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The use of the hammett acidity function, H0, for estimating the hydrogen ion concentration in acids
I. Im~rg. Nuc.L Chem., 1961, Vol. 17. pp, 302 to SlI. ~
Press Ltd. Primed in Northern Ireland
THE USE OF THE HAMMETT ACIDITY FUNCTION, Ho, FOR ESTIMATING THE HYDROGEN ION CONCENTRATION IN ACIDS E. H~FELDT Department of Inorganic Chemistry, Royal Institute of Technology, Stockholm 70 (Received 8 June 1960)
Abstraet~A method is described where the hydrogen ion concentration in acid solutions is evaluated from H0-data. For CFaCOOH, good agreement is found between this and the NMR methods below 6 M. For the acids HF,CHsCICOOH and CHC1,COOH, pK-values in rather good agreement with those in the literature are obtained. Application to HCI and HBr gives the following tentative Kvalues for these acids: 1.05 < log Kact < 2.60; 2.15 < log K~Br < 4"65 Only upper and lower limits can be given for K because of the difficulty in making a sound extrapolation to pure water. It has also been found that, because of a large variation of the activity coefficient of the undissociated acid, Raoult's law cannot be used for estimating its concentration in aqueous solutions of strong acids. REcmcrLY WYATrt~'~ as well as BASCO~mEand BELL~xb~have shown how acidity functions can be used for estimating the degree of dissociation, = in acid solutions. In the following, a slightly different approach to the same problem will be described and applied to a number of different acids. METHOD The Hammett acidity function, H0, is related to the activity of free unhydrated protons, aa+, by: ~*~
where
log ~ aa+ = --Ho
(1)
~P = yBlyBa+
(2)
~0is the ratio of the activity coefficients on the molarity scale of the basic form, B, and conjugate acid, BH +, of the indicators used for measuring H0. Introducing the stoicbeiometric molarity activity coefficient, ),x+ of H + we get from (1) log~ya+ + log Ca+ ------fir,
(3)
For a monobasic acid, HA, which dissociates into H + and A-, ~, is defined by: Ca+ = Cx- = ~C
(4)
where C is the stoicheiometric molarity of acid. Equation (3) can be used for estimating Ca+ provided the function ~0ya+is known. Concerning~ it is generally accepted for solutions of strong acids in water that, at any given concentration, ~ is independent of the indicator used.**~ This assumption has recently found a striking confirmation. When plotting H0 vs. aa,o for HNO,, HC1, H,SO, and HCIO,, WYATte'~ found that all data fell on a Single curve. Also HBr follows the same curve, c'~ This behaviour implies that, when compared at the same water activity, ~0 as well as the hydration equilibria of the proton are the same in all these different acids.~1''~ In Fig. 1, log ~ya+ is plotted vs. H, for three acids: HNO,, HCIO4 and H,SO,. The experimental c1~(a) P. A. H. WYATT,Disc. Faraday Soc. 24, 162 (1957). (b) K. N. BMCOI~E and R. P. BeLLIbid. 158 (c) O. RZnLICHand G. C. HOOD.,Ibid. 87. ~s~ L. P. ~ , Physical Organic Chemistry p. 267 McGraw-Hill New York'(1940). (s~ M. A. PAOL and F. A. LONe, Chem. Rev. 57, 1 (1957). ~ E. H6OI~LVT, Acta Chem. $cand. 14, 1627 (1960). 3O2
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
303
log f'YH" "
3
^~
. H N Os
~/ o
~HCI~
2
I
"6Ho F~c~. 1.--log ~ + ) , vs. Ho for: @ HNOs A HCIO, 0 HaSO,.
A
O.6
• HN03
,,HCIO~
oH2SQ • HCI
O.2
f/
0
"-'-t• I
I
•~
,3
.2
- Z
I
.i
I
I
o
-1
.... I
-2Ho
FIG. 2.--108 ~yu÷ VS. H e in relatively dilute solutions for: @ HNOs A HCIO, O H,SO 4 • HCI.
data are those used in ref 4. As may be seen, all the data fall on a single curve within the limits of experimental error, which can be estimated to be around -t-0.05. This behaviour is in agreement with WY^~T'S finding. In dilute solutions, ~0ya+ approachesunity because pure water is here taken as the reference state for all activity ooeflicients and He will therefore approach --log Ca+. In Fig. 2, log ~ya+ is plotted vs.
304
E. H6G~LVT
Ho for relatively dilute solutions. The HCI data are those of H6OFBLDT.~s~ As shown in a later section, HC1 is likely to be completely dissociated below 3 M i.e. for H0 > --1. As seen in Fig. 2, logeya+ ~ 0 for H0 > +1 i.e. for C < 0.1 M. This may have the 'foilowing reasons (a) a cancellation of the electrostatic interactions on H + and BH + and Co) the water activity varies too little to cause any appreciable displacement of the hydration equilibria. In the following we shall assume as a working hypothesis that eyn+ is the same function of H0 for all acids as is indicated in Figs. 1 and 2 and evaluate Ca+ from equation (3). It would be expected, however, that for relatively weak acids where the dielectric constant attains low values in concentrated solutions, ion pair formation involving the indicator ions will cause individual variations in 9 and the above assumption will break down. ts~ TABLE 1.--TI~ FtrNc'nON Io : ~PyH+ FOR ROUND H0-VALUES
H0
log cya+
He
log ~ya+
>1 +0"75 +0"50 +0"25 0 --0"25 --0"50 --0"75 --1-00 --1"50
0 0"02 0"05 0.09 0-15 0"22 0"30 0"41 0"53 0"85
--2.00 --2-50 --3.00 --3-50 --4.00 --4.50 --5.00 --5,50 --6.00
1"30 1"70 2"15 2.62: 3.10 3.56 4.04
4.52 4.98
In Table 1 average values for log q~ya+ are tabulated for round He-values. They are collected from the best Raman and Nuclear Magnetic Resonance (NMR) data available to the present writer. They will need revision when a body of more complete and more accurate data has been assembled. APPLICATIONS
(a) Weak and moderately strong acids In the following the method outlined in the previous paragraph has been used for finding the thermodynamic dissociation constants of some acids, and the results are compared with the values in the literature. The thermodynamic dissociation constant is obtained by extrapolation of some suitable function to infinite dilution. Two different functions have frequently been used, the stoicheiometric dissociation constant, Q, or the function Kyax, where YHx is the activity coefficient of undissociated acid. These two functions will now be discussed. For the reaction: H A ~ - H + -k A -
(5)
Q is given by C~+C~Q :
C=~ = Ky.A
Cn-"-"~ :
1 -
~t
(6)
y~+yx-
Since a l l activity coefficients a p p r o a c h u n i t y u p o n dilution, the e x t r a p o l a t e d Q-value will give the t h e r m o d y n a m i c dissociation constant, K. A n a l o g o u s l y we find f r o m (6): a~+ax-
Ky,~x = C(1 -- =) ~s) E. H~K3FELDTalad J. BI~3ELEiSBN,./'. Amer. Chem. Soc. 82, 15 (1960).
