Use and Abuse of pH Measurements

N inth Annual Summer Symposium-Analytical Problems in Biological Systems

Use and Abuse of pH Measurements ISAAC FELDMAN Division of Pharmacology, Department o f Radiation Biology, School o f Medic ine and Dentistry, . University o f Rochester, Rochester, N. Y.

The development of the concept of pH from its invention by Serenson in 1909 to the present-day National Bureau of Standards pH scale is reviewed. The liquid junction potential problem involved in the use of the pH meter is discussed in detail. The magnitude of the error due to the junction potential depends upon the pH, ionic strength, nature of the solutes, nature of solvent, pres- ence of colloids, temperature-in fact, anything that affects the mobilities of charged particles. Whether the junction potential error can be ignored depends upon the use to which the pH measurement is put. Some estimates from the literature as to the probable magni- tude of this error are presented and discussed. Under certain relatively simple conditions an approximate relationship may be considered to exist between the pH meter reading and the hydrogen ion concentration. Under other conditions, the pH meter reading must be considered as simply a reproducible reference point having.no theoretical significance. Under some condi- tions, the meter reading cannot even be considered as a reproducible reference point. These points are de- veloped in detail.

4 original hydrogen ion exponent, pH, was invented by THE Sgrenson (S4) in 1909, for the sake of convenience in tabu- lating and discussing hydrogen ion concentrations. He defined i t by thr equation:

CH+ = l o - p H (1)

PI3 - log10 CH (2)

or

I n calculating p~ for a number of buffers, Sgrenson made use of the then-current form of the Nernst equation,

(3)

for the following cell:

P t ; Hf, buffer IiCl salt bridge(0.11- calomel electrode (4)

vheie E , and E , are the electromotive forces, corrected for liquid junction potrntials, for cells containing buffers having hydrogen ion concentrations Cn: and CH;, respectively, and a hydrogen pressure of 1 atm. When CH: equals unity, this equa- tion becomes

E, - Eo p H z ~- 2.3 RT/F ( 5 )

Sgrenson determined Eo by measuring the electromotive force for cells containing hydrochloric acid-sodium chloride mixtures. In calculating the hydrogen ion concentrations in these mixtures, he erroneously applied the Arrhenius belief tha t even strong electrolytes dissociate incompletely. I n addition, a t t ha t time it was not known tha t the electromotive force of cells depends on activities rather than concentrations. Consequently, there is

no direct relationship between Sgrensons p~ and the hydrogen ion activity or concentration.

ACTIVITY pH, OR paH

I n 1924, Sgrenson and Linderstrgim-Lang (56) made use of the concept of activity by defining a new p H term,

paH = - loglo U H + (6)

where U H + represents the hydrogen ion activity. Although the introduction of paH represents a great advance

in chemical thinking, much confusion is encountered in its inter- pretation. The chief difficulty is the impossibility of measuring the activity of a single ionic species without resorting to non- thermodynamic assumptions. Indeed, many assertions are made tha t single ion activities have no physical significance and are therefore meaningless.

Consider the cell, Pt; Hg (1 atm.), HCL, A4gC1; Ag (7)

for which the electromotive force, E, is given by

where EO is the electromotive force of the cell when the activity product, aa+aci-, is unity.

Combining Equations 6 and 8 gives

+ loglo ac1- paH = ___~ 2.3 RT/F E - EO It is obvious from Equation 8 that the electromotive force

of cell 7 depends on the product of the individual ionic activities and that the contribution due to each ion individually cannot be evaluated. Conversely, for this type of cell the electromotive force can be used to determine only the product (aH+aci-), which by definition equals the square of the mean activity, a*. Hence, Equation 9 is of no value in determining QUH vithout some independent means of determining aci -.

S o r is i t possible to determine individual ionic activities simply by separating the hydrogen ions to be measured from the chloride of the reference electrode as in cell 4 This device intro- duces a neu- source of trouble-namely, the liquid junction po- tentials a t the ends of the salt bridge.

The liquid junction potential which exists a t the boundary of two solutions differing in composition is due to a difference in the rates of diffusion of ions of opposite charge. Thus, Then hydrochloric acid diffuses across a boundary, the hydrogen ions move much faster than the chloride ions and tend to produce a positive charge in the solution into n-hich they diffuse. On the other hand, if sodium hydroxide is the migrating substance, the solution into which it migrates tends to become negatively charged, because the negative hydroxyl ions diffuse much faster than do sodium ions. Diffusing salts also produce junction po- tentials but not t o as large an extent as do acids or bases, because hydrogen and hydroxyl ions have much greater mobilities than do other ions.

For cell 4, the electromotive force is given by,

1859

1860 A N A L Y T I C A L C H E M I S T R Y

where E, is the algebraic sum of the two liquid junction poten- tials at the ends of the salt bridge; E: is E , for the hypothetical cell in which both CLH; and UCI; are unity; and the subscripts b and 7 indicate that UH; refers to the hydrogen ion activity in the buffer solution v-hereas acl; refers to the chloride ion activity in the calomel half cell.

Combining Equations 6 and 10 gives:

E = Eo - F l o g l o C & , , ~ R - ~ C I - = Eo - 2 3 RT log,, CHCI f2

It is impossible to determine any one of the terms, E,, UCI;J or U H , ~ without previously knox-ing two of these quantities.

