Raman Spectra from Saturated Water Vapor to the Supercritical Fluid

But simple logic demands that energy must be added to break the hydogen bond, but not to make it. Hence, the dimer cannot be the true high-energy stat...
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J. Phys. Chem. B 1999, 103, 1332-1338

Raman Spectra from Saturated Water Vapor to the Supercritical Fluid G. E. Walrafen,* W.-H. Yang, and Y. C. Chu Chemistry Department, Howard UniVersity, Washington, D.C. 20059 ReceiVed: July 22, 1998; In Final Form: December 18, 1998

Raman OH stretching peak frequencies from saturated water vapor decrease by about 20-30 cm-1 as monomers (OH peak frequency, 3657 cm-1) are replaced by dimers (diagnostic frequency e 3628-3638 cm-1) with temperature rise from 298 K to Tc ) 647 K. Dimerization increases with temperature in the saturated vapor because of the increase in the pressure/density, and this pressure/density driven dimerization extends into the supercritical fluid until additional polymerization produces trimers, etc., at higher densities. Increasing H-bonding (dimerization) in the saturated vapor equals the decreasing, mainly T-driven, liquid H-bonding when T rises to Tc, where the H-bonding is roughly 17%. This means that the mole fraction of (H2O)2 molecules XA is ≈0.7, and the mole fraction of H2O molecules XB is ≈0.3, at the critical point. H-bonding in water at the critical point is not abnormal, because it is essentially independent of the correlation length divergence. The H-bond ∆H of the dimer is discussed.

Supercritical water is of considerable importance, e.g., it has environmental, industrial, and synthetic applications, and is of geologic and oceanographic interest, see discussion and references cited in ref 1. Accordingly, it is currently being very intensely studied, both theoretically2,3 and experimentally.4,5 The intense interest in supercritical water has led to some unusual claims about its hydrogen bonding. For example, a claim that hydrogen bonds in water are absent at the critical point6 has been refuted by its authors and others.2,4,7 A recent article by Ikushima et al.8 also concludes that the hydrogen bonding in water is peculiar at the critical point. This peculiarity refers to the claim that there is a marked decline in the extent of H-bonding at the critical point, compared to the sub and supercritical regions. The procedure of previous workers8-11 was to follow the Raman spectrum of the liquid to the critical point and then to continue into the supercritical region at higher temperatures and pressures. This procedure only required the use of moderate laser power, e.g., 100-500 mW. However, we applied a different procedure made possible by high laser powers (4 or 9 W at 488 nm for observations of ν1 or ν3, respectively, of the monomer). We followed the Raman spectrum of the saturated Vapor to the critical point, and then we examined the supercritical fluid at higher temperatures and pressures. It was not necessary to examine the Raman spectrum of subcritical liquid water in this work, because we were able to employ reported Raman peak frequency data.8 We obtained Raman OH-stretching peak frequencies (and spectra) for the saturated vapor from 25 °C to the critical temperature and above. An Inconel Raman cell with two (or three) sapphire windows was used for 90° scattering, employing 4 W of 488 nm laser excitation.12 This cell was used for quantitative Raman intensity measurements between 255 and 314 °C, where a spectral resolution of 8 cm-1 was employed. We also confirmed our measurements by means of backscattering (4W, 488 nm) using a massive in-line one (or two) window Raman cell made of stainless steel, where the spectral resolution was 5 cm-1. (This latter cell was designed for use to 10 kbar; we could easily reach temperatures of 400 °C and above with it.)

