4242
W. A. SENIORAND R. E. VERRALL
Spectroscopic Evidence for the Mixture Model in HOD Solutions
by W. A. Seniorla and R. E. Verralllb Mellon Institute, Pittsburgh, Pennsylvania
(Received May 9, 1960)
The significance of isosbestic points in spectral studies is established and the results of this investigation applied to the ir spectrum of HOD in solution over the temperature range 25-90’. The results suggest the existence of an equilibrium between two states of the OD group (“bonded” and “nonbonded”) and a van’t Hoff AH value of 2.3 & 0.4 kcal/mol corresponding t o the bond breaking process is derived.
Introduction
seems to support the continuum mode111~18 we believe this to be a hasty conclusion for two reasons. Firstly, The idea that liquid water may be a mixture of a t least the Raman results on the same system show very defitwo species of water molecules was apparently first nite evidence of at least two components14 and secondly put forward by Rowlands and elaborated by Roentgen the ir band exhibits an isosbestic point when several in 1892.2 Since that time, a great deal of effort has curves, obtained at different temperatures, are supergone into calculating theoretically the thermodynamic imposed. This isosbestic behavior is most generally properties of liquid water using a variety of mixture found in two component systems although, as we shall models.3-8 it may occur in other systems. show, At the same time a number of authors have adhered I n an attempt to clarify the appearance of the OD to an essentially “uniformist” average m ~ d e l . ~No ~l~ band contour and its temperature behavior, we shall clear-cut decision between the two paths has been possifirst investigate the general significance of isosbestic ble because of the lack of direct evidence to indicate points and later look in detail a t the ir band contour of the simultaneous presence of more than one type of HOD at 2500 cm-1. water molecule in liquid water.l’ I n general, the mixture models postulate an equilibrium between two or more types of water molecules roughly corresponding to a transition from a nonhydrogen-bonded species to a hydrogen-bonded type such as might be found in an ice structure. I n model terms, this is usually visualized as corresponding to the formation of a “cluster”6 with a limited lifetime probably between lo-” and lO-’sec. If these lifetimes are essentially correct (and it is difficult to visualize a molecular rearrangement process taking place in much less than lo-” sec), then spectroscopic studies a t frequencies in excess of .u10l2 cps should show the presence of more than one component in the system. The evidence at present available is extensive but not conclusive. The work of Luck12 in the near ir shows that the temperature behavior of the spectra is consistent with the idea of a mixture of components in equilibrium but the complexities of assignment in the overtone region leave room for doubt. Similarly, the Raman and ir studies of the fundamental region of liquid water also depend on a particular band assignment. The one region which appears to be accepted by most spectroscopists to be relatively free of doubt in assignment is the OD stretching vibration of the HOD molecule in solution in HzO. With the proper concentration conditions, this band offers the best opportunities for deciding between continuum and mixture models. Although at first sight the appearance of a single band The Journal of Physical Chemistry
A. Isosbestic Points The occurrence of a point on the absorption spectrum of a two component system (in which the two components are in chemical equilibrium) which is invariant with respect to the relative concentration of the two components was apparently first noted in 1924 by Thiel16 who called this point the “isosbestic point” (1) (a) On leave of absence from Unilever Research Laboratory, Port Sunlight, Cheshire, England. (b) University of Saskatchewan, Saskatoon, Saskatchewan, Canada. (2) H. 8. Frank, First International Symposium on Water Desalin& tion, Washington, D. C., 1965. (3) (a) G. Nemethy and H. A. Scheraga, J . Chem. Phys., 36, 3382 (1962); (b) H. 5. Frank, Fed. Proc., 24, No. 2, Part 111, Supp. 15, 1 (1965). (4) R. P. Marchi and H. Eyring, J . Phys. Chem., 68, 221 (1964). (5) V. Vand and W. A. Senior, J . Chem. Phys., 43, 1869, 1873, 1878 (1965). (6) H. S. Frank and W. P. Wen, Discussions Faraday SOC.,24, 133 (1957). (7) C. M. Davis and T. A. Litovitz, J. Chem. Phys., 42, 2563 (1965). (8) M. D. Danford and H. A. Levy, J. Amer. Chem. Soc., 84, 3965 (1962). (9) J. Lennard-Jones and J. A. Pople, Proc. Roy. SOC.,A205, 155 (1951). (10) J. A. Pople, Proc. Roy. SOC., A205, 163 (1951). (11) T. T. Wall ahd D. F. Hornig, J . Chem. Phys., 43,2079 (1965). (12) W. A. P. Luck, Ber. Bunsenges Phys. Chem., 67, 186 (1963). (13) M. Falk and T. A. Ford, Can. J. Chem., 44, 1699 (1966); 46, 3579 (1968). (14) G. E. Walrafen, J . Chem. Phys., 48,244 (1968). (15) A. Thiel, Fortschr. Chem. Phys. Phys. Chem., 18, 38 (1924).
