Formulation of Isohydricity Conditions in Acid-Base systems - A Review

The pH change resulting from addition of a strong acid HB (C) into a weak acid HL (C₀), is characterized by equation for titration curve

V = V₀ ⋅ (α-δ⋅C₀)/(C- α ) ….(1)

where:

α = [H⁺¹] – [OH^-1] = [H⁺¹] – K_W/[H⁺¹] = 10^-pH – 10^pH-14, δ = [L^-1]/([HL] + [L^-1]) = K₁/(K₁ + [H⁺¹]), K_W = [H⁺¹][OH^-1],

and K₁ = [H⁺¹] [L^-1]/[HL] …..(2)

As results from Figure 1, the decrease of pH value (dpH/dV < 0) occurs at higher C values, whereas the dilution effect, expressed by dpH/dV > 0, predominates at low C-values. Generalizing, the related effect depends on the dissociation constant K₁ value and on the relative concentrations (C₀, C) of both acids: HL and HB. Under special conditions, expressed by the set of (C₀, C, pK₁) values ⁷, we have pH = const (i.e., dpH/dV = 0) when mixing the solutions in different proportions C₀/C; it is just the subject of the next section.

Figure 1.The pH-effect of addition of V mL of strong acid HB (C) into V0 = 10 mL of C0 = 0.1 mol/L HL (pK1 = 4.0). The titration curves are plotted here for indicated pC = – logC values, pC = 1, 2, 3, ∞ 7.

Generalizing, the D+T mixture may appear pH = const. during the titration T(V) ⇨ D(V₀) only at defined relation between molar concentrations of components in D and T, as presented below.

Example 1. HB (C) ⇨ HL (C₀) and HL (C₀) ⇨ HB (C)

The simplest isohydric system is composed of a strong monoprotic acid HB and a weak monoprotic acid HL with K₁ expressed by Eq. 2. We derive first the isohydricity relation for the titration HB (C,V) ⇨ HL

(C₀,V₀), where V₀ of C₀ mol/L HL is titrated with V mL of C mol/L HB; V is the total volume of HB (C) added up to a defined point of the titration. From charge and concentration balances

[H⁺¹] – [OH^-1] = [B^-1] + [L^-1] ….(3)

[B^-1] = CV/(V₀+V) ….(4)

(HL) + (L^-1) = C₀V₀/(V₀+V) ….(5)

we get

[H⁺¹] – [OH^-1] ….(6)

where

…….(7)

i.e.,

…..(8)

(see Eq. 2).

Mixing the solutions according to titrimetric mode can be made in quasistatic manner, under isothermal conditions; it enables some changes in equilibrium constants, affected by thermal effects, to be avoided. As will be seen later, the ionic strength (I) of the related mixture is also secured; it acts in favour of constancy of K₁ (Eq. 2) and K_W = [H⁺¹][OH^-1] values. This way, the terms: [H⁺¹] – [OH^-1] = [H⁺¹] – K_W/[H⁺¹] and (Eq. 8) in Eq. 5 are constant at any V- value. In particular, at the start for the titration, V = 0, from Eq. 6 we have

[H⁺¹] – [OH^-1]= (1-n)⋅C0 (9)

Comparing the right sides of Equations 6 and 9, we get, by turns:

From Equations 9, 10

[H⁺¹] – [OH^-1]= C….(11)

From Equations 8, 10

K₁/[H⁺¹] + K₁ = C/C₀ ⇨ [H⁺¹] = K₁ ⋅ (C₀/C-1) …..(12)

[OH^-1] = K_W/K₁⋅ (C₀/C-1)^-1 ….(13)

Assuming [H⁺¹] >> [OH^-1] in Eq. 11, from [H⁺¹] = C, and Eq. 12 we get

K₁ ⋅ (C₀/C-1)= C ⇨ C₀ = C+C²/K₁⇨ C₀ = C+C² ⋅ 10^pk₁ …...(14)

Alternately, after insertion of Equations 12 and 13 in Eq. 11 we have

K₁⋅ (C₀/C-1) - K_W/K₁⋅ (C₀/C-1) = C

Denoting K₁⋅(C₀/C-1)= y, we have: y – K_W/y – C = 0 ⇨ y² – C⋅y – K_W = 0 ⇨ y = C/2 ⋅ (1+(1+4K_W/C²)^1/2) ...(15)

as the positive root. At 4K_W/C²<< 1, from Eq. 15 we get y = C, i.e.

K₁⋅ (C₀/C-1) = C …..(16)

and then we obtain Eq. 14 again.