A volume of liquid water sufficient to guarantee saturation was placed in the Raman cell, prior to measurement, e.g., the amount of water needed to produce saturation at 314 °C was exceeded by 100% in the quantitative Raman intensity measurements between 255 and 314 °C. The horizontal laser beam passed far above the liquid water, which, therefore, was not excited. However, air was not excluded from the cell, because it was convenient to check our intensity measurements via the 2329 cm-1 N2 peak intensity. Some experimental peak frequencies, ∆νMAX, in cm-1, are shown from 120 °C to about 400 °C in Figure 1, saturated vapor and supercritical fluid, upper curve. (Peak frequencies from 25 °C were obtained but not plotted in Figure 1.) Recently reported liquid and supercritical ∆νMAX values8 are included, lower curve, where P ) 226 bar (i.e., just above Pc ≈ 220.6 bar). (We replotted the experimental points and curve of Ikushima et al.,8 but our plot followed the data somewhat more closely than theirs just below the critical point.) ∆νMAX values from the saturated vapor begin near 3656 cm-1 at 120 °C and decline to the critical point where ∆νMAX ≈ 3628-3638 cm-1. In contrast, liquid ∆νMAX values8 rise rapidly, as seen from Figure 1. Our measured ∆νMAX value for the saturated vapor at 25 °C is 3657 cm-1. This value was obtained by 90° scattering. Also the differences between the 90° and back-scattering data shown, respectively, by circles and squares, Figure 1, are thought to arise mainly from the use of different sapphire windows, whose preparation and attendant polarization properties yielded different peak frequencies. Ikushima et al.8 employed the well-established criterion that a rise in ∆νMAX qualitatively indicates H-bond breakage. They thus concluded that there is a peculiar decline in H-bonding of the liquid near the critical point. However, with the same criterion8 we conclude just the opposite for the saturated Vapor, namely, that H-bonds are formed in the vapor as the temperature rises to the critical point. However, there is no disparity between these two conclusions, as demonstrated below. The complete vapor, liquid, supercritical fluid picture presented by Figure 1 indicates an entirely normal situation in that phase separation obviously occurs below the critical point, which

10.1021/jp9831233 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/10/1999

Raman Spectra from Vapor to Fluid

Figure 1. Raman OH stretching peak frequencies from water vapor and the supercritical fluid, upper curve, x, this work, 90° scattering, and !, this work, back-scattering. Liquid and supercritical fluid, lower curve, 4, were obtained from ref 8. Circles guide the eye. Open circles connect our data, whereas filled circles connect the data of ref 8. Tc, arrow.

causes the ∆νMAX values to split into two diverging branches, one for the saturated vapor, and the other for the corresponding liquid. Ikushima et al.8 observed an inflection and subsequent rise of slope in the liquid OH-stretching peak frequency versus temperature, see Figure 1, lower curve, near 3580 cm-1 and 350 °C. This is discussed below, but even the slope change and rise are entirely normal and expected, see NBS/NRC Steam Tables, pp 157 and 15.13 Saturated vapor ∆νMAX values reflect the dimerization equilibrium from P ≈ 1 atm to Pc ≈ 221 bar. The declining saturated-vapor, ∆νMAX curve, Figure 1, signals that the monomer, ∆νMAX ) 3657 cm-1, is being replaced by the dimer, peak frequency, ∆νMAX e 3628-3638 cm-1, as the temperature rises to the critical point. But the temperature rise seems subordinate to the concomitant and very rapid rise in the vapor pressure, which forces monomers to polymerize via hydrogen bonding into dimers or higher aggregates (le Chaˆtelier’s principle). The rapid pressure increase along the saturation line overcomes H-bond breakage which would ordinarily be expected with temperature rise at constant pressure, e.g., for the lowtemperature liquid at 1 atm. Data are presented below which demonstrate that the polymerization (dimerization) in saturated water vapor is directly, and very strongly, pressure/density driven. A situation analogous to the pressure/density driven dimerization in saturated water vapor has recently been reported for the vaporization of 50 mol % HF-H2O,12 where HF dimers are formed at the elevated vapor pressures produced by rising temperatures. Also because dimers appear to be the prevalent