(7)
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
305
where the activity product an+aA- can be computed from stoicheiometric mean activity coefficients tabulated in the literature. It is known that for most strong electrolytes activity coefficient products like yH+YX - have a minimum near I M. On the other hand one would expect yr~x to follow a function. log Yax = k C (8) where the constam k is called the salting out constant. This implies that Q might go through a maximum making any extrapolation of Q most uncertain. The possible behaviour of Q can be illustrated by the following example. Assume: log YHx = 0"05C log y a + = log Yx- =
1
~/c
2" 1 + a/-------~+ 0" 1C
(9a, b)
where 0"05 is a tyl£W,ai value for the salting out constant of a non-electrolyte (e) and 0-1 a typical value for an electrolyte.(7) Inserting (9) into (6), differentiation yields: Qm~ for C ~ 0"83 M (10) On the other hand, a maximum in the extrapolation path for KyHx is not likely to appear and this function should always be preferred whenever good activity data are available. Unfortunately, such data are lacking for several of the acids for which He has been measured, so at present it is necessary to resort to the Q-extrapolation in most cases. For CFsCOOH a direct comparison can be made between N M R (s) and He measurements. (') The N M R measurements give directly the ~-values while the H o method employs equation (3)and the data in Table 1. In Fig. 3, ~ from the two sets of measurements are compared. When evaluating ~ from the Ho data o f I~NDLm and T~OER the measurements for the two indicators o-nitroaniline and 4-chloro-2nitroaniline have been used with the pK-values ---0.29 and --1.03 as recommended by PAL-. and LON6(s) and in good agreement with recent measurements.(S, xe) The densities necessary for transformation from the mole fraction scale used by RANVLES and T~DEl~ to the molar scale were determined by the present writer in connection with some N M R measurements. (11) The latter show good agreement with those of REt~ICH et al. (s) below 10 M. From Fig. 3 it is seen that there is rather good agreement up to 6 M. Above that concentration deviations start to occur which are probably an example of a failure of the working hypothesis due to ion pair formation involving the indicator ions. (3~ With knowledge of ~ and the mean activity of acid the function K y a x can be computed and extrapolated to C = 0 in order to give a value for K. When doing so the activities of CFsCOOH-solutions used were the same as those of RE)UCH et alJ s) The agreement between K~rm~ = 1"8 and KH, = 1.4 is encouraging in view of the experimental uncertainty in H0 and the crude treatment of the NMR-data. For the other acids where no direct comparison can be made, Q has been computed from (6) and in Fig. 4 log Q is plotted vs. C for these acids. In Table 2 the pK-values obtained are compared with the literature. (') ~7)
F. A. LONO and W. F. McDm~Tr, Chem, Rev. 51, 119 (1952). C. W . D A r n s , 1. Chem. $oc. 2093 (1938).
(sJ G. G. HooD, O. R ~ t a e H and C. A. I~ILLY, i. Chem. Phys. 23, 2229 (1955). ~') J. E. B. R ~ D L ~ and J. M. " I ~ D n , J. Chem. Soc. 1218 (1955). ~le) K. N. B~COMn and R. P. BY&L,I. Chem. $oc. 1096 (1959). (la) M. BlmoQvlsr and E. H6OVELDT,J. Inorg. NucL Chem., to be published (1960).
306
.
E. H~I~LDT
For HaPO s and HsPOo H0 has not been determined in sufficiently dilute solutions to permit any reasonably good extrapolation. For HsPOs the measurements in very dilute solutions by KOLTHOFF(ls) are also marked in Fig. 4 and they seem to o( o NI"IR
jO~8 @
'0.6
.
o @@
,a4
. o 0
•
~2
O0
I
I
I
I
I
I
I
I
l~e
I
2 ~ 6 8 10 C Fro. 3.--The degree of dissociation, ct, for CFsCOOH vs. stoicheiometric molarity of acid. O NMR-data, • H0-data. TABLE 2.--pK-vXLU~S FROM// Acid
Symbol in Fig. 4.