SPECTROSCOPIC EVIDENCE FOR
THE
(the point of same extinction). Since that time, such a point has been found in many studies on equilibrium systems, notably those involving the spectra of indicators in systems of variable pH.15316 More recently, apparent isosbestic points have been found in the spectra of water and aqueous s 0 1 u t i o n s ~ ~as~ ~the ~ J tem~ perature is varied. Although some of the conditions necessary for isosbestic points to occur have been establishedlg only a limited range of conditions has been covered. We believe our investigations to be of a more general nature although by no means completely comprehensive. We shall show that in general an isosbestic point may occur in a system of any number of components but that the chance of it occurring in a system of other than two components is remote. For a system which obeys Beer’s law, the total optical density y(v) a t any frequency v in the absorption spectrum of a system of n components can be written i=l
(1)
where xI = mole fraction of the ith component and the g,(v) is the optical density of the ith component a t frequency v. If the n components are involved in an equilibrium with one another, then their concentrations must obey the conservative condition n
c x t = 1 i= 1
Combining equations 1and 2 we obtain (3)
i=2
I n general, to observe an isosbestic point we require that for a t least one particular frequency vc, the optical density of the system be independent of some external variable t (t may be temperature, pressure, pH, etc.) i.e., we require
[SI.... =
~dgl(v) dt
O
from which it follows quite generally that)
o or gl(v)
= constant
for at least one frequency v = v,. this requires et
exp
-
For a Gaussian band
{y} -
2
= a = constant
etAvt = y = constant
(6)
Combining ( 5 ) and (6), the more restrictive condition is found to be
- vt =
&Avt{In y
- In (aAvt)]l”
(7) which is probably somewhat more difficult to satisfy than ( 5 ) . We conclude, therefore, that although an isosbestic point is possible in a pure material, it is extremely unlikely. More probably a pseudo-isosbestic region will be found and this region will grow larger as the range of variable t is extended. II. Two-Component Systems. For a two-component system we must return to the general isosbestic eq 4. Although there are a number of ways in which (4) may be satisfied, only one of these is of real importance. This is the case when dgi(v)/dt = 0, i.e., the intrinsic band shapes of each component are independent of t. Because, in general, we know that the equilibrium constant willdepend on t, it follows that dx,/dt # 0 and therefore to satisfy (4) we must have gI(v,) = g2(v,) which for Gaussian bands requires el exp
-
{y} - {2
=
e2
exp
vcA;2
(8)
and thus the frequencies vc of the isosbestic points are the solutions of v,2
-}
-- 1 {Ai22
Av12
- --}
- 2.,{” Avt2
v1
Avi2
+
}I:[
- ~n for a t least one frequency Y = Y,. Equation 4 is the general equation governing isosbestic behavior in systems of any complexity. We shall, however, be concerned with only certain special cases governed by (4). I . One-Component Systems. In these cases, eq 4 reduces to
(5)
which may be satisfied if e t , vz, and Avz are all constant or if they vary in the manner prescribed by ( 5 ) . Such behavior is difficult to achieve. Equation 5 is the least restrictive one to allow an isosbestic point and more usually it is probably necessary to impose the additional condition that the integrated area remains constant (i-e., constant oscillator strength) which requires