J. Phys. Chem. B, Vol. 103, No. 8, 1999 1333 H-bonded water species at pressures below the critical pressure, we suggest that the approach of the heat of vaporization to zero at the critical point means that liquid water and its saturated vapor are both largely composed of dimers just below Tc. Phase separation below the critical point yields a saturated vapor density less than the liquid density.13 Accordingly, Raman vapor and liquid ∆νMAX values are unequal below the critical point. The heat of vaporization ∆Hv demands that vapor H-bonding be less than liquid H-bonding, which produces higher vapor peak frequencies than those of the liquid. The large vapor peak frequencies correspond to a small density and vice versa for the liquid. Phase separation, as portrayed by Raman ∆νMAX’s, must yield the highly unequal, frequency bifurcation seen from Figure 1. This frequency bifurcation is an entirely normal and expected consequence of phase separation. We emphasize that the liquid ∆νMAX values of Figure 1 refer to a constant pressure of 226 bar and not to the saturation pressures. Accordingly the peak frequency values of Figure 1 for the liquid should differ from the corresponding peak frequency values obtained at the saturation pressures, although the difference should decline as the critical point is approached. The OH stretching peak frequency has long been known10 to fall with increasing density above the critical temperature, e.g., at 400 °C, see also ref 9. The peak frequency at 400 °C falls from about 3638 cm-1 near the critical density, to roughly 3550 cm-1 when the density is 1 g/cm3.10 Comparisons with known dimer frequencies,14,15 suggest that higher aggregates, e.g., trimers, etc., might become important at densities well above the critical density. This observation, plus the important and very obvious flattening of the peak frequency versus temperature curve from the critical point, up to just below 500 °C, Figure 1, suggests that the dimer concentration is roughly maximal at, or just above, the critical temperature. The H-bonding would be 25% if the critical fluid only contained dimers. There are four H-bonds per two water molecules in a fully H-bonded tetrahedral network, whereas the dimer has one H-bond per two water molecules; hence, the H-bonding for the dimer is 25%, relatiVe to the fully H-bonded tetrahedral network. An estimate of roughly 17% H-bonding, obtained below, is in qualitative agreeement with NMR data,1 and with theoretical calculations.2 Good agreement with the X-ray work of Yamanaka et al.5 and with the neutron scattering results of Tassaing et al.4 also results. Ikushima et al.8 also mention dimerization at the critical point. Quantitative, density-normalized, Raman vapor intensities were obtained in this work from 255 to 314 °C by 90° scattering and are presented in Figure 2. Density normalization means that the quantitative Raman intensity at each value of ∆ν is divided by the density.13 This normalization is essential because the mass of water in the excitation volume of the laser beam increases with temperature along the saturation curve, whereas the intensity data should refer to a constant mass of water in a fixed volume. The raw, unnormalized Raman intensities obtained from 255 to 314 °C were quantitatively intercomparable to within about 1%. This was accomplished by continuous monitoring of the laser power level during and between runs and by establishing, visually, that the sapphire windows were absolutely free of water droplets. The raw, unnormalized quantitative Raman spectra did not and could not display an isobestic point, because the vapor density increases with temperature. However, the densitynormalized Raman spectra display a well-defined isobestic point

1334 J. Phys. Chem. B, Vol. 103, No. 8, 1999

Walrafen et al.

Figure 2. Density-normalized Raman OH stretching vapor intensities obtained from 255 to 314 °C. The isobestic point is indicated on the figure, see arrow, above. Density normalization means that the intensities were divided by the density of the vapor.

at ≈3648.5 cm-1, as seen from Figure 2. The experimental uncertainty in this isobestic point is about (0.5 cm-1. We believe that the isobestic point displayed by Figure 2 separates monomers, whose symmetric OH stretching frequency occurs at 3657 cm-1, from dimers, etc., whose diagnostic frequency is e3628-3638 cm-1. This latter frequency of e 3628-3638 cm-1 corresponds to the critical, or supercritical plateau, frequency value, see Figure 1. We emphasize that we do not make detailed vibrational assignments here, but we instead infer that the diagnostic frequency value of e36283638 cm-1 involves an envelope of unresolved rotationalvibrational features, which refer to symmetric, as opposed to asymmetric, OH stretches from dimers, etc. This inference comes from our observation that the Raman intensity of the monomeric ν3 vibration was observed to be extremely weak compared to the ν1 symmetric OH stretching vibration, and a similar weakness would be expected for asymmetric stretching in the dimer or higher polymers. The 3657 cm-1 symmetric stretching contour of the monomer displays low-frequency asymmetry at 25 °C. This complicates contour deconvolution using symmetric components (Gaussians). Therefore, we separated the integrated intensities according to areas either above or below the isobestic point of 3648.5 cm-1, as opposed to Gaussian analysis. (Work on contour deconvolution is in progress, and supports the above-described separation, but is not described here.) We consider that the NHB, non-H-bonded (symmetric) OH stretching mode corresponds to the integrated intensity above 3648.5 cm-1, and that the HB, H-bonded (symmetric) OH stretching modes, correspond to that below, that is, the monomer corresponds to the integrated intensity above the isobestic frequency, and the dimer, etc., to that below. (The asymmetric stretches are discussed below. They are much too weak in the Raman spectrum to invalidate the 3648.5 cm-1 contour separation). However, for brevity in subsequent figures, we designate the NHB or monomer, density-normalized, integrated intensity above the isobestic frequency as B, and the HB or dimer, density-normalized, integrated intensity below the isobestic frequency as A. In summary, NHB ) B ) monomer, and HB ) A ) dimer. A plot of the A and B intensities versus F2 is shown in Figure 3. A intensities in part A and the B intensities in part B. The important and major conclusion resulting from Figure 3 is that the A intensity is linear in F2 (or in P2). This finding