CH,CICOOH I-IF*
[]
H,PO, Hd'O,
A @
CHCIsC00H CCI,C00H CHsSOsH
-DATA ODMPARED WITH THE IATIFALATURE. t = 2 ~ ° C
Ho-data Ref 10 3 3, 16 10 10
pg(H,
pK(Li0
2"8 3"1
2"86 2"91-3"27 2"12 1"70-1"80 1.06 0"64 --0"6
1"1
10, 2o 0
10
--1"1
* When evaluating 0 for HF he value K = 4 for the reaction F- + H F ~ H F s was employed. (u-xj)
Lit Refs 12 13, 14, 1.' 17 18, 19 3 21 10
(II)
(is) D. J. G. I v ~ and I. H. PRYOR,J. Chem. Sac. 2104 (1955)., (m C. B~tossrr, Naturwissenschaften 29, 455 (1941). qls) C. B. Woorreg, J. Amer. Chem. Sac. 60, 1609 (1938). (Is) H. H. BiOnNe and T. DE V ~ , J. Amer. Chem. Sac, ¢D, 1644 (1947); L OAvATrA Private communication (1960) (x,) A. N. GeLastc~m, G. G. SHCtmOLOVAand M. N. TeurdN, J. lnorg. Chem. U.S.S.R. 1, 282 (1956) ¢17) Stability Constants Part lI, p~ 57-58. The Chemical Society, sptm. publ., London (1958). (xs) I. M. KOLTtIOt~, Rec. Trao. Chtm. 46, 350 (1927). o.0) K. TAKARASmand N. YuI, Bull. Inst. Phys. Chem. Res. (Tokyo) 20, 521 (1941) ca0) H. Buitrd~r, R. MURPHYand D. YARIAN, Indlalla Aead, S¢i. 66, S6 (1957). (st) O. R.BnUCH, Chem. Rev. 49, 333 (1946).
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
307
fit well into the He-data. For CClaCOOH log 0 must have a maximum if the present literature value is accepted as correct. This illustrates the possible drawback of the log 0-plot pointed out earlier. For CHsSOsH the value K = 4, suggested in the literature, was estimated from He-data by BELL(xe) and offers no independent check. The discrepancy between K = 4 and the present estimate of K , ~ 10 from the same data is likely to be due to the different approximations inherent in the two approaches. log( 2 o
. . . . . . . . . . . . . . . .
o
.(I
o
o
C~S~H
~CC~COOH
o
-H,PO~ CHCI,~COOH -2
--
...o-HF
.-CHf/COOH
~ 0
I
I
I
I
I
2
3
,~
C
FlO. 4.--The function log Q vs. the stoicheiometric molarity o f acid for: O CHtSO,H • CCltCOOH, • HgPOs, x HtPOt OKolthofl'), A HaPO4, • CHCIsCOOH , [2 HF, • CHaCICOOH.
For the acids HaPOs, CCIsCOOH and CHaSOaH, 0~ was found to become constant, or to increase at about 7"5, 5 and 9 M respectively. These results indicate that at high Concentrations the method of estimating = from He measurements is not applicable as found for CFsCOOH above 6 M. When good activity measurements are available, the plot of 9~as+ vs. a•.o will provide a criterion for each acid of how far this method of estimating 0~can be applied.
Co) The strong acids HCI and HBr For HCI, Raman methods have indicated measurable a m o ~ t s o f HCl.molecules at 9 M (') and analogous conclusions have been drawn from NMR measurements of lu) L. O c t , I. Out,tON and M. MAOAT,]. Pk)'s. Rad. 1, 85 (1940).
308
E. HSGI~LDT
HCI, HBr a n d HI. (u) Since H0,measurements exist for HC1 and HBr and the curve o f H o vs: aH,O is the same for these two acids as for HNO3, HCIO t and HsSO4, it m a y be legitimate as a first approximation to use the data in Table 1 even in concentrated solutions in contrast to the weak acids discussed in the preceding paragraph. The H o data used are those given in the review by PAUL and LONG¢3~together TABLE3.--TENTATIVE~t-VALUF~mR HCI FROMEQUATIONSO) AND(4) C~el 1
2 3 4 4"5 5 5"5 6 7 8
9 10 11 12 13
log~pya+
log et
0"20 0"38 0"57 0"82 0"96 tif0 1"23 1"39
o +0-Ol o -0.02 -0.03 -0-04 -0.04 -0.05 -0.05 -0.07 -0-08 -0.12 -0.13 -o.15 -0.16
--0~20 -0.69 -1-05 -I.40 --1.58 -1.76 - I "93 --2"12 --2"56 --2"86 --3,22 --3.59 --3.99
1"76
2"03 2"35 2~71 3"08 3"48 3'87
--4"#1 --4.82
0C
1
,~1 1
006 0-93 0.91 o.91 0.89 0.89 0-85 0.83 0.76 0.74 o.71 0.69
TABLE 4.--TENTATIVE ~-VALUI~ FOR H B r ~ O M EQUATIONS (~)AND (4)
Cmlr
H0
log ~ y ~
0.1 0"5
0"98 0.20 --0-20 --0"71 --1"11 --1.50 --1.93 --2-38 --2-85 --3"34 --3"89 --4"44
0 0.10 0-20 0.39 0.60 0.89
1
2 3 4 5 6 7 8
9 10
1"23
1"60 2.02 2.47 2"98 3"50
log +0"02 0 0 +0"02 +0"03 +0.01 0 0 --0"02 --0"03 --0.04 --0,06
Of.
--d 1
1 ,~1 ,~1 ,~1 1 1
0-96 0.93 0.91 0.87
with those o f H~3FELDT(5) for dilute solutions o f HCI. The resulting 0c-values are given in Tables 3 and 4. These ~-valucs can only be regarded as tentative because log u is obtained as a difference between two rather uncertain numbers. Since no other estimates o f 0~in these acids seem to exist, no direct comparison can be made. The N M R measurements o f REDLICH et Q[.(24) c a n be used for a consistency test. From proton resonance measurements ~ is currently computed from
= s~
+ s~(1 -
~)
~u~ y . MmUDA and T. KANDA, J~ Phys, $oc. Japan9, 82 (1954) tu) G. C. HOOD,O. RVaaLICtland C. A. Reltz.Y, 3". Chem. Phys. 22, 2067 (1954)
02)
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
309
where S is the corrected resonance shift and p is a function of the stoicheiometric mole fraction of acid, X, defined by: 3X (13)
P=2--X
St and Ss are two empirical constants, which thus have to be evaluated from the experiments. By extrapolation to pure water, REDLICHet al. c~ found the following value for $1: S~ = 11.43 (14) For $2 no value has been obtained because it requires knowledge of the dissociation in some reference solution, which is difficult to get for HC1 because pure HCI is not available at the ordinary temperature and pressure.
