Intermediate Equiligraph

It is complicated,

but not as complicated as finding the roots

of an equation 4th order in [H] with 5 free variables!

The equiligraph can be used to find the pH for a given composition, or it can be used to find the composition for a give pH. The latter analysis is quite tricky for polyprotic acids because the solution involves finding the roots of equations of the order [H]n+2   where n is the number of protons.

Closed form solutions only exist for cubic equations, limiting closed for solutions to monoprotic acids or diprotic acids, if a simplifying assumption can be applied.

Figure 1: Equiligraph showing buffer calculation for a solution of [HAc]=  8.0x10 -3 (M)  and [NaAc] =2.0x10- 3 (M).  The alkalinity line deviates from the [Ac] line only at high pH.

Figure 3: Sulfuric acid solutions with A: CT = 0.1 and  B: CT = 0.001 (M).  The effect of the second proton on the solution pH is shown as correction A and correction B. The correction becomes a factor of 2 when CT < K2 - 1.3. The effect of the second equlibrium on the first endpoint ( pH of [NaHSO4] is shown by comparing the intersection of [H2SO4] with [OH] with the arrows indicating the intersection which would occur for a monoprotic acid with the same Ka.

Figure 3 is a very busy graph, however it shows the effects of dilution on the pH of pure acids and salts.  Figure 4 focuses on the pure acids (e.g. H2B) and figure 5 focuses on pure salt solutions ( e.g NaHB).

The charge balance equation reminds us that there are 2 protons released for every SO42- molecule present.

[H] + [Na] = [OH] + [HSO4] + 2*[SO4]

This factor of 2 needs to included when it becomes significant, which generally occurs when [SO4] > 5% CT

Figure 3 shows that the effect of the second equlibrium is significant for the pH of salt solutions, such as the pure monobasic salt NaHSO4.  There are two arrows, one for each concentration, which extrapolate the slope of the [H2SO4] line to the intersection with [OH]. This intersection occurs at the pH of a pure monobasic salt. This is also the pH of the first endpoint of a titration.

Figure 4:  The data shown in figure 3, expanded to emphasize the effect of dilution on the pH of a dilute solution of a pure polyprotic acid.

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.

Buffer Solutions and Alkalinity

Figure 1 shows the equiligraph for a solutions of acetic acid and sodium acetate where the total concetration of all acetate species, CT,  is 1.0x10 -2 (M).

CT = [HAc] + [Ac] =  1.0x10 -2

Superimposed on the graph is the Alkalinity line, which shows how much strong base needs to be added to a solution of pure acetic acid to achieve the desired pH. Note that we neglect [OH] because CT >> [OH].

[Na] = [OH] + CT * α1 - [H]

Figure 1 also shows a horizontal line representing an addition of enough sodium hydroxide to make the soluiton [Na] = 2.0x10 -3 (M) .

The pH of a solution is read from the intersection of the [Na] line with the [Ac] line.

The composition of a solution with a desired pH can be read direcly by finding the [Ac] intersecting a veritical line from the desired pH.

Dilute solutions of weak monoprotic acids

For solutions of pure weak acids ( e.g no added base) , one generally  finds the pH by locating the intersection of the [B] line with the [H] line.  When the concetration of pure acid is smaller than the equlibrium constant, then the acid begins to behave like a strong acid.

This analysis is shown in figure 2.

Figure 2: Effect of dilution on pH of a weak acid. note that a change in concentration greater than 3 orders magnitide (1x10 -2  --> 6.3x10 -6) does not result in a similar pH change (3.4 --> 5.3).

Dilute solutions of weak polyprotic acids

Many, if not most, molecules that have practical value are not monoprotic!

Extending the analysis of monoprotic acids to polyprotic acids requires only that we be mindful of the multiple protons.  Figure 3 shows sulfuric acid, as an example of a polyprotic acid. It exhibits both strong and weak acid behavior.  One proton always appears strong in aqueous solution because it’s pK1 is -3.0 (K1 = 1000 ) and the maximum aqueous concentration it can have is 18 (M). The concentration is always less then K1.

This is not true for the second proton which has pK2 = 2.0 (K2 = 0.01 ).

This analysis is shown in figure 3, where two sulfuric acid solutions are compared. One has a concentration above K2 ( CT = 0.1 (M)) and the other has a concentration below K2 ( CT = 0.001 (M)) .

Figure 4 shows the same data as figure 3, but on a different scale to show the correction more clearly.  The correction factor is always, for sulfuric acid, between 1 and 2 and changes smoothly.  The correction is exaclty 2 when the dominant form is [SO4], which occurs when CT < K2 -1.3.   (log10 (0.05) = 1.3).

When CT > CT < K2 +1.3, the correction is 1.  When  CT = K2, the correction factor = 1.5.  When K2 -1.3 < CT < K2 +1.3, one obtains the correction factor by reading the [HSO4] and [SO4] from the graph and computing the correction.   Frequently one can estimate the correction factor by inspection, noting the distance from the intersection of [HSO4] and [SO4].

Figure 5:  The data shown in figure 3, expanded to emphasize the effect of dilution on the pH of a monobasic salt ( first endpoint of a titration).

Polyprotic acids and bases have more than two forms present in solution. The multiple equilibria affect each other.

The validity of the monoprotic assumption frequently fails. If a molecule has two equilbrium constants which are close ( e.g pK2 - pK1 <= 2), then the effect is extreme, however the effect can be significant even when the equilibria constants are not close.

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