After mixing isohydric solutions of HL and HB at any proportion, the degree of HL dissociation (see Eq. 8)

(see Eq. 1) is not changed.

The property, expressed by Eq. 14, was formulated first by Michałowski for different pairs of acid-base systems ^{3, 4}, then generalized on more complex mixtures, and extended on mixtures containing basal salts and binary-solvent media ^{4, 17}. Moreover, the isohydricity concept was the basis for a very sensitive method of determination of dissociation constants values ^{3, 4}.

Identical formula is obtained for reverse titration, HL (C₀,V) ⇨ HB (C,V₀), where V₀ of C mol/L HB is titrated with V mL of C₀ mol/L HL. From Eq, 2 and [B^-1] = CV₀/(V₀+V) , ^HL + [L^-1] = C₀V/(V₀+V), we get, by turns,

at [H⁺¹] >> [OH^-1]. Then we have Eq. 10, and then Eq. 14. It means that the isohydricity condition is fulfilled for the set (C₀, C, pK₁), where Eq. 14 is valid, independently on the volume V of T added; it is identical for titrations: HB (C,V) ⇨ HL (C₀,V₀) and HL (C₀,V) ⇨ HB (C,V₀).

The related curves expressed by Eq. 14 are plotted in Figure 2, for different pK₁ within (pC, pC₀) coordinates. The curves appear nonlinearity for lower pK₁ values and are linear, with slope 2, for pK₁ greater than ca. 6. This regularity can be stated from Eq. 14 transformed as follows:

Figure 2.The plots of pC0 = – logC0 vs. pC = – logC relationships obtained on the basis of Eq. 14, for different pK1 values indicated at the corresponding lines 3.

C₀ = C₂/K₁ ⋅ (1+K₁/C) ⇨

pC₀ = 2 ⋅ pC - pK₁ - log (1+10^pC-pK₁)….(17)

and valid for K₁/C <<1 .

It can also be noticed that ionic strength (I) in the isohydric system (HB, HL) remains constant during the titration, i.e., it is independent on the volume V of the titrant T added. Namely, at [H⁺¹] >> [OH^-1], from Equations 3, 11 we get [7]

I = 0.5([H⁺¹] + [B^–1] + [L^–1]) = C…..(18)

It is the unique property in titrimetric analyses, exploited in the new method of pK₁ determination, suggested in ^{3, 4}. According to Debye–Hückel theory, the constancy in ionic strength (I) is, apart from constancy in temperature T and dielectric permeability Ԑ, one of the properties securing constancy of K₁ and K_W values. The systems of isohydric solutions (HL, HB) have then a unique feature, not stated in other acid-base systems; it is the constancy of ionic strength (I), not caused by presence of a basal electrolyte ^{19, 20, 21, 22, 23, 24}.

The isohydricity concept can be extended on other T (V) ⇨ D (V₀) systems, exemplified below.

Example 2. H₂SO₄ (C) ⇨ HCl (C₀)

From the balances:

α - [HSO₄^-1] - 2[SO₄^-2] - [CI^-1] = 0;

[HSO₄^-1] + [SO₄^-2] = cv/(V₀+V); [CI^-1] = C₀V₀/(V₀+V)

we get the relation

….(19)

where is the mean number of protons H⁺¹ attached to SO₄^-2

At V = 0, from Eq. 19 we have C₀ = [H⁺¹] at H⁺¹] >> [OH^-1]. Then we obtain, by turns,

Example 3. NaHSO₄ (C) ⇨ HCl (C₀)

From the balances:

We get the relation

At V = 0, from Eq. 22 we have C₀ = [H⁺¹] at [H⁺¹] >> [OH^-1]. Then we obtain, by turns,

Example 4. Ba(OH)₂ (C) ⇨ NaOH (C₀)

From the balances:

We get the relation

…..(24)

where

At V = 0, from Eq. 24 we have C₀= [OH^-1] at [H⁺¹] <<[OH^-1]. Then we obtain, by turns,

Example 5. Na₂CO₃ (C) ⇨ NaOH (C₀)

From the balances:

we get the relation

where :

K₁ = [H⁺¹][HCO₃^-1]/[H₂CO₃], K₂ = [H⁺¹][CO₃^-2]/[HCO₃^-1]

At V = 0, from Eq. 26 we have C₀ = - α = [OH^-1] at [H⁺¹] << [OH^-1]. Then we get, by turns,

Example 6. CClH₂COOH (C₁) + CClH₂COONa (C₂) ⇨ HCl (C₀)

From the relations:

we have, by turns:

At V = 0, from Eq. 28 we have C₀ = α = [H⁺¹] at [H⁺¹] >> [OH^-1]. Then we get, by turns,

For example, at pK₁ = 2.87, C₁ = 0.1, C₂ = 0.05, from Eq. 30 we get C₀ = 0.002505. For pK₁ = 2.87, C₀ = 0.025, C₂ = 0.05, from Eq. 29 we get C₁ = 0.0998.