v,
n
c SdV).Xt
Y(.> =
4243
MIXTUREMODELIN HOD SOLUTIONS
= 0
(9)
Although eq 9 can be solved for v,, the isosbestic frequency, the form of the solutions does not easily indicate what happens when the band parameters are altered. A more convenient method is to inspect the (16) R. P. Bauman, “Absorption Spectroscopy,” John Wiley and Sons, Inc., New York, N. Y., 1962, p 419 ff. 17) K. A. Hartman, Jr., J.Phys. Chem., 70,270 (1966). (18) J. D. Worley and I. M. Klotz, J. Chem. Phys., 45, 2869 (1966). (19) J. Brynestad and G. P. Smith, J. Phys. Chem., 7 2 , 296 (1968). Volume 78, Number 12 December 1969
4244
W. A. SENIOIE AND R. E. VERIEALL
part, Q, of the solution which occurs under the square root sign and see when this leads to real and imaginary solutions. This quantity Q is simply
Q =
(v2
- v d 2 + ( A V I-~ Avz2) In
(~z/EI)
(10)
the solutions proper being of the form
a
bQ1” (11) where a and b are constants determined by the band parameters. A number of different cases must now be considered. (i) Two Bands of Identical Shape v, =
€1
=
€2;
Avi = AVZ
:. In { E Z / ~ ]I = 0 and Q is a perfect square and it can be shown that there is only one isosbestic point which lies midway between the two bands. (ii) Two Bands of the Same Extinction Coeficient €1
=
€2;
Avl # AVZ
Again, In { ez/al ] = 0 and Q is a perfect square, but since Avl P AVZthere are now always two isosbestic points if VI # v2, although in practice for bands of not too different shape, AVI II Av2 and one isosbestic is well away from the main band area and in a region of very low absorption. If V I = v2 only one isosbestic occurs and this a t the frequency of maximum absorption, i.e., v, =
VI
= v2.
(iii) Bands of Equal Width AVI = A v ~ ;€1 #
€2
Again, Q is a perfect square and it is easily shown that in this case there is only one isosbestic point which may lie between or outside the two maxima, depending on the relative intensity of the two bands. (iv) Bands of Equal Area (e.g., constant oscillator strength) Equation 10 can now be rewritten as = (VZ
- v d 2 4-
(Av12
- A V Z ~ln(AvllAv2) )
< Av2 and € 1 < €2 or A V I> AVZand €1 > e2
The first term in (10) is always >O and the second term is always O, = 0 or O two isosbestics occur exactly similar in form to cases (iv) and (v). If Q = 0 only one isosbestic occurs very similar to case (iii). If Q < 0 no isosbestics occur. This happens when
-
VI)^
< (Avt2 - A V I ~In) ( ~ z / E I )
(13) As a simple example consider the case € 2 = eel In E Z / Q = 1 and if (v2 - VI)^ < (Avz2 - Av12) there will be no isosbestic point. For two typical bands one might have AVZ= 100 cm-’ and Avl = 50 cm-I so that if the band centers are separated by O and hence Q is always >O and there are two isosbestic points for all values of ( v 2 - ~ 1 ) ~ As . vl + vz they become more symmetrically placed on each side of the band maximum. (v) One Strong Sharp Band and One Weak Broad Band Avl
< AVZand el > €2 or AVI> AVZand el < €2
The second term in (10) is always >O and thus Q is always >O. Thus there are always two isosbestic points. Again, as v1 + v2 they become more symmetrically placed around the band maximum. (vi) One Weak Sharp Band and One Strong Broad Band The Journal of Physical Chemistry
(14)
i.e., there must be a t least one frequency at which all components have the same optical density. This
elAv1 = EZAVZ
Q
Av1
FREOUENCY
Figure 1. Nonoccurrence of an isosbestic point in a two-component equilibrium system. Both components obey Beer’s law.