THERMODYNAMIC pH, OR ptH

I n an effort to get around this impasse i t was advocated that pH be defined solely in terms of the most convenient method used for its determinatioIi-i.e., in terms of the electromotive force of a galvanic cell such as cell 4. Such a definition would be thermodynamically rigorous and therefore may be referred to as the thermodynamic pH, or ptH.

Bjerrum and Unmack (9) and Guggenheim and Schindler (14 ) proposed as the fundamental definition,

E - Eo' - E , 2.3 RT/F p tH =

in which Eo' is an experimentally determined constant. This 2.3 RT

is equivalent to setting Eo' equal to (EO - E: - __ F log,, ucl;) of Equation 11. Thus, it would be recognized that, even though they are not rigorously determinable, E: and acl- may be con- sidered as constants for a given cell a t a given temperature. Hence, p t H would be a reproducible number dependent on the hydrogen ion activity if E , could be obtained. They proposed that E , be calculated by the Henderson equation (21). In addi- tion to the fact that it would have to be performed for each pH measurement, such a calculation furnishes only a laboriously obtained approximation of questionable validity (4).

The present-day thermodynamic definition (20) of p H lumps the junction potential term, E,, in with Eu'-i.e.,

E - EEH p tH ~- 2.3 RT/F 2.3 RT + E , - 7 logioacl; where E& = EO' + E , = Eo -

As Harned and Owen have stated (ZO), ' I . . .this equation com- pletely defines a useful p H number, about which no confusion need arise unless an attempt is made to interpret it in terms of. . ." the hydrogen ion concentration. Unfortunately, however, the need does frequently arise nhen one nishes to know the hydro- gen ion concentration. Hence, the p H cannot always be re- garded simply as a variable which must be kept constant during a series of experiments. For this reason, beginning in 1928 with a suggestion by Cohn, Heyroth, and Nenkin (10) as to a possible procedure, there have been several revisions of the pH scale in an attempt to equate, a t least approximately, the hydro- gen ion concentration with the thermodynamic ptH.

MacIY'IES pH

T o the theoretical chemist it would be most pleasing if the thermodynamic hydrogen ion activity coefficient, ft, in the formal expression

p tH = - loglo U H = - log,, CH-.~S~ (14)

were equal to some type of mean activity coefficient, Si. This attitude is understandable, as cell measurements provide only mean activity coefficients. For instance, the electromotive force and f* are accnrately connected by Equation 15.

(15)

This equation holds for cell 7 and also for cell 16,

P t ; H? (1 atm.), HCl(C1) I KC1 salt bridge I HCI(C2), AgCI; Ag (16)

if the hydrochloric acid concentrations C, and CI are equal. When, however, CI # C2 or the right half cell contains a chloride other than hydrochloric acid, ~H:~cI; # f:. The failure to realize this latter inequality has probably led some chemists to the misconception that p tH equals -log,, UH -f*.

Undoubtedly, also many chemists hold the mistaken belief that hIacInnes set up a pH scale in such a way that ft and j , are synonymous. It is true that RIacInnes, Belcher, and Shed- lovsky (27 , 29) , following Cohn's suggestion, used thermo- dynamic dissociation constants as a basis for their pH scale, but they did so in such a way that their p tH is only approximately equal to pH, or -log,, G"+ff,.

RIacInnes and associates found empirically that, for certain dilute bufler solutions in cell 4, the quantity E:H could be selected such that a plot of the left side of Equation 17 u s . the square root of the ionic strength, p, was linear and gave an intercept equal to pK,, the negative logarithm of the thermodynamic dissociation constant of the buffer acid, HS .

where S is an empirical constant, and Cx- and CHX are, respec- tively, the stoichiometric concentrations of buffer salt and acid corrected by the addition and subtraction, respectively, of the computed CH +.

This Equation 17 is similar in form to 18, the rearranged loga- rithmic form of the dissociation constant of a n-eak acid.

where f,, is the activity coefficient of the undissociated weak acid. Hence, for their solutions of ionic strength S0.01, if p tH = -log,, C H ~ ~ * the slope S of Equation 17 should equal the con- stant, A , of the Debye-Huckel equation, 19 and 20 (11, 20):

- AZ: 4; 1 + B a d ; loglofi =

where Zi is the charge on any given ion, and 2, and Z - are, re- spectively, the charges on the cation and anion of a binary elec- trolyte; -4 and B are constants dependent on the temperature and dielectric constant of t,he solvent; and a, is an empirical con- stant related to the distance of closest possible approach between oppositely charged ions.

The S values obtained by MacInnes and associates for acetate and chloroacetate buffers were only 12% higher than those pre- dicted by the Debye-Hiickel theory. This discrepancy would imply a difference of less than 0.01 between p tH and pH, when p jO.01. Hence, it might appear true that their method '' yields pH values n-hich are not equal to -lOgloc'H+f, but are probably as close as they can be adjusted to such equality using. . ." (2.9) the thermodynamic definition of pH. HoiT-ever, one must question their implication that their method ". . .will yield values of p H which are as useful as appears to be possible in the determination of ionization and other equilibrium con- st,ants," for in the ionic strength range of their solutions, to 10- *, the Debye-Hyickel equation gives the same values for logmfk for a uni-univalent electrolyte as for the individual ionic activity coefficient term, lOgia.fH +. That is, in the region, p $0.01, where

V O L U M E 28, N O . 12 , D E C E M B E R 1 9 5 6 1861

the Debye-Huckel limiting law (numerators of Equations 19 and 20) is valid, pH, is equal t o paH. Their work, therefore, must not be interpreted as evidence for the validity of equating p t H with pH, a t higher ionic strengths.