Figure 3. Density normalized intensities, A intensities below the isobestic frequency of 3648.5 cm-1, part A, and B intensities above the isobestic frequency, part B, plotted versus F2, with units of g/l for F. The A intensities shown in part A were fitted by linear least squares. B ) NHB ) monomer, and A ) HB ) dimer.

very strongly implicates the dimer, because the probability of dimerization varies as F2 (or P2). The intensity of B, in contrast, declines nonlinearly, as expected, for the monomer. The same

Raman Spectra from Vapor to Fluid

J. Phys. Chem. B, Vol. 103, No. 8, 1999 1335 TABLE 1

Figure 4. Plot of ln(A/B) vs the saturation pressure P. The rise in the HB or dimer to NHB or monomer intensity ratio, i.e., ln(A/B), with rise of P indicates that the dimerization is P-driven.

general shapes as those of Figure 3 were obtained from plots of A (density-normalized intensity of A) and B (densitynormalized intensity of B) versus P2. The formal expression for the dimerization reaction is

2H2O ) (H2O)2

(1)

The mole fraction equilibrium constant corresponding to eq 1 is: Kx ) XA/(XB)2, where XA refers to dimers, (H2O)2, and XB refers to mononers. The concentration equilibrium constant Kc is proportional to the ratio A/B2. Values of ∆H and ∆V obtained from Kx, Kc, as well as (A/B), are discussed in the Appendix. The ratio (A/B) is useful despite the fact that it does not correspond to an equilibrium constant for eq 1. A plot of ln(A/B) versus the saturation pressure P is shown in Figure 4. If (A/B) were an equilibrium constant, ∆V would correspond to the formal volume change for the reaction, B f A, and it would be given by (∂ln K/∂P)T ) -∆V/RT, if the temperature were constant. But the temperature is not constant. Nevertheless, we have found that the behavior of dln(A/B)/dP corresponds to a ∆V which is negative, and also to a ∆V which declines to a very small negative value below Pc. (Moreover, negative ∆V’s which also decline to small negative ∆V values at high temperatures result for Kx or for Kc, when they are used instead of (A/B)). Declining ∆V values relate to a maximum in the dimer concentration in the vicinity of the critical point, because there is no further decline in ∆V to favor P driven dimerization. (Higher polymerizations, of course, may occur at higher pressures.) The rise in ln(A/B) with P means that the dimerization is favored, P-driven, by rise of the saturation pressure. This implies