SPECTROSCOPIC EVIDENCE FOR
THE
means that the band profiles of all components present must intersect in one unique point and this condition is almost impossible to satisfy even for only three component systems. Systems of this type containing more than three components must be completely unable to meet this condition. Another type of multicomponent system which is of bD E B C interest is of the form A F . . . . .. This type of system can exhibit isosbestic points under special conditions, e.g., since there is only one equilibrium constant and if only one component on each side of the equilibrium is absorbing, an isosbestic may exist-mathematically it behaves as a two component system. This type of equilibrium can also be varied by addition of one or more of the component materials. I n this instance, the law of mass action plays a part and introduces certain special relationships between the amounts of the various components. If, a t the same tirne, other special relations are observed, e.g., keeping the total number of moles of material constant, isosbestic points can be made to occur.2o If, however, the equilibrium is not constrained by very special conditions and is varied only by altering the equilibrium constant (or constants), Le., by varying P and T then isosbestic points will only occur if relations of the form (14) hold between the band profiles. I n multicomponent systems, one might observe a pseudoisosbestic region if the equilibrium conditions are altered only by a small amount. However, the extent of this region would increase as the range of variation of the equilibrium conditions was widened. If the band parameters vary as the equilibrium conditions are altered, perfect isosbestic behavior would only be observed if very stringent requirements were obeyed. I n general, it appears that a dependence of this type would again lead to the existence of an isosbestic region, if the band parameters varied only slightly. I V . XigniJicanceof Isosbestic Points. In sections II11 of this paper we have examined some of the most relevant conditions under which isosbestic points might occur. Somewhat surprisingly it is possible, although rather improbable, to observe isosbestic behavior in a system containing only one pure component so long as the band shape is reasonably sensitive to the external variable t. I n a two component system, a genuine isosbestic is probable only if the band shapes and positions are very insensitive to t and the equilibrium constant is dependent on t. I n addition, in two component systems one should point out that it is entirely possible that no isosbestic will occur even though the equilibrium is rigidly obeyed and even though both components have absorption bands which obey Beer's Law. I n systems of high complexity (three or more components) it seems extremely unlikely that an isosbestic point would be found, except in rather special circumstances. Bearing in mind the foregoing, it seems reasonably safe to state the case as follows. The existence of an
+ +
4245
MIXTUREMODELIN HOD SOLUTIONS
9
+ +
isosbestic point implies the presence of either a one or two component system. If, in addition, it can be established that the observed band is composed of two subbands then the existence of the isosbestic point is virtual proof of a two component equilibrium system. On the other hand, the absence of an isosbestic point has little or no significance in any type of system. One should point out that we have been concerned only with theoretical conditions necessary for isosbestic points to occur. I n many practical cases, a genuine isosbestic point may be present theoretically but may not be observable if, for instance, it occurs in a region of very low absorption. Again, in real situations, any isosbestic is almost certain to occur as a pseudo-isosbestic region rather than a mathematically precise point. I n these circumstances, it is essential to use a wide enough range of conditions to establish that the isosbestic region has a limited area. Our investigations lead us to believe that if the area of the isosbestic region is greater than 1% of the total band area, it would be dangerous to conclude that any equilibrium exists without other information to support the conclusion.
B. Infrared Investigations of the 0-D Stretching Vibration in H20-D20 Solutions Having established the significance of isosbestic points in spectroscopic investigations, we are now in a position to examine in detail the experimental results we have obtained on the 0-D stretching vibration of the HOD molecule in solution over a range of temperature.
I. Experimental Section The spectrometer used was a Beckman IR-9 which was continually purged with dry nitrogen to minimize any spurious absorptions due to water vapor and COz. The spectra were recorded with the instrument operating in the double beam mode using a pen response time of 0.1 sec a t a scanning speed of 60 cm-'/min. Samples were prepared in RIIC cells (Model KO. FHO1) fitted with LiF windows and used with an electrically heated thermostat jacket supplied by RIIC. Cell temperatures were recorded continuously and were kept constant to & l oover the 10-min period of each run. Spectra were recorded on both the heating and cooling cycles to establish that no loss of sample occurred a t the elevated temperatures. The concentration and cell thickness conditions are, we believe, very important. The band to be observed has its maximum absorption a t -2500 cm-l and because of the excess water (H2O) in the system, there are additional bands about 2150 (fairly weak) and a t 3420 cm-I (very intense). The result is that if a very low D20concentration is used with a compensating path (20) C . F. Timberlake and P. Bridle, Spectrochim. Acta, 23A, 313 (1967).
