March, 1862
ACID-BASEEQUILIBRIA OF METHYL RED
hindrances. These molecules deviate considerhbly from sphericity and appear to have molecular envelopes in the shape of a large tetrahedron. However, although tetrahedrally-shaped rnolecuies often show the onset of the plastically crystalline phase, this phnomenon does not appear to be present in either of the two methyl-substituted polythiaadamantanes studied. It is of‘ interest to note that the hexathia compound has been reported to exist in a crystal lattice of rather low symmetry. l1 The monoclinic
527
space group Cth-P21/c is said t o characterize the lattice and the molecules do not appear to possess any element of crystallographic symmetry. Acknowledgment,-The authors acknowledge with gratitude the assistance of Elfreda Chmg and H. Gary Carlson in the experimental measurements, and the financial support of the Division of Research of the United States Atomic Energy Commission in the performance of these studies. (11) G. Higg and B N ~ g a r d hreported by A Fredga, ref
0.
ACID-BASE EQUILIBRIA OF 3IETHYL RED BY RICHARD W . RAMETTE, EDWARD A. DRATZ, AND PRESTON W. KELLY Leighton Hull of Chemistry, Curleton College, North$eld, Minnesota Recoived
OCfObs3T
30, I961
-
The cation of methyl red, or o-(p-dimethylaminopheny1azo)-bennoicacid, has an acid dissociation constant IL 0.0040 according t o spectrophotometiic measurements as well as solubility studies in HCLKCI buffers in which S = 6.0 X 1.54 x 10“3[Ha0*]. Distribution studies using these bufferawith carbon tetrachloride show that the aqueous/CCL concentration ratio for methyl red follows the relationship E = 0.00295 0.585[H30+I, indicating the higher value far K1 of 0.0050, perhaps because of the dissolved CC14. Spectrophotometri? studies and solubility measurements in acetate buff ers, in which S = 6.0 x,X0-6+ 8.9 X I0;l1/[H30+], showthatthedissociatioii constant for the zwitterion Kz = 1.50 X 10-8, The intrinsic solubility of the zwitterion is %bus 6.0 X lo’* according to bath solubility studies. Thme values refer to a temperature of 2 6 O and ,are calculated for aero ionic strength, but the work was carried out a t ionic strength equal to 0.020.
+
+
Introduction determined Q values if the values of the activity Methyit red, or o-(p-dimethy1aminophenylazo)- coefficients can be estimated reliably. benzoic acid, has been the subject of ionization Experimental constant studies for half a The colReagents.-lilethyl red (Eastman 431) was purified by lected reaults refer to several temperatures, the slow recry&allization from redistilled toluene, resulting in effects of ionic strength often were not considered, large crystals. For the spectrophotometric work ethanolic
and some of the experimen’cal approaches were inherently inaccurate. The purpose of the present work has been to use refined spectrophotometric techniques for the determination of accurate ionization conRtant values, and to examine the usefuIness of immiscible solvent extraction and solubility studies for the same purpose. In aquleous solution methyl red exists in three forms; a cation, HsMfl which is in%enselyred, a species H A 4 which is undoubtedly a zwitterion because of the similarity of its absorption spectrum to that of H2M+ (see Fig. I), and a yellow anion, M-. Thle famiIiar use of methyl red as an acidbase indicator is based on the equilibrium between IM-and IIM, In neutral solvents such as benzene and carbon tetrachloride HM in yellow, indicating that the switterion reverts to the non-ionic structure; (C~Es)2NCsH4;”\JZNCsH4COOR(o). The cation may be regarded as a diprotic acid, and two equilibrium quotients (in terms of rnolarities) may be defined iii the usual way &I = [H~O+l[HMl/[H&f+l Qz = [Ha.O+l[M-I/[HMl The corresponding equilibrium constants, Kl and
Kz may be calculated from the experimentally (1) H. T. Tizard, J . Chem. Sue.. 97, 2477 (1910). (2) A. Thiel, F. Wulfken, and A. Dassler, 2. anorg. u. allgem. Chem., 186, 393 (1924).