XA

T, K

0.208 0.211 0.217 0.230 0.251 0.282 0.328 0.396 0.479 0.570 0.648 0.694 0.696

372.8 393.4 416.8 443.6 468.2 492.7 517.3 543.1 568.2 593.3 617.9 643.0 646.9

by le Chaˆtelier’s principle that the volume of the products is less than that of the reactants, see eq 1. We emphasize that the Figure 4 result is useful for determining the H-bonding up to and at the critical point, despite the fact that (A/B) is not the correct equilibrium constant expression. The ln(A/B) versus P function is simply a useful form for extrapolation to the critical point. The data of Figure 4 were fitted with a quadratic polynomial equation of the form: Y ) ln(A/B) ) a + bP + cP2. (The a, b, and c least-squares coefficients are a ) -0.664 303 5, b ) 0.019 317 66, and c ) -0.000 042 679 17.) The maximum pressure from this least squares polynomial fit was found to be PMAX ) 226.31 bar which is only 2.6% larger than the critical pressure Pc ) 220.55 bar. This agreement is well within the accuracy of the Raman intensity data and the extrapolation from 314 to 374 °C. The (A/B) ratios corresponding to PMAX and Pc were then calculated from the quadratic equation, namely, 4.580 and 4.573, respectively. These ratios were then used to approximate the % H-bonding at the critical point employing a mode intensity assumption discussed next. Examination of the data of Huang and Miller14 indicates that there are only two symmetric OH stretching modes of the dimer well below the 3648.5 cm-1 isobestic frequency. Moreover, the only symmetric OH-stretch of the monomer occurs well above the isobestic frequency, at 3657 cm-1. The monomer asymmetric OH stretch; the dimer, acceptor, asymmetric OH-stretch; and, the dimer, donor, free-OH stretch14 occur above the range of Figure 2 and/or are so weak that they make no significant contribution whatever to the current analysis. See the minor isobestic frequency near 3665 cm-1 in Figure 2, and the slight base-line rise with elevated temperature above this second isobestic frequency. We assumed that the HB intensity is proportional to the number of symmetric OH stretching modes, namely, two, below the isobestic frequency, that is, 2f[dimer] ) A, where f is a constant experimental detection factor. Similarly we assumed that the NHB intensity is proportional to the one symmetric OH stretching mode of the monomer above the isobestic frequency, i.e., 1f[monomer] ) B, where f is the same constant experimental detection factor. Accordingly, the Raman intensity ratio is (A/B) ) [2f(nA/V)]/[1f(nB/V)], where nA and nB are the number of moles of A and the number of moles of B. It follows that nA/nB ) (A/2B) ) XA/XB. Because XA + XB ) 1, substitution of (4.580/2)XB or (4.573/2)XB for XA gives a value of XB ) 0.304. Hence, XA ) 0.696. It thus follows from the previously described value of 25% H-bonding for the hypothetical gas composed solely of water dimers that the H-bonding at the critical point is (0.696)(25%) ) 17.4%. A list of XA values at various temperatures is given in Table 1.

