The authors have declared that no competing interests exist.

The isohydricity conditions are formulated for D+T systems composed of titrand D and titrant T, mixed according to titrimetric mode; only acid-base equilibria are involved there. The original method of dissociation constants determination, based on the isohydricity principle, is presented and confirmed experimentally. The pH titrations in the system of isohydric solutions are also put in context with conductometric titrations.

Titrimetric methods of analysis are commonly involved with mixing the solutions of two substances endowed with opposite properties, e.g., acid with base (B ⇨ A), or _{1} ⇨ A_{2}) or two bases (B_{1} ⇨ B_{2}) according to titrimetric mode is usually not practiced. However, the special, isohydricity property is attributed here to solutions of two different acids, A_{1} and A_{2}, or two different bases, B_{1} and B_{2}, having equal pH values, i.e., the term “isohydric” refers to solutions of the same hydrogen-ion concentration, [H^{+1}]. After Arrhenius, such a pair of solutions is termed as isohydric solutions

However, the Arrhenius’ statement

As an example _{1} = 10^{-2.5} ≈ 0.003 mol/L HCN and C_{2} = 1 mol/L AgNO_{3}. From the approximate formulae: [H^{+1}] = (C_{1} ⋅ K_{1})^{1/2} for HCN (C_{1}) and [H^{+1}] = (C_{1} ⋅ K_{1}^{oH} ⋅K_{W})^{1/2} for AgNO_{3} (C_{2}) we get pH=5.85, for both solutions; (AgOH) = K_{1}^{OH} [Ag^{+1}][OH^{-1}]), logK_{1}^{OH }= 2.3; [H^{+1}][CN^{-1}] = K_{1}(HCN), pK_{1} = 9.2; K_{W}=[H^{+1}][OH^{-1}], pK_{W} = 14. However, as were stated in ^{+1} ions when added into HCN solution act as a strong acid generating protons, mainly in the complexation reaction Ag^{+1} + 2HCN = Ag(CN)_{2}^{-1} + 2H^{+1}, and pH of the mixture drops abruptly. The degrees of dissociation of HCN is then changed, contrary to Arrhenius’ statement

Despite some appearances arising from the wording, the isohydricity concept introduced by Arrhenius was not involved with hydrogen ions, but with conductivity, K

Correct equations, expressing the isohydricity property, were formulated by Michałowski and presented, for different systems, in a series of papers

The pH change resulting from addition of a strong acid HB (C) into a weak acid HL (C_{0}), is characterized by equation for titration curve

V = V_{0 }⋅ (α-δ⋅C_{0})/(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}]) = K_{1}/(K_{1} + [H^{+1}]), K_{W} = [H^{+1}][OH^{-1}],

and K_{1} = [H^{+1}] [L^{-1}]/[HL] …..(2)

As results from _{1} value and on the relative concentrations (C_{0}, C) of both acids: HL and HB. Under special conditions, expressed by the set of (C_{0}, C, pK_{1}) values _{0}/C; it is just the subject of the next section.

Generalizing, the D+T mixture may appear pH = const. during the titration T(V) ⇨ D(V_{0}) only at defined relation between molar concentrations of components in D and T, as presented below.

_{0}) and HL (C_{0}) ⇨ HB (C)

The simplest isohydric system is composed of a strong monoprotic acid HB and a weak monoprotic acid HL with K_{1} expressed by Eq. 2. We derive first the isohydricity relation for the titration HB (C,V) ⇨ HL

(C_{0},V_{0}), where V_{0} of C_{0} 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/(V_{0}+V) ….(4)

(HL) + (L^{-1}) = C_{0}V_{0}/(V_{0}+V) ….(5)

we get

[H^{+1}] – [OH^{-1}]

where

i.e.,

(see Eq. 2).

Mixing the solutions according to titrimetric mode can be made in _{1} (Eq. 2) and K_{W} = [H^{+}^{-}^{1}] values. This way, the terms: [H^{+1}] – [OH^{-}^{1}] = [H^{+1}] – K_{W}/[H^{+1}] and

[H^{+1}] – [OH^{-1}]= (1-n)⋅C0

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

K_{1}/[H^{+1}] + K_{1} = C/C_{0 }⇨ [H^{+1}] = K_{1} ⋅ (C_{0}/C-1) …..(12)

[OH^{-1}] = K_{W}/K_{1}⋅ (C_{0}/C-1)^{-1} ….(13)

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

K_{1} ⋅ (C_{0}/C-1)= C ⇨ C_{0} = C+C^{2}/K_{1}⇨ C_{0} = C+C^{2} ⋅ 10^{pk}_{1 } …...(14)

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

K_{1}⋅ (C_{0}/C-1) - K_{W}/K_{1}⋅ (C_{0}/C-1) = C

Denoting K_{1}⋅(C_{0}/C-1)= y, we have: y – K_{W}/y – C = 0 ⇨ y^{2} – C⋅y – K_{W} = 0 ⇨ y = C/2 ⋅ (1+(1+4K_{W}/C^{2})^{1/2}) ...(15)

as the positive root. At 4K_{W}/C^{2}<< 1, from Eq. 15 we get y = C, i.e.