HITCHCOCK-TAYLOR pH

Hitchcock and Taylor (22) used an extended form of the Debye- Hhckel equation,

loglof = - Av$ - B p (21)

to set up a pH scale on the basis of E g H values obtained over a higher ionic strength range, 0.01 to 0.1, than in the work of RIacInnes and associates. hloreover, the former workers studied polyvalent electrolytes, such as the tartrate, citrate, and phosphate buffers, in addition to the uni-univalent buffers studied by the latter.

Replacing the last term of Equation 17 with the right side of Equation 21 and rearranging give Equation 22

CX - E - k p K t - k loglo - - kA& = EO,, + kBp CHX (22)

where k = 2.3 R T / F .

Culciilating A from the Deb!-e-Hiickel theory, Hitchcock and Taylor plotted the left side of Equation 22 against the ionic strength. If such a plot \\-ere linear, the intercept would be E : H . They obtained linear curves for the acetate and the glycollate buffers, but the phosphate and borate buffer plots showed curvature. The accuracy of the EgH extrapolation for the latter buffers is therefore questionable, but to an extent of less than 0.2 mv., corresponding to only 0.003 p H unit.

The meaning of coefficient f in Equation 21, and therefore of the Hitchcock-Taylor p H scale, depends upon the means of obtaining d and B. B is an empirical quantity, which may or may not be constant. I ts value is a function of all the ions af- fecting E and its use therefore gives some sort of a mean activity coefficient nat,ure to f in Equation 21. For the uni-univalent buffers, the value of d would be equal t o the Debye-Huckel constant, A , whether one attempted to compute f, or f+. However, for the phosphate buffer, Hitchcock and Taylor set A = 3.~1. This latter equality would be applicable if Equation 21 w i s used to express the individual-ion activity coefficient ratio iri log,, ( ~ H P o ; / ~ H ~ P o : ). Such a calculation n-odd throw fT, rather than f,, into the E ~ H term. At least for this particular biiffer, it ~ -ou ld seem then that their assigned p H is closer to paH than to pH,, for the effect of the last term of Equation 21 becomes very small as p approaches zero. Hitchcock and Taylor gave no data, other than their assigned pH, for the other poly- vderit buffers xhich they studied.

NATIONAL BUREAU OF STANDARDS pH, OR pHs

The p H scale established by the National Bureau of Standards workers, referred to as the pH, scale, is based upon measurements on cells without liquid junction (2-4, 6 , 16-18). The liquid junc- tion problem will, of course, arise when the pH, scale is used as the reference by other workers 1% ho employ cells containing liquid junctions.

The National Bureau of Standards method of assigning pH, valries consists of three steps (2 , 4). First, for each of three or more portions of the buffer solution with different small concen- trations of added soluble chloride

is determined by measuring the electromotive force of hydrogen- silver chloride cells without liquid junction. By combining Equations 8 and 23 it becomes evident that:

PWH = - loglo f~ +fc 1 - p ~ (24)

Secondly, these pwH values are plotted against the molality of added chloride to give a straight line, the intercept of Rhich corresponds to pwH when the buffer is infinitely dilute with re- spect to chloride (16). This intercept is designated as pwHo. If j & - represents the chloride activity coefficient in a solution of ionic strength equal to that of the buffer when infinitely dilute with respect to chloride, then for the buffer,

-log10 f H -CH = PI1 Ha log!, f E l - (25) Thirdly, the pH, is finally defined as ( -hglofE-CH) in

Equation 25 and is calculated from PRHO by introduction of a conventional, even though nonthermodynamic, scale of in- dividual-ion activity coefficients.

Several reasonable definitions have been proposed for a con- ventional individual-ion activity coefficient. Xone of them may be rigorously tested for validity. In the words of Bates ( 4 ) , the choice should be based for the moat part on convenience and ieasonableness in the light of what is known from the theory of ionic solutions.

On the basis of electrical conductance data, JIacInnes (28) concluded that it would be reasonable to assume (1) that the chloride activity in a solution of any univalent chloride is the same as in a solution of potassium chloride a t the same total concentration and (2 ) that in the latter solution the potassium and chloride ions have the same activity. Combining these two MacInnes assumptions with the definition of the mean activity coefficient of hydrochloric acid, one might estimate hydrogen ion activity coefficients by:

f~ + = f&Jfci- = f & c l / f ~ c ~ j where ~ H C I and fmi refer t o mean activity coefficients of hydrochloric acid and potassium chloride in the particular solution under study.

According to Guggenheim (13) the activity coefficient of an ion of a strong electrolyte, which dissociates into V + positive ions of charge 2, and V - negative ions of charge 2-, is given by:

By this convention, then, the ionic activity coefficients of a single binary electrolyte are equal to each other and also to the mean activity coefficient, f,, of the electrolyte.