12 o
o
II I0
9 g 1.0
I
I
I
I
~29 08 0.70~ FIG. 5.--The function Sip vs. ~H0 for HC 1.
Equation (12) indicates that Sip will be a straight line when plotted versus ~. Using the 0~-values in Table 3, SIp from reference 24 is plotted against ~ in Fig. 5, As can be seen, it is possible to draw a straight line through the data giving St = 11.4
and
S~ = 5-3
(15 a, b)
The agreement between $2 ---- 5.3 for HC1 and $2 = 5.32 for HC10~ (~) is interesting but must be considered as fortuitous in view of the large spread of the data. It is worth emphasizing that combination of accurate H 0 and NMR measurements might provide a route for finding the dissociation of the hydrogen halides. For the time being we shall consider the x-values in Tables 3 and 4 as the best at present available, and proceed to compute KyHAfrom (7) in an attempt to obtain values for the dissociation constants with the aid of the activities in HCI and HBr solutions discussed in reference 4. In Fig. 6 log Kyax is plotted against C for HCI, HBr, HNOs and HC104. The KyHA-values for HNOa and HCIO 4 are those of reference 4. The currently accepted K-values for HNO 8 (23.5) (1°) and HC104 (38)(U)are also shown. Analogous extrapolations can be performed for HC1 and HBr, but they must be considered as 'highly uncertain. In fact, the curve for HCIO4 in Fig. 6 can be fitted by the following three functions using least square procedures: log Kyaclo, = 1.58 + 0.224C + 0-04321C 9" log Kyaclo, = 0"255 + 0.564C + 0.02237C ~ log Kyaolo, ----0"037 + 0-7759C
(16a) (16b) (160
310
E. H6onuoT
In (16a) the value for log K = 1.58 given by REDLICH et al. was assumed to be known and the other two constants obtained by the method of least squares. The other two curves are first and second degree polynomials through the experimental points. The three expressions in (16) correspond to quite different K-values. This
6
5-
-
2
'o
// //// I
1
,b
,'2
,'6 c'
FlO. 6.--The function log K]nt vs. the stoicheiometric concentration of acid for: • HNOs C) HCIO, A HCI • HBr.
uncertainty in the extrapolation procedure should also apply to HC1 and HBr. At present only upper and lower limits can be assigned to their K-values by the following argument. As can be seen from Fig. 6, log KylxA is a linear function of C for HC1 and HBr and theline has a slope of ,--0.35. Such a high value for the salting-out constant of an uncharged molecule seems rather unusual. It is more likely that the curve will bend towards the ordinate as suggested in the extrapolation for HCIO4 by R ~ u c ~ et al. For HCI and HBr a lower limit can thus be obtained by a straight line extrapolation of the experimental data. It also seems highly unlikely that Y~x should pass through a minimum somewhere below the range of measurable concentrations, and the lowest experimental values for KyHA can thus be supposed to give an upper limit for K. In this way we find 1.05 ~ l o g K , ~ <2.60; 2.15 ~ logK~s, <4-65
(17a, b)
Use of Hanm'~ett acidity function for estimating hydrogen ion concentration in acids
311
The dissociation of the hydrogen halides has recently been discussed by B~LL~ with reference to the pertinent literature. Appficafion of Kaoult's law to HC1, HBr and HI has given the following K-values :c~ log KHcI ~ 6,
log KHBr ----8;
Iog KHI = 9
(18a, c)
The discrepancy between the values in (17) and (18) amounts roughly to I0~. The main reason may be, as indicated by Fig. 6, that Raoult's law cannot be used for estimating the concentration of undissociated acid. The application of Raoult's law in this instance requires that Y~A should be practically constant. In fact,YHA varies considerably for all acids, from about one to two powers of ten for H N O 8 up to about six for HCIO~ and analogously for HCI and HBr. In view of this behaviour OfyH~, the K-values in (18) cannot be regarded as final. However, recentlyannounced discrepancies between R a m a n and N M R measurements in HCIO4 cN~ indicate the possibility of a higher degree of dissociation than that obtained from the N M R values used when evaluating the ~yH+-values in Table I. If these Raman data are correct the values for HCIO4 would shifttowards the fight in Fig. I in better agreement with the other acids. This would increase ~ and thus also the K-values for HCI and HBr. DISCUSSION The method of estimating the hydrogen ion concentration from H 0 data seems to offer an alternative to other more elaborate methods such as the R a m a n and N M R methods. As may be seen from Table 2 it is also useful in dilute solutions, but here the emf and conductance methods are superior in accuracy. However, the simple experimental technique involved in measuring Ho makes it attractive. It should also, be possible ;to improve the accuracy of the spectrophotometric technique to make it comparable with the other methods. This is especially important when it comes to the determination of o~in HCI and HBr where these fail. Future measurements should also testthe validityof the assumption of ~y~+ being the same function of H0 for all strong acids. Acknowledgements--The author is grateful to Professor L. G. SnJ~N for some valuable discussions and to the Swedish Atomic Energy Committee for financial support. Dr. V. ~ corrected the
Engtish text. ~uj R. P. BELL, The Proton in Chemistry, Chap. VII. Cornell University Press, New York (1959). iN} G. C. HOODand C. A. REILLY,./. Chem. Phys. 32, 127 (1960).