1336 J. Phys. Chem. B, Vol. 103, No. 8, 1999 The (H2O)2 molecule contribution at the critical point thus corresponds to about 70 mol %, whereas the H2O molecule contribution is only about 30 mol %. This result, of course, is based on the above mode intensity assumption. The value of 17.4% H-bonding at the critical point agrees well with the value of ≈20% estimated from the NMR data of Hoffman and Conradi1 at the critical point. We conclude that the H-bonding of water at and near the critical point is entirely normal. Vapor H-bonding due to dimerization increases with pressure along the saturation curve until it approaches the declining H-bonding of the liquid, which finally yields a single, finite, H-bond concentration of roughly 17% at the critical point, which corresponds to ≈70 mol % (H2O)2 and ≈30 mol % H2O. Frequency bifurcation observed below the critical point is expected from the two-phase system. The well-known correlation length divergence does not seem to produce any special H-bonding effects in the Raman spectrum of water at its critical point. This relates to the fact that the observed Raman OH-oscillator frequencies are modified almost exclusively by nearest-neighbor H-bonding or not, if free, and thus are insensitive to long-distance correlations. The only striking feature of the liquid H-bonding as seen from Figure 1, is the extremely rapid rise in the peak frequency of the liquid between about 350 and 374 °C, at a constant pressure of 226 bar. Note that the slope of the peak frequency versus temperature curve in this region is enormous. This enormous slope may result, microscopically, from the very rapid breakup of trimers, etc., to form dimers, which should be a necessity, provided that the saturated vapor and its liquid both mainly contain dimers immediately below the critical point. In macroscopic terms, the enormous slope is a consequence of the rapid decline in the liquid density with temperature detailed in Tables 1 and 3 of the NBS/NRC Steam Tables.13 Lastly, the pressure/density driVen supercritical polymerization to dimers, trimers, and higher aggregates is simply the continuation of the dimerization in the saturated vapor, which is also pressure/density driven, as the vapor pressure rises to the critical pressure. Dimerization in the saturated vapor begins at the lowest temperatures (120 °C and well below) and continues to the critical point, as depolymerization, i.e., H-bond breakage of the corresponding liquid, which, compared to the vapor is relatively much less pressure sensitive and mainly T-driven, occurs simultaneously with temperature rise. Appendix Determination of the H-bond enthalpy for the water dimer is a difficult problem with the current Raman data because the pressure is not constant. This means that the van’t Hoff method which employs the constant pressure partial derivative, (∂ln K/∂T)p ) ∆H°/RT2, should not be used. Consider Figure 5 in which ln(A/B) ) ln(HB/NHB) is plotted versus 1/T. Here we use an Arrhenius-like approach, instead of the correct equilibrium constant, and we also incorrectly disregard the inconstancy of the pressure. A linear least squares fit of the data of Figure 5 is shown on the figure. This fit is of the form ln (A/B) ) β/T + R, and the ∆H corresponding to this fit is 4800 cal/mol, with a conservative uncertainty of (800 cal/mol. The H-bond enthalpy value of 4800 ( 800 cal/mol agrees satisfactorily with a theoretical value of ≈4700-4800 cal/mol reported by Hankins et al.16 and with a recent molecular dynamics value of about 5500 cal/mol obtained by Mountain.3 However, this agreement occurs when the sign of the ∆H is changed to account for the pressure-driven nature of the reaction.

Walrafen et al.

Figure 5. Plot of ln(A/B) versus 1/T for the vapor along the saturation curve. Unfilled circles correspond to a least squares fit of the data shown in the inset. Circles with crosses refer to data.

The sign of the ∆H obtained from the linear fit of Figure 5 is positiVe for B f A. This positive sign occurs because rise of temperature on the saturation curve increases the vapor pressure thus increasing the dimer concentration. This pressure-driving effect means that A is the high-energy state. But simple logic demands that energy must be added to break the hydogen bond, but not to make it. Hence, the dimer cannot be the true highenergy state. We will reverse the sign of the ∆H values at the conclusion of this Appendix. This sign reversal is correct and necessary to go from pressure-driven, to temperature-driven, conditions. It makes the H-bond energy positive in agreement with the fact that energy is needed to dissociate the dimer. Nonetheless in the following treatment we must retain the measured enthalpy Values with the positiVe sign, that is, the thermodynamically determined sign, throughout our calculations. Adequate treatment of the Raman data requires the use of the full derivative dln K/d(1/T), which allows the temperature and pressure to vary along the saturation curve. Consider that

ln K ) f(P,T)

(1A)

The exact differential formalism for dln K then gives

dln K ) (∂ln K/∂T)pdT + (∂ln K/∂P)T dP

(2A)

Division of eq 2A by dT gives, dln K/dT ) M + N(dP/dT), where M refers to the first partial derivative on the right hand side (RHS) of eq 2A, and N refers to the second partial derivative on the RHS of eq 2A. Substitution of ∆H/RT2 for M, -∆V/RT for N, and further substitution of the Clapeyron equation for dP/dT, where dP/dT ) ∆Hvap/T∆Vvap, leads to the result,

-R dln K/d(1/T) ) -RS ) ∆Hreaction ∆Vreaction(∆Hvap/∆Vvap) (3A) which we subsequently apply to the dimerization reaction in the vapor along the saturation curve. ∆Hreaction and ∆Vreaction are characterized in ref 17. The subscript, reaction, refers to the reaction 2B f A. ∆Hvap/∆Vvap is the ratio of the enthalpy change upon vaporization to the corresponding volume change. S is the slope of lnK versus 1/T for the vapor along the saturation curve. Pressure, temperature,