Strictly speaking, neither of these assumptions is readilj, applicable to a mixture of electrolytes, such as a buffer solution, in which the mean activity coefficients are not usually known. On the other hand, one may apply the Lewis and Randall hypoth- esis (26) that, in dilute solutions the activity coefficient of any ion depends solely upon the total ionic strength of the solution. Of course, it is now knoyn that this hypothesis may be used onl>- for approximation purposes. For instance, in estimating activity coefficients in mixtures in which mean activity coefficients are unknown, it is frequently of value to employ as a first approsi- mation the mean activity coefficient of a given substance in its pure solution having the same ionic strength as the mixture.

The terms RIacInnes assumptions and Guggenheim assump- tion are therefore used hereinafter to describe the applied mathematical device only. Khether the mean activity coefficient of a given substance in its pure solution or in the given mixture is used n-ill be evident from the folloning definitions: ~ E C I ( m i x t u r e ) refers to the mean activity coefficient of hydrochloric acid in the particular mixture indicated, whereas ~ H C I : H C I , refers to the mean activity coefficient of hydrochloric acid in pure hydrochloric acid solution having the same ionic strength as the mixture under etitdy.

The pH, values for the equimolal phosphate buffer, K H 2 P 0 4 : Na,HPO, = 1 : 1, have been calculated by Bates ( 2 ) as a function of ionic strength-i.e., molality of each phosphate X 4-using each of these three conventions. His results are shown in Figure 1. Using the Guggenheim assumption, jz.1- = ~ H C ~ , H C I , , curve G resulted. Curves 0 3 , 0 8 , and DL were obtained by applying the Debye-Hiickel equation using ( a i = 3) for curve

1862

0 3 , (a , = 8) for 0 8 , and the limiting-lam approximation (jgl- = ~ H ~ P O , - = j%o;- )for DL. Application of the MacInnes assump- tions, using . ~ E C I ( H C I ) and SKCI(KCI), by the present author gave curve M . The values assigned as the pH. values are represented by the dots and were calculated by Bates from Equation 19 using a value of 4.4 as the a, for chloride. The National Bureau of Standards ( 4 ) recommends using as a standard reference solution the mixture xhich is 0.02551 with respect to each phos- phate and which therefore has The assigned pH, value is 6.86 zk 0.01 a t 25".

E 0.1.

A N A L Y T I C A L C H E M I S T R Y

using values of 3.76 and 0.01 for the parameters ai and p", re- spectively (16). The Sational Bureau of Standards ( 4 ) recom- mends as a standard either 0.05X or 0.05 molal potassium phthal- ate, both of which are assigned a pH, of 4.01 i. 0.01 a t 25' C.

Other Kational Bureau of Standard?-recommended buffers and their assigned pH, values can be found in Bates' book (4).

Correspondence from Beckman Instruments, Inc., Fisher Scientific Co., and Hartman-Leddon Co., has informed this author that the Kational Bureau of Standards standards are used by them as the basis for assigning pH numbers to their buffers.

MOLALITY OF EACH PHOSPHATE

Figure 1. Relation of molality to pH of 1 to 1 phosphate buffers at 25' C.

Computed with various estimates of ionic activity coef- ficient (8 , p. 657). See text for explanation of symbols

A t p 5 0.1 all three activity coefficient conventions (Guggen- heim, MacInnes, and Debye-Huckel) furnish pH, values xhich agree within 10.01. The use of bromide or iodide cells (6) in place of chloride cells gave pH, values within 0.015 unit of the National Bureau of Standards assigned values a t p S 0.1. How- ever, above p = 0.1 the pH, values depend significantly on the activity convention used. In Figure 1, however, even at p = 0.2-Le., 0.05 molal in each phosphate- the National Bureau of Standards pH, differs by only 0.005 unit from that calculated using the MacInnes convention, but it is 0.02 unit lower than that obtained using the Guggenheim assumption. These dif- ferences result from the fact tha t the a, used by the National Bureau of Standards to calculate .f& - is much closer to the ai for potassium chloride, 4.1, than to the ai for hydrochloric acid, -6. It should be evident that, in interpreting pH values in terms of C=+, it is necessary to decide which activity conven- tion is to be used.

Bates has plotted the pH, values for the phthalate buffer as a function of molality ( 2 ) . His graph is reproduced here as Figure 2. It is seen that even for 0.1 molar phthalate, applica- tion of the MacInnes and Guggenheim conventions gives pH, values which differ from the Kational Bureau of Standards pH, by only 0.002 and 0.01 unit, respectively. The National Bureau of Standards values, represented by dots, n-ere obtained with the aid of the Huckel extension (83) of the Debye-Huckel equation:

PRACTICAL pH

For most of the pH measurements made by analytical or biological chemists, the glass electrode is employed as the indica- tor electrode and is connected by a potassium chloride salt bridge to a calomel reference electrode. Without going into the theory of the glass electrode, one may consider that it operates as a hydrogen ion electrode in that the surface potential of the glass,

RT Ec, is equal to (Eo - - log, U H + ) ; SO that for the cell used in F commercial pH meters,

Glass electrode 1) standard buffer or test soh . I satd. KC1, HgzC12; Hg (28)

the electromotive force is given by Equation 10. When a p H meter is standardized Kith a standard buffer, actu-

ally a potentiometer circuit is adjusted so that it is in balance when the reading on the output meter equals the pH number assigned to the buffer. The standard buffer is then replaced with a test solution-Le., the solution whose pH is desired-and the practical pH of the test solution is obtained, which is defined by

Et - E , Practical p H = pH, + -__ 2.3 RT/F (29) where E t and E, represent the electromotive force when test solution and buffer solution, respectively, are in the potentiom- eter circuit.