Press Ltd. Primed in Northern Ireland
THE USE OF THE HAMMETT ACIDITY FUNCTION, Ho, FOR ESTIMATING THE HYDROGEN ION CONCENTRATION IN ACIDS E. H~FELDT Department of Inorganic Chemistry, Royal Institute of Technology, Stockholm 70 (Received 8 June 1960)
Abstraet~A method is described where the hydrogen ion concentration in acid solutions is evaluated from H0-data. For CFaCOOH, good agreement is found between this and the NMR methods below 6 M. For the acids HF,CHsCICOOH and CHC1,COOH, pK-values in rather good agreement with those in the literature are obtained. Application to HCI and HBr gives the following tentative Kvalues for these acids: 1.05 < log Kact < 2.60; 2.15 < log K~Br < 4"65 Only upper and lower limits can be given for K because of the difficulty in making a sound extrapolation to pure water. It has also been found that, because of a large variation of the activity coefficient of the undissociated acid, Raoult's law cannot be used for estimating its concentration in aqueous solutions of strong acids. REcmcrLY WYATrt~'~ as well as BASCO~mEand BELL~xb~have shown how acidity functions can be used for estimating the degree of dissociation, = in acid solutions. In the following, a slightly different approach to the same problem will be described and applied to a number of different acids. METHOD The Hammett acidity function, H0, is related to the activity of free unhydrated protons, aa+, by: ~*~
where
log ~ aa+ = --Ho
(1)
~P = yBlyBa+
(2)
~0is the ratio of the activity coefficients on the molarity scale of the basic form, B, and conjugate acid, BH +, of the indicators used for measuring H0. Introducing the stoicbeiometric molarity activity coefficient, ),x+ of H + we get from (1) log~ya+ + log Ca+ ------fir,
(3)
For a monobasic acid, HA, which dissociates into H + and A-, ~, is defined by: Ca+ = Cx- = ~C
(4)
where C is the stoicheiometric molarity of acid. Equation (3) can be used for estimating Ca+ provided the function ~0ya+is known. Concerning~ it is generally accepted for solutions of strong acids in water that, at any given concentration, ~ is independent of the indicator used.**~ This assumption has recently found a striking confirmation. When plotting H0 vs. aa,o for HNO,, HC1, H,SO, and HCIO,, WYATte'~ found that all data fell on a Single curve. Also HBr follows the same curve, c'~ This behaviour implies that, when compared at the same water activity, ~0 as well as the hydration equilibria of the proton are the same in all these different acids.~1''~ In Fig. 1, log ~ya+ is plotted vs. H, for three acids: HNO,, HCIO4 and H,SO,. The experimental c1~(a) P. A. H. WYATT,Disc. Faraday Soc. 24, 162 (1957). (b) K. N. BMCOI~E and R. P. BeLLIbid. 158 (c) O. RZnLICHand G. C. HOOD.,Ibid. 87. ~s~ L. P. ~ , Physical Organic Chemistry p. 267 McGraw-Hill New York'(1940). (s~ M. A. PAOL and F. A. LONe, Chem. Rev. 57, 1 (1957). ~ E. H6OI~LVT, Acta Chem. $cand. 14, 1627 (1960). 3O2
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
303
log f'YH" "
3
^~
. H N Os
~/ o
~HCI~
2
I
"6Ho F~c~. 1.--log ~ + ) , vs. Ho for: @ HNOs A HCIO, 0 HaSO,.
A
O.6
• HN03
,,HCIO~
oH2SQ • HCI
O.2
f/
0
"-'-t• I
I
•~
,3
.2
- Z
I
.i
I
I
o
-1
.... I
-2Ho
FIG. 2.--108 ~yu÷ VS. H e in relatively dilute solutions for: @ HNOs A HCIO, O H,SO 4 • HCI.
data are those used in ref 4. As may be seen, all the data fall on a single curve within the limits of experimental error, which can be estimated to be around -t-0.05. This behaviour is in agreement with WY^~T'S finding. In dilute solutions, ~0ya+ approachesunity because pure water is here taken as the reference state for all activity ooeflicients and He will therefore approach --log Ca+. In Fig. 2, log ~ya+ is plotted vs.
304
E. H6G~LVT
Ho for relatively dilute solutions. The HCI data are those of H6OFBLDT.~s~ As shown in a later section, HC1 is likely to be completely dissociated below 3 M i.e. for H0 > --1. As seen in Fig. 2, logeya+ ~ 0 for H0 > +1 i.e. for C < 0.1 M. This may have the 'foilowing reasons (a) a cancellation of the electrostatic interactions on H + and BH + and Co) the water activity varies too little to cause any appreciable displacement of the hydration equilibria. In the following we shall assume as a working hypothesis that eyn+ is the same function of H0 for all acids as is indicated in Figs. 1 and 2 and evaluate Ca+ from equation (3). It would be expected, however, that for relatively weak acids where the dielectric constant attains low values in concentrated solutions, ion pair formation involving the indicator ions will cause individual variations in 9 and the above assumption will break down. ts~ TABLE 1.--TI~ FtrNc'nON Io : ~PyH+ FOR ROUND H0-VALUES
H0
log cya+
He
log ~ya+
>1 +0"75 +0"50 +0"25 0 --0"25 --0"50 --0"75 --1-00 --1"50
0 0"02 0"05 0.09 0-15 0"22 0"30 0"41 0"53 0"85
--2.00 --2-50 --3.00 --3-50 --4.00 --4.50 --5.00 --5,50 --6.00
1"30 1"70 2"15 2.62: 3.10 3.56 4.04
4.52 4.98
In Table 1 average values for log q~ya+ are tabulated for round He-values. They are collected from the best Raman and Nuclear Magnetic Resonance (NMR) data available to the present writer. They will need revision when a body of more complete and more accurate data has been assembled. APPLICATIONS
(a) Weak and moderately strong acids In the following the method outlined in the previous paragraph has been used for finding the thermodynamic dissociation constants of some acids, and the results are compared with the values in the literature. The thermodynamic dissociation constant is obtained by extrapolation of some suitable function to infinite dilution. Two different functions have frequently been used, the stoicheiometric dissociation constant, Q, or the function Kyax, where YHx is the activity coefficient of undissociated acid. These two functions will now be discussed. For the reaction: H A ~ - H + -k A -
(5)
Q is given by C~+C~Q :
C=~ = Ky.A
Cn-"-"~ :
1 -
~t
(6)
y~+yx-
Since a l l activity coefficients a p p r o a c h u n i t y u p o n dilution, the e x t r a p o l a t e d Q-value will give the t h e r m o d y n a m i c dissociation constant, K. A n a l o g o u s l y we find f r o m (6): a~+ax-
Ky,~x = C(1 -- =) ~s) E. H~K3FELDTalad J. BI~3ELEiSBN,./'. Amer. Chem. Soc. 82, 15 (1960).