K_{1}⋅ (C_{0}/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

Identical formula is obtained for reverse titration, HL (C_{0},V) ⇨ HB (C,V_{0}), where V_{0} of C mol/L HB is titrated with V mL of C_{0} mol/L HL. From Eq, 2 and [B^{-1}] = CV_{0}/(V_{0}+V) , ^{HL} + [L^{-1}] = C_{0}V/(V_{0}+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 (C_{0}, C, pK_{1}), where Eq. 14 is valid, independently on the volume V of T added; it is identical for titrations: HB (C,V) ⇨ HL (C_{0},V_{0}) and HL (C_{0},V) ⇨ HB (C,V_{0}).

The related curves expressed by Eq. 14 are plotted in _{1} within (pC, pC_{0}) coordinates. The curves appear nonlinearity for lower pK_{1} values and are linear, with slope 2, for pK_{1} greater than ca. 6. This regularity can be stated from Eq. 14 transformed as follows:

C_{0} = C_{2}/K_{1} ⋅ (1+K_{1}/C) ⇨

pC_{0 }= 2 ⋅ pC - pK_{1 }- log (1+10^{pC-pK}_{1})….(17)

and valid for K_{1}/C <<1 .

It can also be noticed that ionic strength (^{+1}] >> [OH^{-1}], from Equations 3, 11 we get [7]

^{+1}] + [B^{–1}] + [L^{–1}]) = C…..(18)

It is the unique property in titrimetric analyses, exploited in the new method of pK_{1} determination, suggested in _{1} 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 (

The isohydricity concept can be extended on other T (V) ⇨ D (V_{0}) systems, exemplified below.

_{2}SO_{4} (C) ⇨ HCl (C_{0})

From the balances:

α - [HSO_{4}^{-1}] - 2[SO_{4}^{-2}] - [CI^{-1}] = 0;

[HSO_{4}^{-1}] + [SO_{4}^{-2}] = cv/(V_{0}+V); [CI^{-1}] = C_{0}V_{0}/(V_{0}+V)

we get the relation

where ^{+1} attached to SO_{4}^{-}^{2}

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

_{4} (C) ⇨ HCl (C_{0})

From the balances:

We get the relation

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

_{2} (C) ⇨ NaOH (C_{0})

From the balances:

We get the relation

where

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

_{2}CO_{3} (C) ⇨ NaOH (C_{0})

From the balances:

we get the relation

where :

K_{1} = [H^{+1}][HCO_{3}^{-1}]/[H_{2}CO_{3}], K_{2} = [H^{+1}][CO_{3}^{-2}]/[HCO_{3}^{-1}]

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

_{2}COOH (C_{1}) + CClH_{2}COONa (C_{2}) ⇨ HCl (C_{0})

From the relations:

we have, by turns:

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

For example, at pK_{1} = 2.87, C_{1} = 0.1, C_{2} = 0.05, from Eq. 30 we get C_{0} = 0.002505. For pK_{1} = 2.87, C_{0} = 0.025, C_{2} = 0.05, from Eq. 29 we get C_{1} = 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)}] = K_{13}[L_{(3)}H^{+1}]

_{(}_{2)} (C) ⇨ HL_{(1)} (C_{0})

From the balances:

we get

For V = 0, at [H^{+1}] >> [OH^{–1}], from Eq. 31 we have [H^{+1}] =

_{(}_{3)}HB (C) ⇨ HL_{(1)} (C_{0})

From the balances:

we get

For V = 0, at [H^{+1}] >> [OH^{–1}], from Eq. 33 we have [H^{+1}] =

_{(}_{2)} (C) ⇨ ML_{(1)} (C_{0})

From the balances:

we get

For V = 0, at [OH^{–1}] >> [H^{+1}], from Eq. 35 we have [OH^{-1}] =

_{(}_{3)} (C) ⇨ ML_{(1)} (C_{0})

From the balances:

we get

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

_{n}L (C_{0}) and (b) H_{n}L (C_{0}) ⇨ HB (C)

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

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

In particular, for q = n = 1, K_{1} = 1/K_{1}^{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

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 _{i}, for ionic species composing the HB + HL mixture considered in Example 1, we have the formula

k = a_{1}⋅[H^{+1}] + a_{2}⋅[B^{-1}] + a_{3}⋅[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 = (a_{1}+a_{2})⋅[B^{-1}] + (a_{1}+a_{3})⋅[L^{-1}]…..(46)

Assuming, for a moment, thata_{2} = a_{3}, from Eq. 46 we have

K = (a_{1}+a_{3})⋅[B^{-1}] + [L^{-1}]) = (a_{1}+a_{3})⋅[H^{+1}] = (a_{1}+a_{3})⋅10^{-pH} = const…..(47)

at pH = const. At constant ionic strength _{1} and a_{3} are not changed during the titration/mixing. However, the assumption a_{2} = a_{3} is not valid, in general

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

The isohydricity property can be perceived as a valuable tool applicable for determination/validation _{1} 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

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

The principle of the method is illustrated in _{0i}^{*}) related to different C_{0i}^{*} values at constant C value. As we see, a misfit DpK_{i} = pK_{1i}^{*} – pK_{1} between real (pK_{1}) 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 pK_{1}^{*} = pK_{1}, at C_{0}^{*} = C_{0} = C + C^{2}·10^{pk}_{1}.