From the point of view of the magnitude of the practical pH measured for a system, the differences betxeen the three modern

4 14

4 I O

4 06

4 02

3 98 "08

3 9 4 0 0 0 2 004 006 008 010

+ G "M 9 D3

Molality

Figure 2. pH of potassium hy- drogen phthalate at 25' C.

Computed with various estimates of ionic activity coefficient ( 8 , p. 656). See text for

explanation of symbols

pH scales ( MacInnee, Hitchcock-Taylor, and National Bureau of Standards) hold almost no signscance for a biological chemist. With one exception, the pH of standard buffers according to these three p H scales agree within the uncertainty of the National Bureau of Standards assigned values-i.e., within 0.01 from pH 3.5 to 9.2; 0.02 below pH 2.1; 0.03 near pH 13-and also within the limit of accuracy of commercial pH meters. The ex-

V O L U M E 28 , N O . 1 2 , D E C E M B E R 1 9 5 6 1863

ception is the 0.1 21 potassium tetroxalate buffer, m-hich was assigned a p H value of 1.45 by Hitchcock and Taylor and 1.52 by the Sational Bureau of Standards investigators. The p H values assigned to standard buffers on the original S@renson p~ scale are about 0.04 pH unit lorn-er than on the three more modern scales.

An understanding of the various standardization scales, hom- ever, is essential for intelligent interpretation of the meaning of the practical pH. The interpretation is discussed in a later section oi this paper.

LIQUID JUNCTION POTENTIAL ERRORS FOR AQUEOUS SOLUTIONS

From theoretical considerations (do), the junction potential across a boundary would be expected to be a function of pH, ionic strength, the nature of the diffusing ions, solvents, tempera- ture, and, in fact, of anything which affects the mobility of an ion in solution.

Because the pH, values assigned to standard buffers are based upon cells without liquid junction, the original standardization of the p H meter is subject to an error due to the junction poten- tial term, E,, - Bys, subscripts referring to the fact that stand- ard buffer is in the p H meter cell. This error is partially com- pensated by the junction potential term, Ej, - which is involved n-hen the test solution is placed in the cell of the meter.

Thus, because E:' = E:,, the residual junction potential E , , - E,t inherent in the use of the p H meter, produces an error Apa( = residual Ej X F/2.3 R T ) in the determination of the prac- tical pH. One cannot calculate the magnitude of this error ac- curately. However, an indication under certain conditions may be obtained.

Bates, Pinching, and Smith ( 7 ) determined the apparent pH, or pHj, for a number of test solutions in the cell with liquid junc- tion;

P t ; HP, test soln. 1 satd. KC1 I NBS phosphate buffer, H,; Pt (30)

For each of the same test solutions they determined pH, using hydrogen-silver chloride cells without liquid junction by the method already described. The difference, p H , - pH,, for each solution is equivalent to t'he 4 p E which would prevail if the p H meter were employed for the p H measurement using the phos- phate buffer as standard.

Their results may be summarized as follom-s: (1) p H j - pH, did not exceed 0.02 unit for any of their 12 buffers having pH, between 2.15 and 10.00; (2) p H j - pH. reached a magnitude of 0.03 unit for four of the six buffers studied having pH, 5 2.10; and (3) p H j - pH, exceeded 0.02 unit (0.05 a t p H 12.62) for three out of the six buffers having pH, 2 11. S o correlation between p H j - pH, and p can be drawn from their data. Hence the larger pH error below pH 2.1 and above p H 10 cannot be at- tributed to an ionic strength effect, but is probably a true p H effert.

Severtheless, ionic strength has a real effect on spa. Its significance, however, depends on the accuracy required of the pH measurement. Some indication of its magnitude can be ob- tained from the electromotive force measurements of Harned (15) using cells I and 11.

(I) Hg; HgZCI?, H C l ( / ~ ~ o ) + l I c l ( ? ? ~ ) , HU; Pt-Ptl Hz, HCl(>?to), HgzC12; Hg

Pt (11) Pt; Hg, HCl(m0) 1 KCl sstd. I H'Jl(?n,) + l ICl (m) , H2; RT

For cell I, E I = - log, (aH-aclA n-here subscripts s F ( a H -aCl - 1 0 ' and o refer t o the hydrochloric acid solutions with and without

salt, respectively. + Ej. RT ( u H - ) ~ For cell 11, EII = 7 log, ~ (aH ' 1 0

By employing known e.m.f. values for cells reversible to potas- sium and chloride ions and the MacInnes assumptions, Harned evaluated the ratio (UCI-)~/(UCI-)~. Inserting this ratio in

EI gave E , = 7 - log, !sq which when subtracted from EII gave values for Ej in cell 11. His results are plotted in Figure 3.