(7)
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
305
where the activity product an+aA- can be computed from stoicheiometric mean activity coefficients tabulated in the literature. It is known that for most strong electrolytes activity coefficient products like yH+YX - have a minimum near I M. On the other hand one would expect yr~x to follow a function. log Yax = k C (8) where the constam k is called the salting out constant. This implies that Q might go through a maximum making any extrapolation of Q most uncertain. The possible behaviour of Q can be illustrated by the following example. Assume: log YHx = 0"05C log y a + = log Yx- =
1
~/c
2" 1 + a/-------~+ 0" 1C
(9a, b)
where 0"05 is a tyl£W,ai value for the salting out constant of a non-electrolyte (e) and 0-1 a typical value for an electrolyte.(7) Inserting (9) into (6), differentiation yields: Qm~ for C ~ 0"83 M (10) On the other hand, a maximum in the extrapolation path for KyHx is not likely to appear and this function should always be preferred whenever good activity data are available. Unfortunately, such data are lacking for several of the acids for which He has been measured, so at present it is necessary to resort to the Q-extrapolation in most cases. For CFsCOOH a direct comparison can be made between N M R (s) and He measurements. (') The N M R measurements give directly the ~-values while the H o method employs equation (3)and the data in Table 1. In Fig. 3, ~ from the two sets of measurements are compared. When evaluating ~ from the Ho data o f I~NDLm and T~OER the measurements for the two indicators o-nitroaniline and 4-chloro-2nitroaniline have been used with the pK-values ---0.29 and --1.03 as recommended by PAL-. and LON6(s) and in good agreement with recent measurements.(S, xe) The densities necessary for transformation from the mole fraction scale used by RANVLES and T~DEl~ to the molar scale were determined by the present writer in connection with some N M R measurements. (11) The latter show good agreement with those of REt~ICH et al. (s) below 10 M. From Fig. 3 it is seen that there is rather good agreement up to 6 M. Above that concentration deviations start to occur which are probably an example of a failure of the working hypothesis due to ion pair formation involving the indicator ions. (3~ With knowledge of ~ and the mean activity of acid the function K y a x can be computed and extrapolated to C = 0 in order to give a value for K. When doing so the activities of CFsCOOH-solutions used were the same as those of RE)UCH et alJ s) The agreement between K~rm~ = 1"8 and KH, = 1.4 is encouraging in view of the experimental uncertainty in H0 and the crude treatment of the NMR-data. For the other acids where no direct comparison can be made, Q has been computed from (6) and in Fig. 4 log Q is plotted vs. C for these acids. In Table 2 the pK-values obtained are compared with the literature. (') ~7)
F. A. LONO and W. F. McDm~Tr, Chem, Rev. 51, 119 (1952). C. W . D A r n s , 1. Chem. $oc. 2093 (1938).
(sJ G. G. HooD, O. R ~ t a e H and C. A. I~ILLY, i. Chem. Phys. 23, 2229 (1955). ~') J. E. B. R ~ D L ~ and J. M. " I ~ D n , J. Chem. Soc. 1218 (1955). ~le) K. N. B~COMn and R. P. BY&L,I. Chem. $oc. 1096 (1959). (la) M. BlmoQvlsr and E. H6OVELDT,J. Inorg. NucL Chem., to be published (1960).
306
.
E. H~I~LDT
For HaPO s and HsPOo H0 has not been determined in sufficiently dilute solutions to permit any reasonably good extrapolation. For HsPOs the measurements in very dilute solutions by KOLTHOFF(ls) are also marked in Fig. 4 and they seem to o( o NI"IR
jO~8 @
'0.6
.
o @@
,a4
. o 0
•
~2
O0
I
I
I
I
I
I
I
I
l~e
I
2 ~ 6 8 10 C Fro. 3.--The degree of dissociation, ct, for CFsCOOH vs. stoicheiometric molarity of acid. O NMR-data, • H0-data. TABLE 2.--pK-vXLU~S FROM// Acid
Symbol in Fig. 4.
