Graphical Correlation between pH Values, Molarities, and Dissociation Constants of Weak Acids NANDOR PORGES a n d THOMAS F. CLARK Agricultural By-products Laboratory, Ames, Iowa' N THE course of studies on the production of various organic acids through the fermentation of glucose by fungi and bacteria (4,11),it has been found desirable to show the approximate relationships of various constants for certain acids, as well as their comparative strengths. This information was available partly from the dissociation constants, partly through calculation for the specific acids, and from analyses of samples a t various concentrations. In order to avoid frequent calculations, curves for the various acids were prepared and a few were assembled in Figure 1. The plotted values of this graph show the relationships that exist in pure solutions between pH, molarity, and the dissociation constant. A scale for pK, the logarithm of the reciprocal of the dissociation constant, has been included. The pH values for each acid solution were obiained through the use of the formula:
I
pH = log
1 = log -
(H+)
2
-K
+ dK' + 4 K M
where (H+) = concentration of the hydrogen ion, K = dissociation constant, and M = molarity of the solution. This formula was derived from the mathemat'cal expression of the dissociation constant: .I+,
(K)(d _ , -
.-formula, the pH p l u e s a t I'
By means of this Z ~ C were calculated for the various m o l a r ~ t ~ e sIt . ~ should be pointed out that the calculated pH value thus obtained is a function of the concentration or dissociation constant and the molarity and is approximated by the value obtained by and defined in terms of a glass electrode assembly and some standard calibrating solution. Hirsch and Schlays (5) reported the dissociation constant for glucose as 7.8 X 10-la a t 25'C. May,
-
1 Established by the Bureau of Agricultural Chemistry and Engineering. U. S. Department of Agriculture, in coiiperation with the Iowa State College. a The authors wish to include the followine - comment made bv JOURNAL reviewers: "For . . . molarities above 0.1 and for values KO7 lo-' (or, in general, for all values of MIK 5 109 equally precise values of DH (within 0.01 unit), mav be obtained hv the use of the simole A formula pH = -log d h M as by that of the more complicated approximation which they (the authors) used. "The shorter formula, in the farm pH = ~ a ~ l e n itself d s 2 i to the construction of a particularly simple form of homogram, or a plot of pH against log M on which the curves or all acids are straiaht lines."
. .
f
F I C 1.-GRAPHICAL ~ RELATIONSHIPS BETWEEN PH VALUES. MOLARIT~ES, AND DISSOCIATION CONSTANTS.INC L ~ I N OPK,FOR SOME ACIDSAT 25T.
Weisberg, and Herrick (10) obtained for d-gluconic Values for acid the average constant of 1.65 X lo-'. . hydrochloric acid solutions are those adapted by Buchanan and Fulmer (1)from the experimental data of Randall and Young (12). Hydrochloric acid was plotted on the graph in order to show the relative position of a strong mineral acid. The other dissociation constants were obtained from the handbooks of Lange (8)a n d Hodgman (6). Variations in temperatures will cause changes in the plotted values. Kraus (7) reported a decrease in ionization with increasing temperature and Buchanan and Fulmer (1)stated that a rise in temperature increases the conductivity of solutions of electrolytes The decrease in viscosity accompanying a rise in temperature increases conductance and thereby causes an apparent increase in hydrogen-ion concentration which is measured as a decrease in pH. At infinite dilutions the pH-molarity curves should approach the value of pH 6.95, the pH of water a t 25'C. which was determined by Lorenz and Bohi (9) and W. M. Clark (2). The use of the formula appears to give erroneous results for very dilute solutions of glucose and aluminic acid,
yet in spite of such incongruities the graph retains its practical significance. These anomalies (the extension of the curves below the pH line for water) may be the result of disregarding the effectof dissociation of water. Continuation of the curves to approach the value of pH 6.95 bas not been plotted in order to avoid confusion in reading the graph. Values a t other temperatures may be obtained throucrh - tlie use of Fulmer's nomo(3).