,4 = mdmaddtd salt

Figure 3. E , estimated for cell I1 by Harned

1 K . 0.1 molal HCI DIU IiCl 1L. 1N. 2 K . 2L . 2'V. 3 L . 3 N . 4.v.

The p H equivalents of these data can be translated into A p a values. For instance, if a p H meter were standardized with 0.01 molal hydrochloric acid, the practical p H measured for a solution 0.01 molal in hydrochloric acid and 0.09 molal in sodium chloride would be in error by 0.045 pH unit. For a solution 0.01 molal in hydrochloric acid and 2 0 molal in sodium chloride, this error would be 0.065 pH unit. However, if the meter were standardized originally r i t h the (0 01 molal hydrochloric acid, 0.09 molal sodium chloride) solution, then A p ~ for the (0.01 molal hydrochloric acid, 2.0 molal sodium chloride) solution would be reduced to 0.065 - 0.045 or only 0 020 p H unit. Such deduc- tions from curves 2K, 2 S , and 4 5 suggest that , if potassium chloride or sodium chloride is the predominant constituent in an aqueous solution having a pH between 2 and 12, the variation in A p ~ due to ionic strength is less than 0.04 pH unit R hen the ionic strength is increased from 0.05 to 3!

The lithium chloride curves ( lL , 2L, and 3L) consistently show greater values for EII - E, than do the corresponding sodium and potassium chloride curves. This phenomenon is undoubt- edly due to the fact that the mobility of the lithium ion differs

1864 A N A L Y T I C A L C H E M I S T R Y

trary convention must be employed. This convention should be consistent with the convention used in the original assignment of pH, to the standard buffers.

I n Figure 4 are presented values of -1og1ofHt calculated by various proposed conventions. Because of the junction poten- tial error, the uncertainty of the assigned pH, of standard ref- erence buffers, and the experimental error in the use of the pH meter, it is very unlikely that a measurement of the practical p H can be considered to be more accurate than within 0.02 unit for the simplest systems. Hence, it is evident from Figure 4 that it is immaterial Thich Convention is used to estimate f ~ t a t p 5 0.05. However, as p increases, the divergence of the curves increases.

Because the niobilitj- of the hydrogen ion is five times the mobility of the chloride ion in aqueous solution, it seems justi- fiable to state that no individual-ion activity convention is rea- sonable that would allotv fH-

VOLUME 28, N O . 1 2 , DECEMBER 1 9 5 6 1865

0 05 0 . 1 0 . 2 0 . 3 0 4 0 5

0 007 0 011 0 016 0 020 0 032 0 035

1 . 5 2 . 6 3 5 4 5 z.; L I

In this table, Ama, is the maximum deviation of loglo . f ~ - as given by curve X from the value indicated by any of the loser hve curves of Figure 4 at a given p . A ~ H - equals 100% times the ratio of the maximum deviation of f~ + to the value of f~ * indi- cated by curve -11.

MAGNITUDE OF APPROXlllATION IN pcH = PRACTICAL pH

In analytical chemistry and biochemistry courses and textbooks it is customary to recognize the existence of activity coefficients and then to ignore them blithely in p H calculations, under the assumption that the hydrogen ion concentration is approximately equal to the hydrogen ion activity. Thus, the practical pH is set equal to -log,,CH-. The magnitude of the error in the de- termination of CH + from the practical pH, where a calculation is valid, is obtained easily from the activity coefficient of the hydro- gen ion by: yc error = 100% ( 1 - f ~ + ) / f ~ * . At the p of most interest in biological chemistry, 0.16, n-here JH + is -0.8 at 25, this error would amount to -255%,.

Large errors may also result from improper use of the Hender- son-Hasselbach equation. For a buffer containing a weak acid, H S , and its salt, S a x , this equation is:

where K is the apparent ionization conrtant of a weak acid, H S . The exact expression is:

(32)

Hence, if in applying the Henderson-Hasselbach equation the practical pH and the thermodynamic dissociation constant are

wed, an error equivalent to log,, - is introduced. In the region

of its validity, the Debye-Huckel equation may be used to esti- mate the magnitude of this error. From Equation 19,

fX . . f H X

(33)

For a simple buffer sii(h as the acetate buffer, lOg~o(fS/fHS), using Kiellands value ( 2 5 ) of ai = 4.5, is calculated to be -0.12 at P = 0.16. For the more complicated phosphate buffer a t the same ionic strength, using Kiellands a , values ( 2 5 ) of 4 for HP0,- - and 4.5 for H,POa -, the term log ( ~ H P O ~ - - / ~ H ~ P O ~ -) is estimated to be -0.18.

Thus for the acetate and phosphate buffers a t p = 0. le j errors of approximately 0.1 and 0.5 pH unit, respectively, ~vould result if the buffer p H were calculated from the Henderson-Hasselbach equation using thermodynamic pK values.

On the other hand, one cannot assume that ph- values found in textbooks are always thermodynamic values. Frequently, i t is the apparent ionization constant, K, as determined experimen- tally using the Henderson-Hasselbach equation, which is listed. This constant ie neither the concentration constant K , nor K t , but is equal to ( K ~ ~ H A / ~ A ) or (KJH -). Obviously, if pK values

are used, the Henderson-Hasselbach equation is the correct ex- pression to employ.

It is veri- important t o emphasize the fundamental fact that i t is the ionic strength due to all the ionic constituents, rather than the buffer ions alone, which affects the activity coefficient of each constituent. Thus, the statement that the Cxi + UH - when it is small! is obviously incorrect, although it is frequently implied even in the latest standard texts.