CH,CICOOH I-IF*
[]
H,PO, Hd'O,
A @
CHCIsC00H CCI,C00H CHsSOsH
-DATA ODMPARED WITH THE IATIFALATURE. t = 2 ~ ° C
Ho-data Ref 10 3 3, 16 10 10
pg(H,
pK(Li0
2"8 3"1
2"86 2"91-3"27 2"12 1"70-1"80 1.06 0"64 --0"6
1"1
10, 2o 0
10
--1"1
* When evaluating 0 for HF he value K = 4 for the reaction F- + H F ~ H F s was employed. (u-xj)
Lit Refs 12 13, 14, 1.' 17 18, 19 3 21 10
(II)
(is) D. J. G. I v ~ and I. H. PRYOR,J. Chem. Sac. 2104 (1955)., (m C. B~tossrr, Naturwissenschaften 29, 455 (1941). qls) C. B. Woorreg, J. Amer. Chem. Sac. 60, 1609 (1938). (Is) H. H. BiOnNe and T. DE V ~ , J. Amer. Chem. Sac, ¢D, 1644 (1947); L OAvATrA Private communication (1960) (x,) A. N. GeLastc~m, G. G. SHCtmOLOVAand M. N. TeurdN, J. lnorg. Chem. U.S.S.R. 1, 282 (1956) ¢17) Stability Constants Part lI, p~ 57-58. The Chemical Society, sptm. publ., London (1958). (xs) I. M. KOLTtIOt~, Rec. Trao. Chtm. 46, 350 (1927). o.0) K. TAKARASmand N. YuI, Bull. Inst. Phys. Chem. Res. (Tokyo) 20, 521 (1941) ca0) H. Buitrd~r, R. MURPHYand D. YARIAN, Indlalla Aead, S¢i. 66, S6 (1957). (st) O. R.BnUCH, Chem. Rev. 49, 333 (1946).
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
307
fit well into the He-data. For CClaCOOH log 0 must have a maximum if the present literature value is accepted as correct. This illustrates the possible drawback of the log 0-plot pointed out earlier. For CHsSOsH the value K = 4, suggested in the literature, was estimated from He-data by BELL(xe) and offers no independent check. The discrepancy between K = 4 and the present estimate of K , ~ 10 from the same data is likely to be due to the different approximations inherent in the two approaches. log( 2 o
. . . . . . . . . . . . . . . .
o
.(I
o
o
C~S~H
~CC~COOH
o
-H,PO~ CHCI,~COOH -2
--
...o-HF
.-CHf/COOH
~ 0
I
I
I
I
I
2
3
,~
C
FlO. 4.--The function log Q vs. the stoicheiometric molarity o f acid for: O CHtSO,H • CCltCOOH, • HgPOs, x HtPOt OKolthofl'), A HaPO4, • CHCIsCOOH , [2 HF, • CHaCICOOH.
For the acids HaPOs, CCIsCOOH and CHaSOaH, 0~ was found to become constant, or to increase at about 7"5, 5 and 9 M respectively. These results indicate that at high Concentrations the method of estimating = from He measurements is not applicable as found for CFsCOOH above 6 M. When good activity measurements are available, the plot of 9~as+ vs. a•.o will provide a criterion for each acid of how far this method of estimating 0~can be applied.
Co) The strong acids HCI and HBr For HCI, Raman methods have indicated measurable a m o ~ t s o f HCl.molecules at 9 M (') and analogous conclusions have been drawn from NMR measurements of lu) L. O c t , I. Out,tON and M. MAOAT,]. Pk)'s. Rad. 1, 85 (1940).
308
E. HSGI~LDT
HCI, HBr a n d HI. (u) Since H0,measurements exist for HC1 and HBr and the curve o f H o vs: aH,O is the same for these two acids as for HNO3, HCIO t and HsSO4, it m a y be legitimate as a first approximation to use the data in Table 1 even in concentrated solutions in contrast to the weak acids discussed in the preceding paragraph. The H o data used are those given in the review by PAUL and LONG¢3~together TABLE3.--TENTATIVE~t-VALUF~mR HCI FROMEQUATIONSO) AND(4) C~el 1
2 3 4 4"5 5 5"5 6 7 8
9 10 11 12 13
log~pya+
log et
0"20 0"38 0"57 0"82 0"96 tif0 1"23 1"39
o +0-Ol o -0.02 -0.03 -0-04 -0.04 -0.05 -0.05 -0.07 -0-08 -0.12 -0.13 -o.15 -0.16
--0~20 -0.69 -1-05 -I.40 --1.58 -1.76 - I "93 --2"12 --2"56 --2"86 --3,22 --3.59 --3.99
1"76
2"03 2"35 2~71 3"08 3"48 3'87
--4"#1 --4.82
0C
1
,~1 1
006 0-93 0.91 o.91 0.89 0.89 0-85 0.83 0.76 0.74 o.71 0.69
TABLE 4.--TENTATIVE ~-VALUI~ FOR H B r ~ O M EQUATIONS (~)AND (4)
Cmlr
H0
log ~ y ~
0.1 0"5
0"98 0.20 --0-20 --0"71 --1"11 --1.50 --1.93 --2-38 --2-85 --3"34 --3"89 --4"44
0 0.10 0-20 0.39 0.60 0.89
1
2 3 4 5 6 7 8
9 10
1"23
1"60 2.02 2.47 2"98 3"50
log +0"02 0 0 +0"02 +0"03 +0.01 0 0 --0"02 --0"03 --0.04 --0,06
Of.
--d 1
1 ,~1 ,~1 ,~1 1 1
0-96 0.93 0.91 0.87
with those o f H~3FELDT(5) for dilute solutions o f HCI. The resulting 0c-values are given in Tables 3 and 4. These ~-valucs can only be regarded as tentative because log u is obtained as a difference between two rather uncertain numbers. Since no other estimates o f 0~in these acids seem to exist, no direct comparison can be made. The N M R measurements o f REDLICH et Q[.(24) c a n be used for a consistency test. From proton resonance measurements ~ is currently computed from
= s~
+ s~(1 -
~)
~u~ y . MmUDA and T. KANDA, J~ Phys, $oc. Japan9, 82 (1954) tu) G. C. HOOD,O. RVaaLICtland C. A. Reltz.Y, 3". Chem. Phys. 22, 2067 (1954)
02)
Use of Hammett acidity function for estimating hydrogen ion concentration in acids
309
where S is the corrected resonance shift and p is a function of the stoicheiometric mole fraction of acid, X, defined by: 3X (13)
P=2--X
St and Ss are two empirical constants, which thus have to be evaluated from the experiments. By extrapolation to pure water, REDLICHet al. c~ found the following value for $1: S~ = 11.43 (14) For $2 no value has been obtained because it requires knowledge of the dissociation in some reference solution, which is difficult to get for HC1 because pure HCI is not available at the ordinary temperature and pressure.