(3) I. M. Kolthoff, Ree. trav. chcm., 48, 144 (1924); 44, 75 (1925). (4) A. hlerstoja, Ann. Acad. Sei. Fennzeae Sec. A, I I , Chsmica, No. 12, 7 (1944). ( 5 ) F. A. F. Vermast, Indonesian J . Nut. Sci., 109, 57 (1953). (6) S. W. Tobey, J . Chem. Eduu., 86, 514 (1958). (7) C. N. Reilley and E. M. Smith, Anal. Chem., 82, 1233 (1960).
stock solutions were prepared by weighing the cnlculated quantity and diluting to make the molarity eQual to 1.00 X 10-8. The carbon tetrachloride wa8 redistilled, solutions were standardized by conventional methods, and all reagents were of high quality. Apparatus -Photometric measurements were made with a Beckman Model B instrument using both 1-em. rectangular and 5-cm. cylindrical Pyrex cells. The 8-cm. cells were calibrated ‘.C
3 4 7 10 13 16 20
I
5 405 G.993
1.208
...
10.48 12.40 14 65 I
I
it
amount)
HM(non4onic)
If
Organic layer
Fig. B,-Determination
0.8 1.2 1.6 2.0 10*[H80*I. of &I through solubhty in HCI-KCI buffers.
8.888
2.106 2.647 3.082 3 681
Solvent Extraction Study.-In principle the solvent extraction approach is very similar to the solubililiymethod. In both cases an aqueous buffer is equilibrated with an “inexhaustible reservoir” of HNI, arid the transfer of methyl red to the buffer goes through a minimum a t the isoelectric point. However, me have not attempted pin extraction study a t the higher pH values because of the compiications which mould be intipoduced by the distribution and dimerization of acetic acid. The hydrochloric acid, potassium chloride buffers of ionic strength 0.0200 were used and a diagram of the equilibrium system may be shown as Aqueous * HM( zwitterion) &I-( slight layer
0.4
01
HM(non-ionic)
The distribution of HM between Ihe two phases has been indicated in two steps, (1) the “true” distribution of the non-ionic form which would have a distribution coeficieiit Kd, and (2) the zwitterion formation which is presumably a p r o p ess having an equilibrium constant K,. In practice it is not possibJe to observe the presence of the non-ionic form in the qquaous phase and we resort to writing a combined distribution coefficient, D = [HM],,/ [HAI],,,, which includes both Kdand K,. Following Sandelill Tye define an ext&,ction coefficient, E , to be the ratio of the total aqueous methyl red concentration t o the total organic layer concentration. When the quotients Ql, Qz, and D are algebraically comrbined we find that The right-hand term of equation 8 is due to the (11) E. B. Sandall, “Colorimetric Determination of Traces of Metals,” 3rd edition, Interscience Publishers, Nem York, N. Y . , 1959, p. 55.
0.8 1.2 1.6 2.0 lO’[HaO+]. Fig, 7.-Determination of Q1 through distribution of methyl red between carbon tetrachloride and HC1-KC1 buffers. Q.4
presence of a slight amount of M- in the series of solutions. This term can be evaluated with ample accuracy because QZ is known from the other studies and D can be determined by successive approximations. In any case, the term is very smaH compared to the others and we write E ‘ = D + - D [Ha0 +I Q1
(9)
which suggests that a plot of E’ vs. [H,O+] (analogous to the solubility study a t higher acidity) would be linear with an intercept equal to D and an intercept/slope ratio equal to Q1. The data are in Table V and Fig. 7 shows the straight-line plot. The least-squares calculations give a value of 0.00295 k 0.00013 for the distribution coefficient. The results are fairly precise and yet the value for Q1 (0.0050 f 0.0002) is significantly higher than those obtained in the spectrophotometric and solubility studies. X possible explanation for this difference is that the HhI form of methyl red tends to associate with the carbon tetrachloride which has dissolved in the aqueous pbqse. This would shift the aqueous equilibrium somewhat in favor of HM, or in other words would make H&f+ appeaz to be a bit stronger as an acid. It might be asked whether methyl red undergoes polymerization iii the carbon tetrachloride solutions. Attempts to study this were inconclusive and this aspect was not pursued because in the e&-action study so little dye was transferred from the organic layer thet the effect of dimerization
532
Vsl. 66
HAROLD 8.JORX\~~TON AND JULIAN HEICKLIW
would be constant, and would mean only that D as reported here was not a true monomer distri-
bution coefficient, Acknowledgment,-We
are grateful to the Re-
search Corporation for a Cottrell Grant, and to E. I. du Pont de Nemours & Company for departmen tal grants which were used partially for support of this work.