In the cases of the acids shown (oxalic acid excepted) the curves of the pH values (calculated from dissociation constants riven in Table 1) meet the one-molar ordinate a t the k values equival;nt to their respective dissociation constants. It is interesting to note that the K scale is logarithmic within each unit. TABLE 1 CONSTANTSOP VARIOUS ACIDSAT 25'C. (From Hodgman (6) and Lange (8) except where indicated) DlUOnATrDN
1.86 X 6.3 X 6.4 X 1.48 X 1.3 X 3.0 X 8.0 X 2.14 X 1.0 X 1.65 X 7.8 X 1.38 X 3.8 X
Acetic Alvminie
Boric Bufytic Carbolie
Carbonic Citric
Formic Fumatic Gluconie ( l o ) Glucose (5) Laetie Oxalie
10-8
lo-'" 10-1' 10." 10-1 10-7 10.1 10-4 10-2 10-8 10." 10-a 10.'
APPLICATIONS OP T H E GRAPH
The dissociation constant (K) of a weak acid being known, the pH value a t molar concentration and the DK eauivalent can be aomoximated bv the oosition of for fumaric On the one-mo1ar acid, having a dissociation constant of 1 X will have a pH value of 1.5 and a pK equivalent of 3. Curves for acids not given in this graph may be approximated. Observation shows that for any two acids the curves are nearly equidistantly apart between the concentrations of 0.05 arie 1.0 molar. As an example, carbonic and butyric acids may be chosen as the reference and unknown acids, respectively. The distance between the dissociation constants of those two acids on the one-molar ordinate will then be the distance between the pH curves a t corresponding molarities, so that for any given molarity the pH of the second acid, butyric, can be readily determined. If the pH and corresponding molarities of an acid are known, its dissociation constant and pK may be readily approximated from such a graph. This method of presenting various acid constants provides a means of indicating apparent acidity or "sourness" of various acid solutions of equivalent molarity, the apparent acidity decreasing from the top toward the bottom of the graph. Thus, hydrochloric is the most activeof the acids plotted and aluminic is the least active. Such a group of curves may be applied in fermentation studies; knowing the pH and molarity of a liquor, the acid of maximum apparent activity in the solution &
may be indicated from the graph. Thus, in the conversion of glucose to gluconic acid by Acetobacter (11). a liquor with an acidity of pH 2 in approximately 0.6 molar solution indicated that the predominant portion of the acid mixture is probably gluconic. This was verified by quantitative analyses. At the same concentration, lower pH values would indicate the presence of acids more active than gluconic, while higher values would indicate the probable absence of this acid. The approximate amount of glucose necessary to produce a concentration of acid of any desired pH may be determined by means of such a graph. The activity of some microorganisms is inhibited below specific pH values, hence acidification offers a means of preventing the growth of some contaminants. The graph may aid in the selection of an acid to give the desired pH value. This graph corroborates the order of successive neutralizations in the conductometric titration of a mixture of acids as reported by Whittemore, Reid, and Lynch (13), these neutralizations occurring in the order of decreasing ionization constants. Displacement of acid radicals from their salts can be predicted from a study of such a graph, an acid of relatively high activity displacing the radical of an acid of lower activity. This fact aids in the preparation of organic acids from their salts by a suitable strong acid. The solutions are adjusted to the proper pH by varying the salt or acid concentration. A comparison of the pH values of a few organic acids as calculated from the formula and as determined with a glass electrode is presented in Table 2. As the pH
.,
&
LAtlLK 2
C~~~P*US,,NS 0. ras PH OI A
F B W OBG*NIC
.A .,
mowlr
,,,c,
1
$I$
E:z Acetic
$ ;::
atti,
2c:;
.+
+ OXaiic
Acme
AND
CoMemAnons 0s
coNsrANrs
DsraaMlNBo
EL&CrPOMBTRICALLV
pH
cancentralion
Colcrloled from.