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J. Phys. Chem. B, Vol. 103, No. 8, 1999 1337

and molal volume or density are all variables along the saturation curve, necessitating full derivatives in eqs 2A and 3A. Equation 3A requires an accurate, empirical, least-squares function for the ratio of the enthalpy and volume changes of vaporization. The function ∆Hvap/∆Vvap ) f2t ) exp[a + bt + ct2 + dt3 + et4 + ft5] was found to be very satisfactory. (a ) -4.353 246, b ) 6.557 021 × 10-2, c ) -2.472 270 × 10-4, d ) 6.697 791 × 10-7, e ) - 1.100 192 × 10-9, and f ) 8.225 660 × 10-13, and t is in °C, whereas f2T is the function after conversion to T in K.) This exponential function fitted the NBS/NRC data13 to better than 0.03% over the temperature range from 100 to 350 °C. Such accuracy is required because the enthalpy and volume changes of the reaction are very sensitive to errors in ∆Hvap/∆Vvap. Our next step was to solve a series of simultaneous equations at 2° increments from 120 to 314 °C, the range of our Raman intensity data. The equations are of the form

f1(T) ) ∆Hreaction - ∆Vreaction[f2(T)]

(4A)

where f1(T) ) -RS, and f2(T) was defined above. More specifically, we solved equations simultaneously as follows:

f1(Ti) ) X - Y[f2(Ti)]

(5A)

f1(Tj) ) X - Y[f2(Tj)]

(6A)

where i and j refer to temperatures separated by 2 K, and X and Y refer to the enthalpy and volume changes of the reaction, respectively. In the simultaneous solutions we assumed that the enthalpy change and volume change of reaction are constant over small increments of T, e.g., 2°. This in effect yielded an average of the enthalpy and volume changes of reaction over these 2 degree increments. A plot of ln(A/B2) (A/B2 is proportional to Kc) is shown versus 1/T in Figure 6. The results of the ∆Hreaction and ∆Vreaction calculations are shown in Figures 7 and 8, where the circles represent points obtained at 2 K increments of temperature, and calculated from the functions f1(T) and f2(T), as opposed to raw data. Note that the calculated results shown in Figures 7 and 8 extend above our upper experimental intensity range, i.e., above 314 °C, to the critical temperature, 374 °C. Our Raman intensity data range was 120-314 °C. Some ∆Hreaction values in this range are 404 cal/mol at 120 °C, 6031 cal/mol at the midpoint temperature of 217 °C, and 9983 cal/ mol at 314 °C. The ∆Vreaction values rise nonlinearly from about -4612 cm3/ mol near 393 K to -142 cm3/mol at 587 K. This rapid change in the ∆Vreaction values, from negatively very large to negatively very small, reinforces the previous conclusions about the pressure-driven nature of the dimerization reaction. It also suggests that the concentration of the dimer is maximal near the critical point, as is evident from the fact that the curve of Figure 8 tends to flatten near the critical point. We emphasize the fact that the same general trends in ∆Hreaction and ∆Vreaction depicted in Figures 7 and 8 were also obtained in extensive simultaneous equation calculations involving the mole fraction equilibrium constant Kx as well as A/B. The trends of Figures 7 and 8 are by no means unique to Kc. Moreover, it is also stressed that the change in the ∆Hreaction and ∆Vreaction with T necessarily involves a concomitant change in P, i.e., plots of the heat and volume changes of reaction versus P would be just as valid as those of Figures 7 and 8. In view of the common trends found from calculations involving both the concentration and mole fraction equilibrium

Figure 6. Plot of ln(A/B2) versus 1/T for the vapor. Circles with crosses refer to data. The fit shown by the stippled circles refers to a least squares fit, namely, ln(A/B2) ) A + B/T + C/T2.

