T H E E F F E C T OF SALTS O K WEAK ELECTROLYTES* 11. Calculation of Overlapping Constants’ BY HENRY S . SIhfMS
Introduction To avoid making the preceding article? too long it seems better to present in a separate paper the potentiometric titration data of polyvalent acids, and to explain the method of calculating the titration indices of those polyvalent acids with overlapping constants. In a previous paper3 a method was given for calculating titration indices which overlap. This method is accurate only when the ionic strength is the
PH FIGI Citric acid titration curve. The dotted curve represents the theoretical curve at zero ionic strength (infinite dilution). Curve I is direct titration h4th SaC1; 2, in the presence of “dilute” MgC12;3, in the presence of “concentrated” MgC12. See Fig. 6.
same for all the data. We will review this method, as applied t o a divalent acid, below. In the study of activity coefficients it was necessary to devise a method to calculate accurate titration indices from data with various ionic strengths. This will also be presented below together with data on several divalent acids and one trivalent acid. An example of such data is found in * From the Department of Animal Pathology of The Rockefeller Institute for Medical
Research, Princeton, K. J. ‘The term “overlapping” constants (or indices) in distinction from “isolated” constants is used when the titration indices are less than 2 . 5 index units apart. Thus PGI‘ - PGI’= 0.66 for azelaic acid (at an ionic strength of 0 . 0 1 ~ )hence ; the constants are “overlapping”. On the other hand the difference between the titration indices of oxalic acid is 3.03, and the constants or indices are “isolated”. 2 Simms: J. Phys. Chem., 32, I I Z I (1928). 3Simms: J. Am. Chem. Soc., 48, 1239 ( 1 9 2 6 ) . See corrections in footnote I of the preceding article.
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Fig. I for citric acid, which illustrates the magnitude of the effect of salts. These data can at present be calculated only with the use of the method described in section 111.
11. Review of Method of Calculation of Titration Indices of a Divalent Acid from Data with Constant Ionic Strength First-Plot the experimental Hf ion indices (PH) against the corrected equivalents of base4 (b’), b’ = b - a + h - o h
(1 34)
C
(as in Fig. I of the above pape?) and estimate graphically the approximate titration indices5 (Po1’ and PO,’). Second-For each experimental point in the buffer range of the titration index which we wish to calculate (say PO*’)subtract the Hfion index (PA) from the other tentative titration index (Pal’) and calculate the corresponding avalue (at) from the equation? a P H - P G = log(1) I - a
subtracting this value of al from b’ we get a2,since (from I 33): b‘ = a2 (2) Fourth-The titration index in question ( P G ~ ’ may ) be calculated from azby Equation I .6 This calculation is made for all points in the buffer range of the titration index in question (Pol’) and the average of these index values may be used (in place of the tentative value) to calculate the other index (PO,)by repeating the process (Le., calculate a2 for points in the lower buffer range; subtract them from b’ to get al;and calculate PG,’from aJ. Third-By
4
+
When great accuracy is not required (or when the correction is small) the activities
H and OH may be used in place of the concentrations h and oh:
bt = b-a + C
+ “
(approximately)
The notation is the same as in the previous articles. b ’ = corrected equivalents of base. ( b - a ) / c = equivalents of base. b = molar concentation of st:yng base. a = ” acid. ion (H = activity). h It ,f ,t ,, hydrogen oh = ” ” ” hydroxyl ion (OH = activity). I
6 This may be done most easily with a paper mold cut the shape of a t y ical monovalent dissociation curve (Equation I ) . Values of log a / ( I -a) for various values o?a may be found in W. M. Clark: “The Determination of Hydrogen Ions”, p. 460 (1922). This paper mold should be applied to the upper and lower parts of the titration curve and the two curves drawn (with pencil) in such a way that the s u m of a1 and az of the two curves should equal b‘ (of the experimental curve) a t each Pn. The center ( a = o . s )of each of these monovalent curves will thus give a tentative titration index which may be used in the succeeding steps. 6 This may be obtained easily from a large scale plot of a against log a/(I- a ) (see footnote s).
x
EFFECT OF SALTS ON WEAK ELECTROLYTES
2.8
FIG.2 Oxalic Acid
1
5.2
FIG.3 Malonic Acid
5.4 56 58
PKzl
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For bases, ampholytes and substances with more than two overlapping indices see the above paper,3p. 1245. If the calculated indices differ materially from the tentative ones, the process should be repeated using the calculated indices to obtain more accurate values.
FIG.4 Succinic Acid
If, as is likely to be the case, the ionic strength is not constant, the titration indices found above will not be correct and the method described below may be used to obtain more accurate values. III. Method of Calculation of Titration Indices from Data at Different Ionic Strengths For this procedure it is advisable to have data over a wide range of ionic strengths (below 0.1p ) in the presence of univalent ions, unless the substance is sufficiently similar to those which we have studied to permit estimating the slopes of the curves. First-Obtain approximate titration indices by the method described above.
EFFECT OF SALTS O S WEAK ELECTROLYTES
Second-Plot
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these indices against the square root of the ionic strength the data for each index (below
(gP) and draw straight line curves through 0.1,~)~.
Third-At the value of d;of each experimental point in the buffer range of a given titration index (say P G 2 ’ ) obtain a tentative value of the other index ( P G , ’ ) from its cume in the above plot. Fourth-With this series of P G , ‘ values calculate the P G l ’ values by the procedure in Section I1 and plot them on the above plot and draw a line through the revised value of PG2’.
FIG.5 Azelaic Acid (The points represent average values).
Fifth-Obtain PG:’values from this line at the values of 4 of the points in the lower buffer range; calculate PG,‘ values from these; plot them and draw a line through them. Sixth-If the above curves are satisfactory, obtain from them the values of PG1’-PG,’ at the 1/i of each point. From these the values of Po,‘ - PK,‘ (or PKl’-PG*’) may be obtained from the lower curve (extended if necessary) of Fig. z of the first paper.3 These when subtracted (or added) to P Q ~(or ’ P G ~ ’give ) P K ~ (or ‘ PK3’). Seventh-The dissociation indices ( P K ~and ’ P K ~ ’when ) plotted against 4; should give straight line curves with slopes agreeing with those presented in this article.8 If the dissociation indices (PK’) 7 These curves should not be perfectly straight lines. are a linear function of d i the titration indices (Po’) will not be linear, since the relation between the dissociation and the titration indices is not linear.3 However there is little error in assuming the curves to be straight in dilute solution. 8 The last two steps may be performed on average Po‘ values for points close together and the average Pp’ values thus calculated may be plotted against average d/Pvalues for these points.
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The slopes indicate the extent of agreement or disagreement with the Debye-Huckel equation. The deviations observed in the data in this article are discussed in the previous article.
IV. Estimation of Ionic Strength If a monovalent acid is titrated directly with strong base the ionic strength is given very closely by the equation: p = b’c (3) The same formula applies to PO,’of an “isolated” polyvalent acid (b’
