_{2} solution would be:

2S = [Cl^{-}] + [HgCl^{+}] + 2[HgCl_{2}] + 3[HgCl_{3}^{-}] + 4[HgCl_{4}^{-2}]

Note that each **mole** of HgCl_{2} dissolving gives two chlorine atoms and therefore the term 2S appears on the left part of the equation. Note also that if a species contains several chlorine atoms (i.e. HgCl_{3}^{-}) we must count this species as many times as there are Cl atoms in it. In this case 3 times.

A special case of mass balance is the so called “**proton condition**”. This is a mass balance on **protons** (**hydrogen ions**) present in a solution of an acid such as HCl of concentration C moles/l.

In this case the proton condition is as follows:

[H^{+}] = C + [OH^{-}]

The above can be obtained by the following reasoning:

“For each OH^{-} one H^{+} is formed by the dissociation of water” and for “each HCl –which is a strong acid and dissociates completely - one H^{+} is formed”.

In the case of HCN solution – HCN is a weak acid and dissociates slightly to produce H^{+} - the proton condition is given by:

[H^{+}] + [HCN] = [OH^{-}]

In this case all the proton species are written on the left-hand side while the proton deficient species on the right-hand side.

What is a charge balance equation?

A **charge balance equation** expresses the fact that a solution containing** ions** must be electrically neutral. This relation is obtained by counting the total number of positive charges per unit volume and setting it equal to the total number of negative charges per unit volume.

Consider for example a solution containing C moles of hydrochloric acid per liter of solution. A **charge balance equation** is given by:

[H^{+}] = [Cl^{-}] + [OH^{-}]

Since all [Cl^{-}] comes from the HCl which dissociates completely, then [Cl^{-}] = C and the above equation becomes:

[H^{+}] = C + [OH^{-}]

For a solution containing C moles of NaCN the charge balance equation is given by:

[H^{+}] + [Na^{+}] = [OH^{-}] + [CN^{-}]

**Mass and charge balance equations**, as the ones shown above, will be used in order to calculate a **general relation for the pH of a strong acid** in the post entitled **“Strong Acids and Bases – Ionic Equilibrium – A general relation for the pH of a strong acid”**.

__Relevant Posts__

Ionic equilibrium - A general expression of the pH of a strong acid

pH of a strong acid – Examples

Strong Acids & Bases: pH Calculations involving mixtures of strong acids and bases

__References__

- J-L. Burgot “Ionic Equilibria in Analytical Chemistry”, Springer Science & Business Media, 2012
- J.N. Butler “Ionic Equilibrium – Solubility and pH calculations”, Wiley – Interscience, 1998
- J.N. Butler “Ionic Equilibrium – A mathematical approach”, Addison-Wesley Publishing Company Inc., 1964

__Key Terms__

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__equilibrium__,__ionic equilibrium__,__mass balance equation__**,**

__charge balance equation__**,**

__equilibrium constant__
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