Volume 73, Number 1,9 December 1969
4246
length to make the band observable, there is a very large curved background which must be taken into account. The desired spectrum is then the small difference between two very large signals and correction procedures are extremely difficult. At the other end of the scale, ie., a high DzO concentration, the solution contains a considerable number of D 2 0 molecules and these absorb radiation a t exactly the same frequency as the HOD molecules. I n the concentration range 5-10 mol % D2O in HzO the contribution from any remaining D 2 0 molecules can be kept below 2% of the observed band and, a t the same time, the background correction can be held to less than 5% of the total band. By comparison with a spectrum for pure H2O obtained under the same conditions it was established that the background in this region is almost exactly a straight line and the correction used was of this type. Under these conditions a cell thickness in the range 1-2p is necessary and this is not too difficult to achieve.21 We have therefore worked in this concentration and cell thickness range. At the D2O concentrations used, the presence of HOD-HOD pairs is substantial. There are, however, a number of reasons why we believe this type of coupling to have no influence on our results. Firstly, Raman results obtained on similar systems by BernsteinZ2and W a l ~ - a f e nindicate ~~ that the asymmetry is independent of the isotope concentration within the range 1-10 mol %. Secondly, the observed splitting of A25 cm-l found by Haas and H ~ r n i g in * ~ice and attributed by them to this coupling effect is likely to be a maximum value as the coupling effects are almost certain to be strongest in the solid state. Our observed frequency difference between the two components is 125 cm-' and it is very unlikely, therefore, that this splitting can be accounted for in terms of HOD-HOD coupling. Finally, the work of Schulte and HornigZ6on ionic solutions indicates that whereas the 0-H vibrations of water arc sensitive to anions, they are very insensitive to cations, suggesting that the 0 atom of the molecule has a very considerable screening effect. Therefore, it seems rather unlikely that the central molecule can distinguish whether its nearest neighbors possess H or D atoms except perhaps in the solid state where the intermolecular coupling will be strongest. We therefore believe our results to be characteristic of the uncoupled OD group. 11. Experimental Results
Figure 2 shows the experimental results obtained in Taken individually, the temperature range 29-87'. the curves show little or no sign of any asymmetry which might indicate the presence of more than one component. The remarkable feature of the set, of course, is the almost mathematically precise isosbestic point which occurs, and in view of our preceding analysis we must take further steps to decide whether or The Journal of Physical Chemistry
W. A. SENIORAND R. E. VERRALL
ts -
FREQUENCY
CM-'
Figure 2. Plot of optical density us. frequency for the ir absorption of HOD at -2500 cm-1 for various temperatures. The presence of an isosbestic point a t -2575 cm-1 is clearly shown.
not our system contains more than one component. The frequency of the isosbestic point is 2575 cm-' which coincides almost exactly with that found by Walrafen14in his Raman studies on the same band. I n the Raman effect also, there seems to be no doubt of the existence of at least two components as the results show a fairly obvious shoulder, -2650 cm-', in addition to the main maximum at -2525 cm-l. Hartman'sL7 results on the OH band of HOD in DzO also exhibit an isosbestic point. In order to try and establish the structure of the ir band we have used a numerical differentiation technique. The observed curves were carefully measured over the frequency range 2200-2800 crn-l at 10 cm-l intervals. Successive differences are then taken between adjacent points and the modulus of the difference is plotted os. the mean frequency of the interval. In this way, numerical differentiation of the observed curves is carried out which has the advantage that no assumption about band shape need be made. Because the differential of a curve is much more sensitive to noise fluctuations on the data than the curve itself, we have found it advantageous to smooth our raw data before carrying out the numerical differentiation. This is done in the following way. Starting from point 1, a parabola is fitted through the first N points (where N determines the degree of smoothing required and, in our case, was 6) using a standard least squares method. The resulting equation is used to calculate the Y value at point 1 and this value replaces the original value. The process is then (21) W. K. Thompson, Trans. Faraday Soc., 61, 2635 (1965). (22) W. F. Murphy, W. Holzer, and H. J. Bernstein, Raman Newsletter, No. 1, 10 (1968). (23) G. E. Walrafen, J. Chem. Phm., 50,560 (1969). (24) C. Haas and D. F. Hornig, ibid., 32, 1763 (1960). (25) J. W. Schultz and D. F. Hornig, J . Phys. Chern., 65,2131 (1961).
