DAVIDH. ROSENBLATT
40
tained from the C constant of the B.E.T. equation, were negative. No calorimetric evidence prior to the present research clearly supported the conclusion that negative net heats of adsorption should be obtained in the case of water adsorbed on high temperature carbon. Previous evidence was based on the separate measurement of the two large, nearly equal, quantities q1 and q~ (or their equivalents, El and EL calories per mole adsorbed). The difference, q1 - q L , was of the order of magnitude of the experimental error.16121 The net heat of adsorption, obtained from heat of wetting data, is a measure of the entire interaction of the surface and its adsorbed film. Since water forms a contact angle on graphite, the adhesion of water to graphite is less than the cohesion of water. To cause the water to adhere to graphite rather than to itself, energy must be provided. This would result in an adsorption of energy, that is, a negative net heat of adsorption. The relation between the formation of a contact angle and negative net heats of adsorption can be shown. Combining equations 2 and 4 one obtains pi
- pL
=
hso
- h m - hLv
(6)
Then using the Gibbs-Helmholtz relation h = (21) F. G . Keyes and (1927).
Y
-
T(67/8t)
(7)
RI. J. Marshall, J . A m . Chem. Soc., 4 9 ,
1.56
Vol. .58
where y is the free surface energy, one can write the equation q1
- qL = ( Y P -
YSL
- YLV) -
The quantity (7s. - YSL - YLV), the spreading coefficient, is negative for liquids that form contact angles and positive for liquids that spread spontaneously. The second term on the right hand side of equation 8 cannot be neglected, yet if either the assumption that the temperature coefficient of the spreading coefficient is positive or that the second term is of lesser magnitude than the first, is valid, the formation of a contact angle (negative spreading coefficient) would result in a negative net heat of adsorption. Mercury forms a contact angle against carbon. It would then be expected that the net heat of adsorption of mercury on carbon would be negative. Coolidgez2found that the net heat of adsorption of mercury on charcoal was -4000 cal. per mole. The conclusion that negative net heats of adsorption can be predicted must be tempered with the reminder that for true predictions the free energy relations must be used rather than the total energy relations. Yet for qualitative predictions the enthalpy can often replace the free energy function. (22) A. S. Coolidge, ibid., 49, 1949 (1927).
A NEW METHOD FOR CALCULATING DISSOCIATION CONSTANTS FROM SPECTROPHOTOMETRIC DATA BY DAVIDH. ROSENBLATT Chemical Corps Medical Laboratories, Armu Chemiral Center, Alar$and Received February 8 , I063
A new method has been developed for the calculation of the apparent dissociation constants of monobasic and (in many cases) dibasic acids from spectrophotometric data. It should prove of value in fiituations where it is inadvisable or impossible to determine extinction coefficients of the pure ionic or molecular forms. The method requires a small number of determinations.
It is often observed that the visible or ultraviolet spectral absorption peaks of an organic compound are shifted when the compound loses or gains a proton. Consider, for example, the monobasic acid equilibrium HA H+ A. (Charges are not attached to the symbols HA and A because the "acid" HA may be the conjugate acid of a base, an uncharged molecule or even a singly ionized dibasic acid.) When the two conjugate forms, HA and A, coexist in solution the observed molar estinction coefficient for the solute at any wave length is equal to the sum of the products of mole fraction and molar extinction coefficient of each form. This has provided a convenient means of determining the apparent dissociation constants of such compounds. The most usual method for calculating dissociation constants from spectrophotometric data has been to make a series of determinations of molar extinction coefficients a t a particular wave length and at various hydrogen ion concentrations. The hy-
+
drogen ion concentration corresponding to the point midway between the maximum and minimum values for the extinction coefficient is then equivalent to the dissociation constant.1 This is true because the mean extinction coefficient represents a condition where the two forms are present in equal concentrations. Since
it follows that p k ' ~= pH when [HA] = [ A ]
(b)
The above method requires the preparation and study of quite a number of solutions-wit,h solutes ranging from pure HA to pure A-if any accuracy is to be obtained. Another approach,2 also applicable only when it is possible to measure the extinction coef(1) N. Bjerrum, Somm. ehem. chemische-tech. Vortrage, 21, 30 (18151. (2) L. A. Flexser, L. P. H a m m e t t and A. Dingwall, J . A m . Chem. Xoc., 6T, 2103 (1935)
Jan., 1954
CALCULATING DISSOCI.4TION CONSTANTS F R O M
ficients of the pure HA and A forms, uses the expression molar extinction coefficient of HA molar extinction coefficient of A = molar extinction coefficient of a mixture a t hydrogen ion activity a,, and containing xo1ut.e concn. of [HA], and [A],
eB = eY = e,
The value of [HAIn/[A], may then be substituted in equation (a), together with t,he pH a t which en was determined, to give PICA. This method involves only three det,erminations of extinction coefficients, but two of these must be made with the pure conjugate forms. Neither the above treatments for a monobasic acid nor any other spectrophotometric methods within the purview of the present author makes any provision for cases where it is inconvenient or impossible to determine eg and/or E? directly. Such a case, for example, might be that of a very weakly acidic substance, the A form of which would exist exclusively only in strongly basic solution, which condition might be impossible to attain in the medium used or might be achieved only a t the cost of an inordinately high ionic strength. The method presented here for the calculation of the dissociation constants of monobasic acids does not require pure conjugate forms. Any three solutions with sufficient differences in pH and apparent molar extinction coefficient can supply the required information. By combination of equation (c) with the common expression
: ;1
a1
a3 a3
a1
€1
KA=
€2
€3
:!