The ionic strength effect on buffer pH may be very strikingly illustrated by Table 111, which lists the practical pH measured for solutions 0.05J1 in potassium acid phthalate and containing varying concentrations of potassium chloride. The previous considerations concerning liquid junction potentials make i t evident that the decrease in pH is due mainly to changing ac- tivity coefficients with subsequent increased dissociation of the week acid.

Table 111. Practical pH of 0.05.M Potassium Hydrogen Phthalate plus Added Potassium Chloride at 25 C.

Added KC1, Molarity Practical pH 0 0.10 0 . 5 0 1 0 1 5 2 0

4 , O O (assigned) 3 . 9 2 3 . 7 3 3 . 6 3 3 . 6 2 3 57

COLLOIDS 4ND SUSPEVSIONS

The existence of the suspension effect on the pH of clavs, soil-, and ion exchange resins has been knoxn for some time In general, the pH of suspensions and pastes appears to be loner than the pH of their supernatant liquids.

For instance, Jenny and associates ( 2 4 ) reported that a pH of 9.2 was obtained for a lOyc potassium bentonite suspenqion, whereas a 1 to 1 potassium bentonite-water paste gave a pH of 5.8. They attributed the suspension effect to the liquid junction potential a t the point of contact between the potassium chloride bridge and the suspension. iis evidence, they presented the following results. For an ion exchange resin sediment in contact n-ith its supernatant liquid, a p H of 6.0 was measured when both the glass and calomel electrodes n ere immersed in the superna- tant, but a p H of 2.0 was indicated when the electrodes were ini- mersed in the sediment. The e.m.f. equivalent of this p H dif- ference, 240 mv., was obtained between two calomel elertrodes when one was suspended in each phase, but when a glaEs electrode was suspended in each phase no potential difference existed be- tween the two glass electrodes.

Although there is disagreement (1, 12,Zl+, 31,SS) as to n hether the suspension effect is due to the liquid junction potential or is a true membrane potential at a Donnan system, there is no question that the effect exists and that, as a result, pH measure- ments on suspensions of highly charged particles are meaningless.

The effect is not significant, however, for solutions containing mobile colloidal ions or proteins of high equivalent weight be- cause of the efficiency of the potassium chloride bridge. For in- stance, the author detected no difference between the pH of whole blood, the sedimented cells obtained on centrifugation, and the supernatant plasma. The meaning of the pH near cell surfares or of dental plaques, however, may be questionable

NONAQUEOUS SOLUTIONS

The junction potential between a nonaqueous solvent and an aqueous salt bridge is probably frequently large and variable, de- pending greatly on the nature of the solvent and on the per- centage composition of the solvent if it is a mixture. For in- stance, Guthezahl and Grunm-ald (16) estimated that E , a t the

1866 A N A L Y T I C A L C H E M I S T R Y

junction (soln. in ethyl alcohol-waterlsatd. aqueous KCl) in- creases exponentially from 6.4 mv. (-0.1 p H unit) when the left boundary solvent is 35% ethyl alcohol t o 139 mv (-2.3 p H units) tvhen i t is 100% ethyl alcohol. They also concluded that the E, values are relatively independent of the nature of the solute in the ethyl alcohol-water, but their calculations involved only solute concentrations for which the Debye-Huckel limiting law is valid-Le., presumably p 7 0.01. In fact, Tan Uitert and Haas (36) showed that for the boundary (HC1-XaC1 in dioxane-water1 satd. aq. KC1) E , increases, not only with the concentration of the organic compound, but also with the electrolyte concentration, reaching a value of about 70 mv. for 0.03 formal electrolyte in 82% dioxane.

Obviously, no fundamental meaning may be attributed to the practical p H of a nonaqueous solution which produces an un- known, but probably large, junction potential when in contact with saturated potassium chloride. Even when approximate junction potentials are known for nonaqueous solutions, they can- not be simply described by equations but only by a number of ex- perimentally determined working curves (36) . Hence, one may only rarely place even a qualitative interpretation on the dif- ference between the practical pH values of two solutions having different solvents, or of two solutions whose solvent is a mixture of the same constituents but in different amounts. That is, with the present lack of data, one cannot even say whether a given p H for an acetone solution indicates a higher or a lower hydrogen ion activity than exists in an ether solution of the same pH, or in another acetone solution containing some water.

When the solvent composition is kept constant, however, i t is probable (15, 36) that with proper precautions E , can be kept fairly constant [deviation about 2 mv. in ethyl alcohol solutions (16) ] . Hence, the practical p H may be used as a reproducible property, interpretable qualitatively, for nonaqueous solutions if the solvent composition 2s not varied. Many analytical separa- tions and determinations involve bringing the practical p H of a partially nonaqueous solution to a stipulated value. It is im- portant, therefore, to emphasize the fact that a change in the solvent composition destroys the validity of considering a given p H as implying constant hydrogen activity.