SPECTROSCOPIC EVIDENCE FOR THE MIXTURE MODELIN HOD SOLUTIONS repeated starting from point 2, etc. Using the corrected points, the first derivative is then obtained as outlined above. We have tested these procedures by truncating a 5-figure error table at the third decimal place. This procedure gives rise t o random errors -0.2% on the standard error curve (at half-height) which, in turn, errors in the first derivative gives rise to -5-10% obtained by our method. When the initial data is smoothed as above and then differentiated, the resultant error in the differential is -0.5%. A judicious choice of the measuring interval and the N value (number of smoothing points) is necessary to obtain best results. This is easily done after one or two trials. Using the above procedure we have obtained the first differentials AYl) for the set of curves in Figure 2 and these results are shown in Figure 3. On the low frequency side of the minimum, all the curves behave as one would expect from the differential of a single component band. However, on the high frequency side of the minimum there is a distinct inflection at -2690 cm-' which occurs at all temperatures and which becomes more pronounced as the temperature is raised. The position of this inflection is consistent with the presence of a second component in the original band at about 2650 em-'. This agrees very well with the observed band at 2650 em-l in the Raman effect. We conclude, therefore, that the -OD ir band of HOD liquid contains at least two components with positions 2525 and 2650 em-l. This is, then, in complete agreement with the observed Raman results. The positions of these bands are also consistent with the idea of two states of the OD group, L e . , a hydrogenbonded OD group having a vibrational frequency -2525 ern-' and a non-hydrogen-bonded OD group with a frequency -2650 em-l. The behavior of the two bands as the temperature is varied is also quite consistent with this assignment. Further analysis indicates that the peak positions and half band width of both components are practically independent of temperature. Again, Walrafen's Raman results14 show exactly similar behavior. Of further interest is the experimental result obtained by Kecki et aL126in a systematic infrared study of the effect of electrolytes on the OD and OH stretching band intensity in HDO-H20 and HDO-D20 solutions. In the case of NaC104, the OD stretching band of HDO is split into two components having frequencies of 2530 em-' and 2640 cm-'. The position of the high frequency component is invariant to perchlorate concentration but its intensity increases at the expense of the low frequency component as the perchlorate concentration is increased. These frequencies agree rather well with those obtained in this paper on the salt free HDO-H20 solutions. Perchlorate anion was the only anion in the above studyZ6which exhibited this effect on the OD stretching band.
,015
4247
1
(1
Figure 3. Moduli of the first differentials of the ir spectra of HOD (Figure 2) as obtained by the numerical differentiation and smoothing technique outlined in the text. The asymmetry a t ~ 2 6 9 0cm-l is clearly shown.
On balance, one might speculate that the effect of the perchlorate anion consists mainly in breaking the "normal" hydrogen bonds and therefore increases the concentration of nonhydrogen-bonded OD component a t the expense of the bonded component. It appears that the unique behavior of perchlorate anion in aqueous solution, compared to other anions of similar size, may be related to its tetrahedral symmetry. The results of this work and ref 26 appear to support the concept of a mixture model for liquid water much better than they do that of the continuum model. It is interesting to note that the peak position of the high frequency component (-2650 cm-l) is well below the &-branch 0-D absorption of HOD vapor at 2720 em-'. This finding further substantiates the results of Franck and RothZ7on the 0-D absorption of HOD under supercritical conditions. If a vmax vs. T curve of their results for p = 1.0 g/cm3 is extrapolated to higher temperatures it approaches an asymptotic value well below 2700 em-', indeed at about 2650 em-'. The implication is that nonhydrogen-bonded molecules in the liquid (or dense fluid) state are in no way comparable to free water molecules in the vapor phase. One other feature of the experimental results is worthy of comment-the broadness of the observed ir and Raman bands. It is certain that this broadness arises from geometrical factors, ie., a t a certain instant in time if one could examine the detailed structure of the liquid it would be found that in the "bonded" material hydrogen bonds of a wide range of lengths existed. This suggestion is supported by the width of (26) Z. Kecki, P.Dryjanski, and E. Kozlowska, Roezniki Chem., 42, 1749 (1968). (27) E. U. Franck and K. Roth, Discussions Faraday SOC.,43, 108 (1967). Volume 78, Number 18 December 1069
4248
W. A. SENIORAND R. E. VERRALL
the peaks found in the X-ray radial distribution function for water, (e.g., $he peak at -2.90 h; has a width a t half-height of -0.5 AZ8and by the fact that the radial distribution function can be derived from the Raman spectra. l 1 Vand and Senior's5 assumption of broadened energy bands is exactly equivalent to the above suggestion. In view of our conclusions in section A.IV therefore, it seems fairly certain that an equilibrium between two states of the OD group does exist, particularly since the Raman evidence entirely supports this proposition.