12 o
o
II I0
9 g 1.0
I
I
I
I
~29 08 0.70~ FIG. 5.--The function Sip vs. ~H0 for HC 1.
Equation (12) indicates that Sip will be a straight line when plotted versus ~. Using the 0~-values in Table 3, SIp from reference 24 is plotted against ~ in Fig. 5, As can be seen, it is possible to draw a straight line through the data giving St = 11.4
and
S~ = 5-3
(15 a, b)
The agreement between $2 ---- 5.3 for HC1 and $2 = 5.32 for HC10~ (~) is interesting but must be considered as fortuitous in view of the large spread of the data. It is worth emphasizing that combination of accurate H 0 and NMR measurements might provide a route for finding the dissociation of the hydrogen halides. For the time being we shall consider the x-values in Tables 3 and 4 as the best at present available, and proceed to compute KyHAfrom (7) in an attempt to obtain values for the dissociation constants with the aid of the activities in HCI and HBr solutions discussed in reference 4. In Fig. 6 log Kyax is plotted against C for HCI, HBr, HNOs and HC104. The KyHA-values for HNOa and HCIO 4 are those of reference 4. The currently accepted K-values for HNO 8 (23.5) (1°) and HC104 (38)(U)are also shown. Analogous extrapolations can be performed for HC1 and HBr, but they must be considered as 'highly uncertain. In fact, the curve for HCIO4 in Fig. 6 can be fitted by the following three functions using least square procedures: log Kyaclo, = 1.58 + 0.224C + 0-04321C 9" log Kyaclo, = 0"255 + 0.564C + 0.02237C ~ log Kyaolo, ----0"037 + 0-7759C
(16a) (16b) (160
310
E. H6onuoT
In (16a) the value for log K = 1.58 given by REDLICH et al. was assumed to be known and the other two constants obtained by the method of least squares. The other two curves are first and second degree polynomials through the experimental points. The three expressions in (16) correspond to quite different K-values. This
6
5-
-
2
'o
// //// I
1
,b
,'2
,'6 c'
FlO. 6.--The function log K]nt vs. the stoicheiometric concentration of acid for: • HNOs C) HCIO, A HCI • HBr.
uncertainty in the extrapolation procedure should also apply to HC1 and HBr. At present only upper and lower limits can be assigned to their K-values by the following argument. As can be seen from Fig. 6, log KylxA is a linear function of C for HC1 and HBr and theline has a slope of ,--0.35. Such a high value for the salting-out constant of an uncharged molecule seems rather unusual. It is more likely that the curve will bend towards the ordinate as suggested in the extrapolation for HCIO4 by R ~ u c ~ et al. For HCI and HBr a lower limit can thus be obtained by a straight line extrapolation of the experimental data. It also seems highly unlikely that Y~x should pass through a minimum somewhere below the range of measurable concentrations, and the lowest experimental values for KyHA can thus be supposed to give an upper limit for K. In this way we find 1.05 ~ l o g K , ~ <2.60; 2.15 ~ logK~s, <4-65
(17a, b)
Use of Hanm'~ett acidity function for estimating hydrogen ion concentration in acids
311
The dissociation of the hydrogen halides has recently been discussed by B~LL~ with reference to the pertinent literature. Appficafion of Kaoult's law to HC1, HBr and HI has given the following K-values :c~ log KHcI ~ 6,
log KHBr ----8;
Iog KHI = 9
(18a, c)
The discrepancy between the values in (17) and (18) amounts roughly to I0~. The main reason may be, as indicated by Fig. 6, that Raoult's law cannot be used for estimating the concentration of undissociated acid. The application of Raoult's law in this instance requires that Y~A should be practically constant. In fact,YHA varies considerably for all acids, from about one to two powers of ten for H N O 8 up to about six for HCIO~ and analogously for HCI and HBr. In view of this behaviour OfyH~, the K-values in (18) cannot be regarded as final. However, recentlyannounced discrepancies between R a m a n and N M R measurements in HCIO4 cN~ indicate the possibility of a higher degree of dissociation than that obtained from the N M R values used when evaluating the ~yH+-values in Table I. If these Raman data are correct the values for HCIO4 would shifttowards the fight in Fig. I in better agreement with the other acids. This would increase ~ and thus also the K-values for HCI and HBr. DISCUSSION The method of estimating the hydrogen ion concentration from H 0 data seems to offer an alternative to other more elaborate methods such as the R a m a n and N M R methods. As may be seen from Table 2 it is also useful in dilute solutions, but here the emf and conductance methods are superior in accuracy. However, the simple experimental technique involved in measuring Ho makes it attractive. It should also, be possible ;to improve the accuracy of the spectrophotometric technique to make it comparable with the other methods. This is especially important when it comes to the determination of o~in HCI and HBr where these fail. Future measurements should also testthe validityof the assumption of ~y~+ being the same function of H0 for all strong acids. Acknowledgements--The author is grateful to Professor L. G. SnJ~N for some valuable discussions and to the Swedish Atomic Energy Committee for financial support. Dr. V. ~ corrected the
Engtish text. ~uj R. P. BELL, The Proton in Chemistry, Chap. VII. Cornell University Press, New York (1959). iN} G. C. HOODand C. A. REILLY,./. Chem. Phys. 32, 127 (1960).