I n addition to the junction potential problem, there is another obstacle which must be overcome before useful relationships can be established betn-een p H values of solutions having different solvents. The activity coefficient for a given solute varies greatly from solvent to solvent because of the effect of dielectric constant of the solvent. The Debye-Huckel equation is of only limited use in furnishing activity coefficients in nonaqueous solutions be- cause of the inadequate knowledge of the ion-size parameter, uz, in nonaqueous solutions. At low ionic strength, however, the Debye-Huckel limiting law can be employed to obtain activity coefficients, f:, m-hich are relative to a value of unity a t infinite dilution of the solute in the given solvent. I n order to connect the activity coefficients of a solute in two different solvents, the primary medium effect, .fm> must be determined. This coefficient measures the free energy of transfer of the solute from one solvent to another, so that f z of a solute in a given solvent relative to a standard state in another solvent equals JFfm.

The primary medium effect can reach tremendous values. For instance, relative to an infinitely dilute aqueous solution as the standard state, f m of hydrochloric acid a t low concentrations in- creases from about 2 in 207, dioxane to about 200 in 82% di-

oxane (20). For an ethkl alcohol solution of hydrochloric acid, fm of hydrochloric acid is about 300 relative to the water standard (30). Even more startling are the ionic f m values estimated by Gutbezahl and Grunn-ald (15). They obtained 5 X lo4 and 2 X lo4 for hydrogen and ammonium ions, respectively, in ethyl alcohol relative to water as solvent but only 3.4 for the chloride ion in ethyl alcohol relative to the same standard state.

McKinney, Fugassi, and Warner (30) suggested that an acidity scale could be set up in any solvent by the same procedures used to establish the pH scale for aqueous solutions. Although an enormous amount of work must be performed before such scales will be useful, a significant start in this direction has been made by Gutbezahl and Grunwald (15) and by Bates ( 4 ) .

LITERATURE CITED

Babcock, K. I., Overstreet, R., Science 117, 686 (1953). Bates, R. G., Analyst , 77, KO. 920, 653 (1952). Bates, R. G., Chem. Revs. 42, 1 (1948). Bates, R. G., Electrometric pH Determinations, Wiley, Yew

York. 1954. Bates, R. G., J . Research Natl . Bur. Standards 39, 411 (1947). Bates, R. G., hcree, S. F., Ibid., 34, 373 (1945). Bates, R. G., Pinching, G. D., Smith, E. R., Ibid., 4 5 , 4 1 8 (1950). Beckman Instruments, Inc., South Pasadena, Calif., Beckman

Bjerrum, N., Unmack, A,, Kgl. Danske Tidenskab. Selskab.

Cohn, E. J., Heyroth, F. F., Menkin, AI. F., J . Am. Chem. SOC.

Debye, P., Huckel, E. , Physik . 2. 24, 185 (1923). Erikson, E. E., Science 113, 419 (1951). Guggenheim, E. 8., J . Phys. Chem. 34, 1758 (1930). Guggenheim, E. A., Schindler, T. D., Ihid., 38, 533 (1934). Gutbezahl, B., Grunwald, E., J . Am. Chem. SOC. 75, 565 (1953). Hamer, W. J., ilcree, S . F., J . Research Natl . Bur. Standards 32,

Bull. 225-A, 100-D.

Math. jus. Med. 9 , 1 (1929).

50, 696 (1928).

215 (1944). Ibid. , 35, 381 (1945). Hamer, W. J. , Pinching, G. D., Acree, S. F., Ibid., 36, 47 (1946). Harned, H. S., J . Phys. Chem. 30, 433 (1926). Harned, H. S., Omen, B. B., Physical Chemistry of Electrolytic

Henderson, P., 2. physik. Chern. 59, 118 (1907); 63, 325 (1908). Hitchcock, D. I., Taylor, A. E., J . Am. Chem. SOC. 59, 1812

Solutions, 2nd ed., Reinhold, Wew York, 1950.

(1937). Huckel, E. , Physib. 2. 26, 93 (1925). Jenny, H., Nielson, T. R., Coleman, N. T., Williams, D. E.,

Science 112, 164 (1950). Kielland. J.. J . Am. Chem. SOC. 59. 1675 (1937). Lewis, G. N., Randall, M., Thermodynamics, 1st ed., p. 380,

AIacInnes, D. A, Cold Spring Harbor Symposia Quant Biol. 1 , 1IcGraw-Hill, Sew York, 1923.

190 (1933). AIacInnes, D. A , , J . Am. Chem. SOC. 41, 1086 (1919). LIacInnes. D. d.. Belcher. D . , Shedlovsky, T., Ibid., 60, 1094

(1938).

JIeasurement, ASTM Tech. Pub. 73, 19 (1947). LIcKinney, D. S., Fugassi, P., Warner, J. C., Symposium on pH

Marshall, C . E. Science 113, 43 (1951). Milasso, G., Elektrochemie, Springer-Verlag, Vienna, 1952. Mysels, K. J., Science 114, 424 (1951). S@renson, S. P. L., Biochem. 2. 21, 134 (1909). Spkenson, S. P. L., Linderstr$m-Lang, K., Compt. rend. traz . lab.

Van Uitert, L. G., Haas, C. G., J . Am. Chem. SOC. 75,451 (1953). Carlsberg 15, No. 6, 40 (1924).

RECEIVED for review M n y 22, 1956. Accepted September 6, 1956. Based on work performed under contract with the U. S. Atomic Energy Commis- sion a t the University of Rochester Atomic Energy Project, Rochester, N. Y . Figures 1 and 2 are reproduced by permission of the Society for Analytical Chemistry from The Analyst , 77, 653 (1952) .

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