1
:I
OPTICAL TIONS
TABLE I DENSITY OF EQUIMOLAR 3-NITROCATECHOL SOLUAT DIFFERENT HYDROGEN ION CONCENTRATIONS
Hydrogen ion activity, moles/l.
6.61
x
10-8
1.86
x
10-8
Optical density At 300 mp At 440 mp
1.276 1.003 0.896
1.07 x 10-7
0.122 .499 .670
VALUES OF K1 FOR 3-NITROCATECHOL BY VARIOUS METHODS Method h74
This is a linear equation in four terms and three unknowns (t,he latter in parentheses), since en and an are measurable quantities. Solution for the unknown KA requires three simultaneous equations which may be solved with determinants ale1
Buffer concentrations should be kept lorn in order that the ionic strength may not change too much the degree of dissociation of the absorbing solute, or, alternatively, thermodynamic dissociation constants may be found by extrapolation to zero ionic strength. The following example illustrates the use of the method. It was desired to determine whether the first apparent dissociation constant for 3-nitrocatechol (considered as a monobasic acid) in water, obtained in this Laboratory by potentiometric titration, was correct. This value had been found to differ appreciably from those of Gilbert, Laxton and Prideauxj3which had been arrived a t by conductometric and color comparison methods. A 10-3 M solution of 3-nitrocatechol was added to each of three 0.005 M phosphate buffers at different pH’s, so that the resulting solutions were 2 X 10-4 M in 3-nitrocatechol and 4 X M in buffer. These solutions were examined in a Beckman Model DU spectrophotometer at wave lengths of 300 and 440 mp. Data are shown in Table I. The values of KAobtained by using these data in equation (e) are shown in Table 11, along with values determined by potentiometric titration and the figures given by Gilbert, Laxton and Prideaux.
TABLE I1
the following general equation is derived
am am
41
SPECTROPHOTOMETRIC DATA
(e)
1
Optical density, D, which is the instrumentally measured quantity, is related to the molar extinction coefficient by the relationship D = E C ~ where , c is concentration of solute and d is the length of the light path of the spectrophotometric cell. If the light path and total concentration of absorbing solute are the same in each measurement, then c and d will cancel out, and D may then be substituted for E in equation (e). A convenient laboratory procedure consists of making a stock solution of the solute to be examined, and diluting equal portions of this solution with equal amounts of buffer. A precise knowledge of the solute concentration is then not necessary.
Conductometric7 Color comparison7 Potentiometric Spectrophotometric, a t 300 mp Spectrophotometric, a t 440 m,u
1 . 9 x 10-6 1 . 1 x 10-6 2 . 0 x 10-7 2 . 2 x 10-7 1 . 0 x 10-7
The close agreement of the present spectrophotometric values with the potentiometric value of KA for 3-nitrocatechol suggests that the conflicting results obtained by Gilbert, Laxton and Prideaux may be due to impurities in their 3-nitrocatechol. The agreement of values of K A in the present work demonstrates that the method of calculation is of practical value. It has been borne in mind, in the present work, that 3-nitrocatechol is really a dibasic acid; but since KJK2 (the ratio of the first to the second dissociation constant) was known to be greater than lo3,the treatment as a monobasic acid4was considered justified. In the case of a dibasic acid for which K1/K2 < 1000, however, the validity of such treatment would be open to scrutiny, inasmuch as the intermediate form HA (of the dissociation H2A G H+ HA G 2 H+ A) will always be accompanied by significant amounts of H2A and/or A which prohibit the direct determination of its ex-
+
+
(3) F.L. Gilbert, F. C. Laxton and E. B. R. Prideaux, J . Cham. SOC., 2299 (1927). (4) B. J. Thamer and A. F. Voigt, Tars JOURNAL, 66, 225 (1952).