h ( ( l - XA)/XA]= In C - AH/RT
Now In [(l - X)/X] = In [X(1
- X)/Xz] In [X(1
*
= GAXA
(15)
Defining the equilibrium constant for the reaction as = XB/XA = (1
- XA)/XA Cexp
or The Journal of Physical Chemistry
- X)] - l n X 2
In [(l
- X)/X]
= lnX(1
- X) - 2 1 n X
(18)
Combining (18) and (17) we obtain In XA = In [XA(l -
XA)]~'~
- ~n c'"
+ AH/2RT
Assuming the conclusions of section B.11 to be correct, our next step is to try and determine the enthalpy change involved between the two states. Normally this is done by decomposing the total band into its two components at a series of temperatures and assuming the band areas (or peak heights) of each component to be proportional to the number of molecules in each state. A simple plot of log A / A Z vs. 1/T then gives the van't Hoff AH for the process from its slope. The main difficulty with this procedure is the band resolution step. This is usually accomplished by more or less refined techniques of curve fitting multiple Gaussian or Lorentzian bands to the observed band profile. At best, this means an arbitrary choice of band shape and, at worst, one has no guarantee that the final solution is unique. It seems likely that this technique is satisfactory if the bands do not overlap too strongly (ie., in the Raman case where two peaks can be distinguished) but there must be serious doubts about it in the type of situation we have in the ir spectrum. However, we have used the above method using both Gaussians and Lorentzians to derive a van't Hoff AH value of 4.5 0.8 kcal/mol for the process OD(H-bonded) -F OD(non-H-bonded) . We have, however, found an alternative method of extracting this information which needs no assumptions about the band shapes. The only requirement is that we must be able to find a frequency range in the observed band at which one component has little or no absorption compared to the other. Consider the system A % B where the mole fraction concentrations of A and B are XA and Xg respectively. The optical density a t the same frequency a t which the absorption of say B is zero is then purely determined by XAand the absorption coefficient of A, GA, giving
K
=
or
111. Thermodynamic Quantities
Y
(17)
- AH/RT
(16)
(19)
or via (15) In Y = In [xA(1 - XA)]"'
+ In [GA/C'/'] + AH/2RT
(20)
The function In [ x A ( 1 - XA)]"' is a very slowly varying function of XA and for values of XA in the range 0.3-0.7 is constant to within * 5 % . For XA in the range 0.2-0.8 it is constant to within *12.5%. Therefore for values of XA in the range 0.3-0.7 a plot of In Y vs. 1/T will be a straight line with a slope AH/2R. If the XAvalues lie far outside the above range then the plot will be a curve not a straight line. Similarly, if a wavelength has been chosen such that condition (15) is not obeyed, this is also obvious from the graphical presentation, a peculiar S-shaped type of curve being obtained. In practice, therefore, the In Y vs. 1/T graphs are plotted for a number of wavelengths covering the range of absorption of the band and a set of
'I
2 5-05
2600\
I
2350
0.
2.5
3.0
3.5 2.5
3.0
' / T U 3
Figure 4. Plot of log (optical density) us. 1/T a t various frequencies across the HOD absorption band ~ 2 5 0 0em-'. The numbers at the left of each curve give the frequency at which the measurements were made. (28) J. Morgan and B. E. Warren, J . Chem. Phys., 6,666 (1938).