In further examples: 7 – 10 we apply the notation

where

[H⁺¹][L_(i)^-1] = K_1i[HL_(i)] (i=1,2) ; [H⁺¹][L₍₃₎] = K₁₃[L₍₃₎H⁺¹]

Example 7: HL₍₂₎ (C) ⇨ HL₍₁₎ (C₀)

From the balances:

we get

For V = 0, at [H⁺¹] >> [OH^–1], from Eq. 31 we have [H⁺¹] = and then, by turns,

Example 8: L₍₃₎HB (C) ⇨ HL₍₁₎ (C₀)

From the balances:

we get

For V = 0, at [H⁺¹] >> [OH^–1], from Eq. 33 we have [H⁺¹] = and then, by turns,

Example 9: ML₍₂₎ (C) ⇨ ML₍₁₎ (C₀)

From the balances:

we get

For V = 0, at [OH^–1] >> [H⁺¹], from Eq. 35 we have [OH^-1] = and then, by turns,

Example 10: L₍₃₎ (C) ⇨ ML₍₁₎ (C₀)

From the balances:

we get

For V = 0, at [OH^–1] >> [H⁺¹], from Eq. 37 we have [OH^–1] = and then, by turns:

Example 11: (a) HB (C) ⇨ H_nL (C₀) and (b) H_nL (C₀) ⇨ HB (C)

Assuming that the acid H_nL forms the species H_iL^+i-n (i = 0, 1, … , q), we get the charge and concentration balances:

….(39)

Applying the function

expressing the mean number of protons attached to the basic form L^-n, where

from Eq. 39 we get , by turns :

Eq. 41 is also obtained for the reverse titration (b), where we get, by turns:

Assuming [H⁺¹] >> [OH^-1], from Eq. 42 we get [H⁺¹] = C. Putting it into (3), from (7) we get

…..(43)

In particular, for q = n = 1, K₁ = 1/K₁^H, from Eq. 43 we get the relation

transformed into Eq. 14.

The diversity in meaning the isohydricity term, referred to pH and conductivities, made an inevitable inconsistency/controversy, indicated above. Conductivity κ = 1/ρ (ρ – resistivity) of a solution is a sum of terms involved with all cationic and anionic species contributing the current passing through the solution ²⁵

…...(44)

where z_i – charge (in elementary charge units), and u_i – ionic mobility for i-th ionic species, X_j^zi, F – Faraday constant; each ion contributes a term proportional to its concentration [X_j^zi]. The property (44) is valid at low concentrations, where interactions between ions can be neglected. Ionic interactions in more concentrated solutions can alter the linear relationship between conductivity and concentrations. Denoting |z_i|·u_i·F = a_i, for ionic species composing the HB + HL mixture considered in Example 1, we have the formula ⁴

k = a₁⋅[H⁺¹] + a₂⋅[B^-1] + a₃⋅[L^-1]…..(45)

From the simplified charge balance [H⁺¹] = [L^-1] + [B^-1], valid at [H⁺¹] >> [OH^-1], from Eq. 45 it results that

k = (a₁+a₂)⋅[B^-1] + (a₁+a₃)⋅[L^-1]…..(46)

Assuming, for a moment, thata₂ = a₃, from Eq. 46 we have

K = (a₁+a₃)⋅[B^-1] + [L^-1]) = (a₁+a₃)⋅[H⁺¹] = (a₁+a₃)⋅10^-pH = const…..(47)

at pH = const. At constant ionic strength I (this property is immanent in such isohydric system, see Eq. 18), a₁ and a₃ are not changed during the titration/mixing. However, the assumption a₂ = a₃ is not valid, in general ⁴.

In experimental part of the paper, the results from pH titrations (in aqueous and mixed-solvent media) will be compared with results from conductometric titrations.

The conjunction of properties: pH = const, I =  const, together with constancy of temperature (T = const), as stated above, provided a useful tool for a sensitive method of determination of pK1 values for weak acids HL, as indicated and applied in ^{3, 4}. This method is illustrated with some examples taken from ⁴.