The validity of some models presented above were verified and confirmed by results of pH-metric and conductometric titrations T (V) ⇨ D (V_{0}), presented in _{2}ClCOOH) and (2) mandelic acid (HL = C_{6}H_{5}CH(OH)COOH) solutions as titrands with HB = HCl (C) as the titrant. All technical details of these titrations are specified therein

(1) pH titration HCl (C) ⇨ CH_{2}ClCOOH (C_{0i}^{*})

(2) pH titration HCl (C) ⇨ C_{6}H_{5}CH(OH)COOH (C_{0i}^{*})

The C and C_{0i}^{*} (i=1,…,5) values are collected in _{11}^{*} = 2,65, we have C_{01}^{*} = 0,051246; at C = 0.00472, pK_{11}^{*} = 3.10 we get C_{01}^{*} = 0.032767. The C_{0i}^{*} values were calculated from Eq. 48 for pK_{1}^{*} values taken from the vicinity of the related pK_{1} value known from the literature data.

HL = chloroacetic acid | ||||

pK_{1i}^{*} |
C | C_{0i}^{*} |
a | b |

2,65 | 0.00965 | 0.05125 | 2.04646 | -0.00942 |

2,75 | 0.00965 | 0.06202 | 2.02252 | -0.00692 |

2.87 | 0.00965 | 0.07868 | 1.95490 | -0.00162 |

2.97 | 0.00965 | 0.09643 | 1.90275 | 0.00664 |

3.10 | 0.00965 | 0.1269 | 1.83071 | 0.01105 |

The pH titrations HB (C) ⇨ HL (C_{0i}^{*}) were made at V_{0} = 3 mL of D and T added up to V = 4 mL. The exact pK_{1}^{o} value was searched here according to interpolation procedure. The results of titrations (_{i} is the slope of the related line (i=1,…,5). The coefficients a and b are calculated according to least squares method from the formulae:

where

For example, linear approximation of the curve in _{11}^{*} = 3.10 (see ^{2})^{1/}^{2},where

HL = mandelic acid | ||||

pK_{1i}^{*} |
C | C_{0i}^{*} |
a | b |

3.10 | 0.00472 | 0.03277 | 2.48438 | - 0.01766 |

3.20 | 0.00472 | 0.04003 | 2.43421 | - 0.01197 |

3.55 | 0.00472 | 0.08377 | 2.28123 | 0.00393 |

3.83 | 0.00472 | 0.15534 | 2.12521 | 0.01417 |

3.93 | 0.00472 | 0.19434 | 2.06462 | 0.02054 |

(where N=200 – number of experimental points (V_{j}, pH_{j})) from the V-interval < 0, 4 > is comparable with precision of pH-measurements. Note that the curve at pK_{11}^{*} = 3.10 (

The slopes b, obtained from the series of n = 5 titrations were applied for evaluation of the true pK_{1} = pK_{1}^{o} value. Assuming the linear relation between b = b_{i} and pK_{1}^{*} = pK_{1i}^{*}, we apply the regression equation

where i = 1,…,n; n = 5. Then we have:

Where _{1}^{*} = pK_{1}^{o} = 0.13954/0.048647 = 2.868 at b = 0.

The pK_{1}^{o} values are related to b = 0. In both cases, an additional, 6th titration made for pK_{16}^{*} = pK_{1}^{o}, obtained by interpolation, confirmed the adequacy of this evaluation (see

The experimental value for pK_{1}^{o} = 2.868 referred to chloroacetic acid agrees with the one cited in literature: 2.87 _{1}^{o} = 3.481 lies within the wide interval: from 3.41

The results of pH titrations presented in _{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 (_{1i}^{*} ca. 2.90. For MA, the parallel course of conductometric titration occurs at pK_{1i}^{*} = 3.55 (_{1}^{*} = 3.481 was obtained (_{1i }=3.55 is not parallel to V-axis (_{2} for Cl^{-1} (see Eq. 47) differs from a_{3} 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 pK_{1} 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| _{V} value. Referring e.g. to addition of V mL of titrant T mol/L HL (C) into V_{0} mL of C_{0} mol/L HB, we apply c = CV/(V_{0}+V); in ideal case β_{V} . The isohydricity is not directly relevant to buffering action; nevertheless, it is

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.

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 (_{1} 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 pK_{1} 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 pK_{1} values, for which the standard method of pK_{1} determination based on inflection point location on the related titration curve obtained for (HL, MOH) system is not applicable