The Equiligraph :  Revisiting an old tool

Six months in the laboratory

will save you an afternoon in the library.

Phase diagrams show us the  equilibrium relationships between distinct and identifiable phases of matter, such as solids, liquids and gases. However, they do not give us much information about the equilibrium relationships among chemical species which are present within a given phase. For example, an aqueous phase would be considered a single phase even if it is a complex mixture of chemical species, such as seawater, blood or a solution used to clean electronics after soldering.

The Equiligraph is a graphical representation of the composition of an aqueous solution and shows explicitly how the equilibrium between components changes with respect to pH.  Three examples are shown here.

Figure 1 shows an equiligraph for a non dissociating chemical species, such as sugar or Chloride ion.  Note the [H] and [OH] lines representing the dissociation of water.

Figure 2 shows an equiligraph for acetic acid, which is a weak acid and dissociates slightly in water.

Figure 3 shows the equilibirum beween 5 forms of  EDTA, which is a tetraprotic acid.

Figure 1: Equiligraph of non dissociating species, such as sugar, in water. The concentration is shown as a horizontal line that moves up and down.

The equiligraph is an old tool, developed before computers were invented.  The set of coupled equations representing a monoprotic acid -base equilibrium is 3rd order in [H].

Most analysis resort to simplifying assumptions to make even a monoprotic acid or base tractable.  The order of the equations rises as n+2 for polyprotic acids and bases, making the equations describing a tetraprotic acid 6th order in [H].

The only assumptions that the equiligraph imposes is that the contribution due to the autodissoction of water is negligible.  Even this effect can be accounted for using the equiligraph. Generally, one constructs an equiligraph assuming ideal solutions with all activity coefficients equal to unity., however this is not a requirement.

Figure 2: Equiligraph of  acetic acid, which dissociates slightly in water.  The concentration of the acid [HAc] and base [Ac-] forms change as a function of both pH and total concentration. Note that the lines represent α0 and α1, shifted by CTotal .

The pH of pure acetic acid is found by locating the intersection of the [Ac-] line with the [H] line, which represents the point where autodissocation of the acetic acid produces one acetate ion for every hydrogen ion:

HAc  <--> H  + Ac

where :

[H] = [Ac]

Similarly, the pH of pure sodium acetate is found by locating the intersection of the [HAc] line with the [OH] line:

Ac  + H2O  <-->  HAc + OH

where:

[HAc] = [OH]

The composition of a buffer solution with a given pH is found simply by finding the intersection of a vertical line representing that pH with the [HAc] and [Ac] lines.

Chemical species which dissociate in water add an extra level of complexity. There are two chemical species present, whose total concentration is fixed, but whose relative concentration changes with pH.  Figure 2 shows an equilgraph of acetic acid at two different concentrations.  Note that each concentration shows two lines, which cross at the point pH = pKa.  This set of lines shifts up and down as the total concentration of acetate  (Ctotal = [HAc] + [Ac-]) changes, however the relative concentration of [HAc] / [Ac-] remains constant for any given pH.

Using figure 2, we can easily find the pH of a solution of  pure acetic acid, pure sodium acetate or any mixture (buffer solution) of the two.

The basic concept is shown in figure 1. The Y axis is the base 10 logarithm of the concentration and the X axis is the pH.  Two light gray lines show the concentration of hydrogen [H] and hydroxide ions [OH]. The concentration line of [A] is perfectly flat because the substance shows no acid -base behavior in this pH range.  Changing the concentration simply moves the log [A] line up and down.

One of the most important questions in any technology is:

What change is significant?

The answer to this question is different for every situation, however, answering the question requires an understanding of the sensitivity of the system to changes in conditions.

Getting a global view of the system can provide great insight as to what conditions need to be tightly constrained and monitored.

Figure 3 shows an equiligraph for EDTA or ethylenediamine tetraacetic acid.  The equiligraph provides a global perspective of the equilibria among the 5 forms: [H4B], [H3B], [H2B], [HB] and [B].  The pH of solutions of various compositions is readily identified by intersections of lines, as shown in figure 2. This approach  accounts for closely spaced equilibrium constants and does not require any simplifying assumptions, save that contribution of [H] from the dissociation of water be negligible.

Figure 3: Equiligraph of  EDTA, showing the complextiy of a tetraprotic acid. Generally EDTA solutions are made from the dibasic salt ( Na2 H2B) and only the tetrabasic form is significant for complexation reactions.  it is used to chelate heavy metals in blood, which has a pH near 7.0, where the concentration of the active form [B] is quite low.

Only two simplifying assumptions are commonly used:

1. *The contribution of [H] and [OH] due to the dissociation of water are neglected

2. *Solutions are ideally diulte ( activity coefficient = 1) .

Corrections for both assumptions are simple to derive and apply. For example, non-ideal solutions are corrected using the activity coefficient, 𝛾. The equilibrium constants are defined in using activities. In most cases we assume ideal solutions ( 𝛾 =1), so the activity is equal to the concentration.

Keywords: Chemistry, Acid, Base, Equilibrium Constant, Ka, Kb, K1, Ionic, Aqueous, Weak, Strong, Titration, Buffer Capacity,  alpha, End point, Equivalence point, pH, pKa, pKb, pK1,Composition, Mass Balance, Charge Balance, Coupled Equations, Graphical, Numerical Solution, Monoprotic, Diprotic, Triprotic, Tetraprotic.

aX= 𝛾 * [X]

Ka = aH * aB / aHB

The activity coefficient is always less than one, so non ideal solutions have a greater concentration of a given species. The equiligraph represents the activity of the chemical species in solution, so one simply finds the concentration result using the equiligraph , then divides that concentration by the activity coefficent to get the true concentration in solution.  Simple!

Second, the concentraion of [OH] is always available from the graph.  Figure 1 shows the [H] and [OH] lines for water.

The assumption is that the concentrations are large enough to neglect the effect of auto dissociation of water on the  resultant pH.

[H] = [OH] + [B] - [Na]

If the effect of water cannot be neglected, one siimply reads  [OH] directly from the graph and adds it to [H] read from the graph. Simple!

The fundametnal basis of the equiligraph is the mass balance equation:

CTotal = [HB] + [B]

which is recast in terms of factions, α.

α0 = [HB] / CTotal

α1 = [B] / CTotal

CTotal  = CTotal * α0 +  CTotal * α1

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