Formulation of the System of Isohydric Solutions

The isohydricity (pH constancy) as the property referred to mixtures of acids or bases, is illustrated on a simple example of the solutions: HL (C0, mol/L) and HL (C mol/L) mixed according to titrimetric mode (pH titration). A new derivation of the Michałowski formula 1 2 0 10 pK C C C    expressing this property is presented, and its applicability for determination of pK1 = – logK1 value is indicated. The principle of the isohydric method of pK1 determination is also outlined.


Introduction
The term "isohydric" refers to solutions of the same hydrogen-ion concentration.According to Arrhenius' statement [1], expressed in more contemporary terms, "if two solutions of the same pH are mixed, pH of the mixture is unchanged, regardless the composition of the solutions".This statement is not valid, however, when referred on any pair of electrolytic systems.
To prove it, let us take, for example, the pair of solutions: C 1 = 10 -2.5 ≈ 0.003 mol/L HCN (pK 1 = 9.2) and C 2 = 1 mol/L AgNO 3 ( = 2.3 for Ag + + OH -= AgOH) OH 1 logK [2].From the approximate formulae: we get pH = 5.85, for both solutions.However, as were stated in [3], silver ions when added into HCN solution act as a strong acid, generating protons mainly in the complexation reaction Ag + + 2HCN = 2 + 2H + , and pH of the mixture drops abruptly.So, the isohydricity property is limited to the systems where only acid-base equilibria are involved.This property was formulated first by Michałowski [4] for different pairs of acid-base systems, then generalized on more complex mixtures of acid-base systems, and extended on mixtures containing basal salts and binarysolvent systems [5].Moreover, the isohydricity concept was the basis for a very sensitive method of determination of dissociation constants values [4,5].

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The present article intends to familiarize the readers with this interesting concept [6] that is in some relevance with buffering action [7,8] and pH-static titration [9,10] principles.Isohydricity concept is also referred to acidbase homeostasis in living organisms [11].

Principle of Isohydricity
It is usually stated that an addition of a strong acid HB (C mol/L) into a weak acid HL (C 0 mol/L) decreases pH value of the resulting solution and, consequently, shifts (retracts) the HL dissociation, according to Le Chatelier's principle.As results from Figure 1, the decrease in pH value (dpH/dV < 0) and then retracting the dissociation of HL occurs at higher C values, whereas the dilution effect, expressed by dpH/dV > 0, predominates at lower C-values.The related effect depends, however, on the HL strength, expressed by its dissociation constant K 1 value.and on the relative concentrations (C 0 , C) of both acids, i.e.HL and HB.Under special conditions, expressed by the set of (C 0 , C, pK 1 ) values [4], pH = const (i.e., dpH/dV = 0) when mixing the solutions in different proportions; it is just the subject of the present note.

Formulation of the Isohydricity Concept
The simplest system of isohydric solutions is composed of a strong monoprotic acid HB and a weak monoprotic acid HL, characterized by pK 1 = -logK 1 value, where K 1 is expressed by Equation (1).In order to derive the relation for the isohydricity concept, let us consider the titration of V 0 mL of C 0 mol/L HL with V mL of C mol/L HB.From charge and concentration balances we get where i.e.
  Mixing the solutions can be made according to titrimetric mode, 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 K 1 and ionic product of water, K W = [H + ][OH -] during the titration in the system of isohydric solutions.This way, the terms: [H + ] -[OH -] = [H + ] -K W /[H + ] and 1n (Equation ( 7)) in Equation ( 5) are constant at any stage of titration in the isohydric system.In particular, at the start for the titration, i.e. for V = 0, from (5) we have Comparing the right sides of ( 5) and ( 8), we get, by turns: Then we get, by turns, Identical formula is obtained for reverse titration, where V 0 mL of C mol/L HB is titrated with V mL of C 0 mol/L HL.It means that the isohydricity condition is fulfilled for the set (C 0 , C, pK 1 ) where the relationship ( 14) is valid, independently on the volume of titrant added.The related curves expressed by Equation ( 14) are plotted for different pK 1 in Figure 2 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 the Equation (14) transformed by turns: and valid for pC significantly smaller than pK 1 .It can also be noticed that ionic strength (I) of the solution remains constant during the titration, i.e. it is independent on the volume V of the titrant added.Namely, from ( 2), ( 10)-( 12) we get It is the unique property in titrimetric analyses, exploited in the new method of pK 1 determination, suggested in [4,5].In the light of the Debye-Hückel theory, the constancy in ionic strength (I) is, apart from constancy in For C 0 = 0.1, pK 1 = 4.0 we have pC = 2.507.

An Isohydric Method of Acidity Constant Determination
The isohydricity property can be perceived as a valuable tool applicable for determination [4,5] of acidity constant (pK 1 = -logK 1 ) for a weak acid HL, see Equation 1.For this purpose, a series of pairs of solutions (HB (C), HL ( )) is prepared, where C and are interrelated in the formula where 1i (i = 1,•••, n) are the numbers chosen from the vicinity of the true pK 1 value for HL.Within each pair (HB (C), HL ( )), the pH titrations HB (C, V)  HL ( C , V 0 ), are made (see Figure 3).In [4,5], at V 0 = 3 mL, n = 5 titrations were made up to V = 4 mL.The results of titrations are approximated by straight lines pH = a i + b i •V (see Figure 4), where b i is the slope of the related line.The plots of the slopes vs.

Final Comments
The new derivation of the formula (14) expressing the isohydricity condition for the simplest case of a mixture composed of strong monoprotic (HB, C mol/L) and weak monoprotic (HL, C 0 mol/L), when mixed according to titrimetric mode, is presented.The roles of titrand and   titrant in such systems can be reversed.The formulae for more complicated systems are given in [4].The isohydricity property can be formulated for the systems where only acid-base equilibria occur Equation ( 14), relates analytical concentrations of strong acid HB and weak acid HL.Such an interesting property is not directly relevant to buffering action.Nevertheless, it is on-line with a general property desired from buffering systems.The systems of isohydric solutions have 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.The conjunction of properties: pH = const, I = const, together with constancy of temperature (T = const) provided a useful tool for a sensitive method of determination of pK 1 values for weak acids, as indicated and applied in [4,5].This method is illustrated with some examples taken from [4].

Figure 2 .
Figure 2. The plots of pC 0 = -logC 0 vs. pC = -logC relationships obtained on the basis of Equation (14), for different pK 1 values indicated at the corresponding lines.
temperature T and dielectric permeability , one of the properties securing constancy in K 1 and K W values. Referring again to Figure 1, from Equation (14) we calculate   line, corresponding to the slope b = 0 (compare with Figure 3) provides an evaluation of the true pK 1 value, 1 pK  = pK 1 .Additional, n + 1 th titration HB (C, V)  HL ( C  , V 0 ), made at 0firmation of this pK 1 value, if b n+1 value obtained in this titration is relatively small.