Vol. 5s
HARRYESSEX
42
tinction coefficient. The type of difficulty here encountered has been met in various ways. VI& and Gex (as corrected by Thamer and Voigt3) elaborated a method in which, the extinction coefficients of H2A and A being known, that of HA is found by successive approximations; it is necessary to use data a t two different wave lengths. A second cumbersome procedure5 also makes use of successive approximations and of the extincjion coefficients of H2A and A. The ingenious method of "indirect colorimet,ryJJ6 necessitates the use of equipment not normally available in the laboratory. Finally, in the treatment by Thamer and V ~ i g t individual ,~ experimental determinations of the extinction coefficients of pure H2A and A must be made; also, the optical density a t a suitable wave length must be graphed against pH and a maximum or minimum found on this curve in a region where HA is the predominant form of the solute. Because these approaches require considerable effort, it is worthwhile to inquire into the feasibility of extending the use of equation (e) into the region where K1/K2 < 1000. The question cannot be given an absolute and all-encompassing answer, since several factors influence the accuracy with which K1 (or Kz) can be estimated. There is one variable source of error, however, which the use of equation (e), rather than (b) or (c), may affect considerably; this is the relative error ( E ) in KI, made by assuming that all (5) E. 8. Hughes, H. H. 0. Jellinek and B. A. Ambrose, THIS JOURNAL, 68, 414 (1949J. (6) L. Sacconi, ibid., 64, 829 (1950).
ionized H2A is in the form HA, and which is pH dependent, as shown in
Nom the hydrogen ion activity (and hence E ) a t which [HA] is maximal is fixed, and best results with equation (c)-and in effect with (b)-are obtained when the extinction coefficient of HA is determined a t maximal [HA]. With equation (e), however, the magnitude of E may be decreased to a considerable extent' by selecting appropriately large values of a,, and this may lead to a smaller over-all error. Another possible treatment for dibasic acids is to use the solution of five linear equations of the form
This form is derived in a manner analogous to that of equation (e) and may be solved for K1 and K z with fifth order determinants. The symbols are the same as those used previously, exce t that ea is the molar extinction coefficient of $A. The difficulty of solving such equations is great, and the present author has made no effort to test their usefulness. (7) Similarly, if the value of Kz is sought, the error from asiuming a monobrsaia acid HA would be E = &KI, and would be reduced by diminishing an.
THE MECHANISM OF GAS PHASE RADIATION-CHEMICAL REACTIONS BY HARRY ESSEX ChemistrQ Department, Syracuse University, Syracuse, N . Y . R ~ c e i v e dApril $1, 1965
From the effect of electric fields on the ion yield, and other experimental data, the extent can be calculated to which a gaseous radiation-chemical reaction is initiated by each of four primary mechanisms. Both published and new data (the details of which will appear subsequently) ar? used t o illustrate the methods for calculating the ion yield attributable to each mechanism. The effect of the nature of the irradiation on the ion yields and the maximum ion yields possible for each mode of initiation are discussed.
Introduction Our studies of the last twenty years on gas phase radiation-reactions,'-'O together with other evidence, indicate that such reactions are initiated by one or more of the following mechanisms. 1. Recombination of oppositely charged ions,ll e.g. (1) H. Essex and D. Fitegerald, J. Am. Chem. Soc., 66,65 (1934). (2) C. Smith and H. Essex, J . Chsm. Phue., 6, 188 (1938). (3) M. J. McGuinness, Jr., and H. Essex, J . Am. Chem. Soc., 64, 1908 (1942). (4) A. D.Kolumban and H. Essex, J . Chem. Phys., 8,450 (1940). ( 5 ) N. T.Williams and H. Eseex, ibid., 16,1153 (1948). (6) N. T.Williams and H. Essex, ibid., 17,995 (1949). (7) J. Efimenko, Thesis, Syracuse University, Syracuse, N. Y.,1949. (8) Y . S. Rudolph, Thesis, Syracuse University, Syracuse, N. Y . , 1951. (9) Philipp H.Klein, Thesis, Syracuse University, Syracuse, N. Y., 1951. (10) Philipp H.Klein, Thesis, Syracuse University, Syracuse, N. Y., 1952. (11) This mechanism was originally proposed by 8. C. Lind, "The Chemical Effects of Alpha Particles and Elertrons," 2nd Ed., Chemiral Catalogue Go., New York, N. Y., 1928.
+
ABf e- +A AB+ AB- +2A
+
+B
+ 2B, etc.
2. Molecule splitting on electron collision with attachment of the electron to a fragment of the molecule and without ionization, e.g.
+ e- +A + B-
AB
3. Molecule splitting on electron collision without attachment of the electron, and without ionization, e.g. AB
+ e- +A + B + e-
4. Molecule splitting on ionization, e.g. a
&
AB+or+A++B-+or a
AB
(a)
e-
e-
+e-
+ ore- +A+ + B + e- +or eCY
(b 1 ,
As indicated above, these four primary mechanisms are different ways of effecting one and the