I
3.5
SPECTROSCOPIC EVIDENCE FOR
THE
MIXTUREMODELIN HOD SOLUTIONS
two or three lines are found in the regions where eq 15 ” constant. A is obeyed and where In [ X A ( ~ X A ) ] ~ is set of such curves obtained from our data is shown in Figure 4. Note that the set of lines lie in the region -2425 cm-l where indeed we would expect to find eq 15 best satisfied. The very low curvature also indicates that the condition In [ x A ( 1 - X A ) ] ~ ”constant ~S is well satisfied. The deviations from linearity at higher frequencies are due to the presence of the 2650 cm-’ component and those at -2350 cm-l and below are almost certainly caused by the presence of small amounts of C02 gas in the instrument. C02 gas has a strong absorption at 2350 cm-’ which is very difficult to eliminate even in double beam instruments. The van’t Hoff AH value obtained using this alternative method is 2.3 i 0.4 kcal/mol and is the van’t Hoff heat associated with the disruption of the 0-D. unit. This value of 2.3 kcal/mol is to be compared with the value of 2.5 kcal/mol obtained by Walrafen14 in his Raman studies of the same process, as well as with the value of 2.4 kcal/mol obtained by Worley and Klotz18 from near infrared studies of HDO in D20. Using this method one cannot obtain the actual concentrations of the two components unless some assumption is made about the relative extinction coefficients of the species. However, the high degree of linearity of the In Y us. 1/T plots in the region where (15) holds indicates that the concentration ratios are probably about 2 : 1 or 1: 1 with the bonded state being the most abundant. The apparent discrepancy between the numerical values of the van’t Hoff AH obtained in this work by two different methods requires further comment. It must be pointed out that, because of differences in the mode of presentation, the comparable results from the two methods of analysis are 2.25 and 2.3 kcal/mol; Le., there is close sgreement between them. However, for reasons mentioned previously in this section, we feel the second method offers a more objective analysis of the data. One other point concerning the variation of the van’t Hoff heat values obtained from spectral studies is worthy of mention. This argument is not presented to account for any difference in the AH values of the present work but, rather as a possible reason for the
-
a
0
4249
general inconsistency among results to date. It has been shownz9that, for cooperative hydrogen bonding phenomena in helix-coil transitions of some synthetic polypeptides, the transition involving hydrogen bonding can take place over a narrow or wide range of temperature depending on the degree of difficulty in starting an ordered region. An “equilibrium constant” constructed in the ordinary way would therefore show a temperature dependence which would vary depending upon the degree of cooperativeness of the process of hydrogen bond formation. FrankSohas suggested that an analogous situation might arise in the 3-dimensional case of the equilibrium between bonded and unbonded species in liquid water, giving rise to different van’t Hoff AH values from studies of different properties. The important implication is that the van’t Hoff heat for hydrogen bond formation may be different from the calorimetric value, and that the latter might well be considerably smaller in absolute value than has sometimes been imagined. S1
C. General Conclusions I n conclusion, it appears to us that the weight of spectroscopic evidence supports the idea of a two state model for the OD group of the HOD molecule in solution in HzO. It follows that at least two states of the H2O molecule must be present in liquid water. The detailed geometrical structures involved remain to be resolved in the future.
Acknowledgments. Both of us are deeply indebted to Professor H. S. Frank for his most helpful encouragement, advice, and friendly criticism during the course of this work. We wish to thank Dr. G. E. Walrafen for providing us with details of his work prior to publication. R. E. V. acknowledges the award of a N.A.T.O. Fellowship from the National Research Council of Canada and W. A. S. is grateful for financial support from the Office of Saline Water, Department of Interior, under Contract 14-01-001-403, which sponsored this work. (29) B. H. Zimm and J. K. Bragg, J. Chem. Phus., 31,526 (1959).
(30) H. 8.Frank, private communication.
(31) Nemethy and Scheraga (ref 3a) have given an illuminating d i s cussion of why this might be expected to be true.
Volume 79, Number 19 December 1960