TUNNELLING CORRECTIONS FOR UNSYMMETRICAL ECKART
POTENTIAL ENERGY BARRIERS BY HAROLD S. JOHNSTON AND JULIAN HEICKLEN Department of Chemistry, University of California, Berkeley 6,California Reoeivsd October $0, 1981
Tunnelling corrections have been evaluated for the unsymmetrical Eckart potential for ranges of parameters expected for ordinary chemical reactions at ordinary temperatures.
I n computing chemical reaction rates using activated complex theory, one must include a correction for quantum mechanical barrier penetration and non-classical reflection, the effects referred to as "tunnelling." For small degrees of tunnelling, a correction was derived by Wigner.' Bell2 worked out the tunnelling problem for a truncated parabola and a Boltzmann distribution of incident systems. Shavitta and Johnston and Rapp4 computed tunnelling corrections for symmetrical Eckarts functions for chemically interesting values of the parameters involved. The present article presents similar calculations for the unsymmetrical Eckart function. Eckart's one-dimensional potential energy function is y =
- exp(2r x/L)
(2)
where 5 is the variable dimension and L is a characteristic length. For the symmetrical function, A is zero. Both a symmetrical and an unsym-
metrical Eckart function are given by Fig. 1. It is seen to be flat at both - w and C O . The maximum value is AVi above the value a t m and AV'z above the value a t m. F" is the second derivative of the potential energy function evaluated a t its maximum. The parameters A , B, and L in eq. 1 and 2 are related to AV1, AV,, and F" of Fig. 1by
+-
+
A
AVI
-
(3)
AVg
B = [(AV,)'/l
+ (AV1)'/:I9
(4)
The inverse relations are
+ B)'/4B
(A
AVi AVz
(A - B)a/4B -F* = rZ(A2 Bz)/2LSBa
-
(6) (7) (8)
A particle of mass,m and energy E approaching the barrier is characterized by the parameters u", aland a2 u* = hv'/kT
P-
(9)
Y*
=
a1
E
(1/2r)(-F*/m)'/g 2nAVi/hv*
a2
E
2rAVn/h~*
(10) (11) (12)
In these variables, the probability K ( E )that a particle starting toward the barrier with energy E a t - a will pass the barrier and appear later at m with energy E is found by solving Schroedinger's equation for the Eckart function, and thc transmission probability is6
+
\
\ \
\
3
'.---.
cash 2 r ( a
K(E) = 1
where
- b ) f cash 2nd
- cos11 2r(a + b ) f
-+-
27fa = 2[a,E]'/2 (a1-1/2
-2
-1.0
-0.5
Fig. 1.-Symmetrical
0.5
0
1.0
A.
and unsymmetrical Eckart function.
E is one example of the variable energy considered in eq. 13. V* is the same as AV1. F* is dV/dzZ evaluated a t the maximum.
(1) E. Wigner. 2. physik. Chem. (Leipzig), Bl9, 203 (1932). (2) R. P. Bell, Trans. Faraday SOC.,66, 1 (1959). (3) I. Shavitt, Theoretical Chemistry Laboratory, University of Wisconsin, Madison, Wisconsin, Report AEC-23, Series 2, 3 (1959). (4) H. S. Johnston and D. Rapp, J . Am. Chsrn. SOL,83, 1 (1961). ( 5 ) C. Eckart. Phys. Rev., 36, 1303 (1930).
cash 2nd
a2-1/2)-1
2 r b = 2[(1 f $)a1 - a4'/z(a1-'/1 2 ~ = d 2[alaZ- 4rz/16I1/a 4 = E/AVi
+
a2-'/2)-1
(13)
(14) (15) (16)
(17)
When d is imaginary, the function cosh 2nd in eq. 13 becomes cos 2 4 dl. The tunnelling correction factor r*is interpreted as mechanical rate r * -- quantum classical mechanical rate
With a Boltzmann distribution of incident particles