0.1931 0.1023 0.2161 0,1081 0.4670*
Canrlntllf of ZSQC. 2.41 2.87 1.84 2.28 1.16 1.34 1.89 a.06 2.53
0.46891 0.6553*
1.72 1.93
0.0341*
1.66
as
Malorily 0.8340 0.0998 0.9830 0.1361
Obsnvrd ol
rndicolcd OC.
Tcnpnolrrrr,
Obrcrocd 6s Carerled lo
25°C.
2.40 (29') 2.84 ( 2 9 7 1.83(27') 2.31 (210) 1.13(29') 1.23(29') 1.86(27') 2.03(27") 1.63(29')
2.47 2.91 1.86 2.34 1.20 1.30 1.89 2.06 1.66
1.53(29')
1.56
hue d u u indicate the eoncentrntion of the individual =ids in the
mixture.
of a solution is affected by variations in temperature, the observed readings were corrected to 25'C. by means of Fulmer's formula (3) and are fairly close to the pH as calculated from the constants or read from the graph. In the given mixtures of acids, the pH value of the mixture appears to be that of the acid that liberates the greatest amount of hydrogen ions; for example, a 0.0341 molar solution of oxalic acid has a
greater concentration o f hydrogen ions or lower pH v a l u e than does a 0.6553 m o l a r solution of formic acid. The most satisfactory use o f such a g r o u p of curves is obtained w h e n p l o t t e d on a l a r g e scale in o r d e r that
the intermediate values may be f o u n d w i t h greater accuracy. A s F i g u r e 1 presents the curves of o n l y a f e w acids, i t is suggested that the investigator p l o t curves f o r the acids w i t h which he is concerned.
LITERATURE CITED
BUCHANAN AND FULMER, "Physiology and biochemistry of bacteria," Williams and Wilkins Co., Baltimore, 1928, Vol. I, Chap. 4. CLARK,"The determination of hydrogen ions," 2nd ed., Williams and Wilkins Co., Baltimore, 1922, p. 29. FULMER,"The relation between pH and the reaction of aqueous solutions a t various t&peratures," Iowa State College J. Sci., 1, 37-42 (1926). GASTROCK. POROES.WELLS.AND MOYER.''Gluconi~ acid production on pilot plani scale," Ind.' ~ n C~k m. . , 30,
.-- ,-" --,.
797" 9 ( 1 Q7Q\
Hrnscn AND ScmAYs, "Die Bestimmung des Laugebindvermiigen der wichtigsten Zucherarten," Z. physik. Chem., 141, 387412 (1928). HODGMAN, "Handbook of chemistry and physics," 23rd ed.. Chemical Rubber Publishing Co., Cleveland, 1939, 2221 PP. KRAUS."The properties of electrically conducting systems." Chemical Catalog Co.. New York City. 1922, 415 pp.
LANCE,"Handbook of chemistry," Handbook Publishers, Inc.. Sandusky. Ohio, 1934, 1265 pp. LORENZAND BoHr, "Die Elektrolytische Dissociation des Wassers," 2. physik. Chem., 66, 733-51 (1909). MAY,WEISBERG,AND HERRICK, "Some physical constants of d-gluconic acid and several of its salts," J. Wmh. Acad. Science.. 19.. 443-7 (19291. . POROES.CLARK,AND GASTROCK, ''Gluconic acid production by Acetobacter, a survey," unpublished data. RANDALL AND YOUNG,"The calomel and silver chloride electrodes in acid and neutral solutions. The activity coefficient of aqueous hydrochloric acid and the single potential of the deci-molal calomel electrode," J. Am. Chem. Soc., 50, 989-1004 (1928). WHITTEMORE, REID, AND LYNCH,''Nitric acid pulping. Analysis of the used-acid pulping liquors," Ind. Eng. Chem.. 30, 1192-8 (1938).
.