Figure 7. Values of the enthalpy change for the reaction 2B f A calculated from simultaneous equations.

constants, and also from A/B, it appears that the T and P dependences of the ∆Hreaction cannot be discarded, despite the fact that they are surprisingly large. Moreover, a detailed

1338 J. Phys. Chem. B, Vol. 103, No. 8, 1999

Walrafen et al. curtailed by the steric effects of other very close molecules, dimers and possibly monomers, as the density rises with increasing temperature, particularly near the critical point. The result should be a strengthening of the H-bonds with rising pressure, as the steric effects of the nearby (≈3 Å) molecules, e.g., in clusters (3), force the dimer into stable configurations involving strong H-bonds. Finally, the ∆H from Figure 5, and the ∆H from Figure 7 at the experimental midpoint temperature of 217 °C yield a plausible range for the ∆H of (T-driven) H-bond formation when their signs are changed as indicated at the beginning of this Appendix, that is, we suggest that the ∆H for forming H-bonds with decline of temperature, and under isobaric conditions, would fall roughly within the range of -4800 ( 800 and -6300 cal/mol, respectively. References and Notes

Figure 8. Values of the volume change for the reaction 2B f A calculated from simultaneous equations.

thermodynamic analysis of d∆H/dT yielded ∆CP values from d∆H/dT ) ∆Cp - (dP/dT)[∆V - T(∂∆V/∂T)p], but such values were not particularly illuminating. Therefore, we offer a mechanistic explanation for the strong T- and P-dependence of the ∆Hreaction, albeit speculatiVe. The lowest energy configuration of the dimer corresponds to the donor water molecule lying in the plane which bisects, and is perpenducular to, the acceptor water molecule.16 However, there is a great opportunity for bending between the two water molecules, as well as for rotation of the free proton of the donor water molecule about the adjacent OH-bond. This bending and protonic rotation should tend to weaken the H-bond of the dimer, and this weakening should be most likely to occur at low pressures, when the dimers are far from other molecules. However, the bending and protonic rotation should be sharply

(1) Hoffmann, M. A.; Conradi, M. S. J. Am. Chem. Soc. 1997, 119, 3811. (2) Kalinichev, A. G.; Bass, J. D. J. Phys. Chem. A 1997 101, 9720. (3) Mountain, R. J. J. Chem. Phys. 1998. Submitted for publication. (4) Tassaing, T.; Bellissent-Funel, M.-C; Guillot, B.; Guissani, Y. The partial pair correlation functions of dense supercritical water. Europhys. Lett. 1998, 42, 265. (5) Yamanaka, K.; Yamaguchi, T.; Wakita, H. J. Chem. Phys. 1994, 101, 9830. (6) Posterino, P.; Tromp, R. H.; Ricci, M. A.; Soper, A. K.; Neilson, G. W. Nature 1993, 366, 668. (7) Soper, A. K.; Bruni, F.; Ricci, M. A. J. Chem. Phys. 1997, 106, 247. (8) Ikushima, Y.; Hatakeda, K.; Saito, N.; Arai, M. J. Chem. Phys. 1998, 108, 5855. (9) Frantz, J. D.; Dubessy, J.; Mysen, B. Chem. Geol. 1993, 106, 9. (10) Franck, E. U.; Lindner, H., Doctoral dissertation of the latter, University of Karlsruhe, 1970. (11) Carey, D. M.; Korenowski, G. M. J. Chem. Phys. 1998, 108, 2669. (12) Walrafen, G. E.; Yang, W.-H.; Chu, Y. C.; Hokmabadi, M. S. J. Phys. Chem. 1997, 101, 3381. (13) Haar, L.; Gallagher, J. S.; Kell, G. S. NBS/NRC Steam Tables, Hemisphere: New York, 1984. (14) Huang, Z. S.; Miller, R. E. J. Chem. Phys. 1989, 91, 6613. (15) In private discussions with Miller, R. E., 1998, it was mutually agreed that one cannot expect anything but rough agreement between lowand high-temperature dimer frequencies because of the change in the rotational-vibrational intensity distributions with temperature rise. Nevertheless, it is still possible to use the results of Huang and Miller as a rough guide for diagnostic purposes. (16) Hankins, D.; Moskowitz, J. W.; and Stillinger, F. H. Chem. Phys. Lett. 1970, 4, 581. (17) The quantities ∆Hreaction and ∆Vreaction are specific to the simultaneous equation method described here. ∆Vreaction, however, is not a simple volume change, but corresponds, instead, to the change in the partial molal volumes for the reaction, if Kx is used.