The isohydricity property can be perceived as a valuable tool applicable for determination/validation ^{3, 4} of pK₁ for a weak acid HL. For this purpose, a series of pairs of solutions {HB (C), HL (C_0i^*)} (i=1,…,n) is prepared, where C and C_0i^* are interrelated in the formula

…..(48)

where pk_1i^* (i = 1,…, n) are the numbers chosen from the vicinity of the true (expected, correct) pK₁ value for HL (compare with Eq. 14). From Equations 14 and 48 we have the relation

…….(49)

The principle of the method is illustrated in Figure 3, where simulated titration curves pH = pH(V) are obtained for titrations HB (C) ⇨ HL (C_0i^*) related to different C_0i^* values at constant C value. As we see, a misfit DpK_i = pK_1i^* – pK₁ between real (pK₁) and pre-assumed (pK_1i^*) values for acidity constant causes a non-parallel, to V-axis, course of the related curve pH = pH(V); the curve/line is parallel to the V-axis only for pK₁^* = pK₁, at C₀^* = C₀ = C + C²·10^pk₁.

Figure 3.The pH vs. V relationships plotted for the titration HB (C) ⇨ HL (C0i*) at pK1 = 2.87 for HL; V0 = 3, C = 0.01, C0i* calculated from Eq. 48 at indicated pK1i* values.

The results of pH titrations presented in Figure 4a and 4b can be compared with results of conductometric titrations specified in ⁴ for the (HCl, HL) systems with (a) HL = chloroacetic acid (CA), (b) HL = mandelic acid (MA). First, one can state that the conductomeric titration curves are arranged in the reverse order than pH titration curves, from the viewpoint of changes in C_0i^* values; this is understandable because higher pH values correspond to lower [H⁺¹] values. Moreover, the conductometric titration curves have more regular course than pH titration curves. At pK_1i^* = 2.87, the pH titration curve for CA is nearly parallel to V-axis (Figure 4a), whereas for conductometric titration the parallel course can be ascribed to the line obtained at pK_1i^* ca. 2.90. For MA, the parallel course of conductometric titration occurs at pK_1i^* = 3.55 (Figure 4b), whereas from results of pH titrations the value pK₁^* = 3.481 was obtained (Table 1); the pH titration curve for MA at pK_1i =3.55 is not parallel to V-axis (Figure 4a). This means that a₂ for Cl^-1 (see Eq. 47) differs from a₃ for anions L^-1 related to CA and MA, respectively. However, the differences are not too large and the conductometric titration, offering more regular course of the respective curves, can be considered as a reasonable alternative to the pH titration made within the isohydric method of pK₁ determination.

Constancy of pH during addition of one of the solutions forming the isohydric system into another one recalls the concepts of buffering action and the dynamic buffer capacity β_V = |dc/dpH| ^{1, 2, 34, 35, 36}. Isohydric systems are characterized by extremely high β_V value. Referring e.g. to addition of V mL of titrant T mol/L HL (C) into V₀ mL of C₀ mol/L HB, we apply c = CV/(V₀+V); in ideal case β_V  . The isohydricity is not directly relevant to buffering action; nevertheless, it is on–line with a general property desired from buffering systems. The isohydricity concept is also in some relevance with pH-static titration ^{37, 38} principle. Moreover, the isohydricity concept is referred to acid-base homeostasis in living organisms ^{39, 40}. In biology, it is used for describing plants that limit transpiration in order to maintain a constant amount of water in the leaves ^{41, 42, 43, 44}. The isohydric principle has special relevance to in vivo biochemistry, where multiple acid-base pairs are in solution.

Some acids involved in redox (e.g. HClO, HBrO) and complexation equilibria do not meet the conditions imposed by the isohydricity property, see e.g. ^{45, 46, 47, 48}, and other authors’ references cited therein.

The formulation referred to the isohydric D+T acid-base systems formed from D and T of different complexity was presented. Particularly, the titration in (HL, HB) system may occur at constant ionic strength (I) value, not resulting from presence of a basal electrolyte. This very advantageous conjunction of the properties provides, among others, a new, very sensitive method for verification of pK₁ value for HL. The method was tested experimentally on (HL, HCl) systems in aqueous and mixed-solvent media, and compared with the literature data. Some useful (linear and hyperbolic) correlations were applied for pK₁ validation purposes.

The isohydric method, formulated on the basis of isohydricity property, is also a proposal for use in physicochemical laboratories, as a sensitive tool for the determination of dissociation constants of weak acids HL, especially ones with small pK₁ values, for which the standard method of pK₁ determination based on inflection point location on the related titration curve obtained for (HL, MOH) system is not applicable ⁴. The reference of isohydricity to conductometric titrations was also discussed in context to the pH-metric titration.