Article pubs.acs.org/JPCA
Active Thermochemical Tables: Water and Water Dimer Branko Ruscic* Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States, and Computation Institute, University of Chicago, Chicago, Illinois 60637, United States S Supporting Information *
ABSTRACT: A new partition function for water dimer in the temperature range 200−500 K was developed by exploiting the equations of state for real water vapor, liquid water, and ice, and demonstrated to be significantly more accurate than any proposed so far in the literature. The new partition function allows the Active Thermochemical Tables (ATcT) approach to be applied on the available experimental and theoretical data relating to water dimer thermochemistry, leading to accurate water dimer enthalpies of formation of −499.115 ± 0.052 kJ mol−1 at 298.15 K and −491.075 ± 0.080 kJ mol−1 at 0 K. With the current ATcT enthalpy of formation of the water monomer, −241.831 ± 0.026 kJ mol−1 at 298.15 K (−238.928 kJ mol−1 at 0 K), this leads to the dimer bond dissociation enthalpy at 298.15 K of 15.454 ± 0.074 kJ mol−1 and a 0 K bond dissociation energy of 13.220 ± 0.096 kJ mol−1 (1105 ± 8 cm−1), the latter being in perfect agreement with recent experimental and theoretical determinations. The new partition function of water dimer allows the extraction and tabulation of heat capacity, entropy, enthalpy increment, reduced Gibbs energy, enthalpy of formation, and Gibbs energy of formation. Newly developed tabulations of analogous thermochemical properties for gas-phase water monomer and for water in condensed phases are also given, allowing the computations of accurate equilibria between the dimer and monomer in the 200−500 K range of temperatures.
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gases, originated as an empirical polynomial fit of experimental observations,11 but subsequent analyses uncovered that virial coefficients have a very appealing interpretation in terms of equilibrium constants for the formation of clusters,12 fueling the idea that real gases could be reasonably well described as an equilibrated mixture of ideal gas monomers, dimers, etc.,13 particularly if a small additional term, often described as “excluded volume”,14 is added to the second virial coefficient. The availability of reliable thermochemistry for water clusters would considerably simplify a variety of tasks, such as, for example, the evaluation of absolute concentrations of water dimers in the atmosphere (which is one of the necessary ingredients in evaluating their significance in Earth’s radiative budget) or, more generally, improving thermodynamic computations involving water vapor by allowing the inclusion of water dimers, trimers, etc. Surprisingly, as we shall see, reliable thermochemistry has not been currently established even for the dimer, let alone for higher water clusters. In order to establish the thermochemistry of an arbitrary chemical species, it is in principle necessary to have two groups of information. The first group consists of determinations that establish the enthalpy of formation ΔfH°T of the target species
INTRODUCTION Water dimer is arguably the epitome of a hydrogen bonded chemical entity and is the simplest building block of bulk water that includes intermolecular interactions. Association of H2O molecules by hydrogen bonding is responsible for numerous anomalous properties of water,1 and a quantitatively correct description of the binary water−water interaction potential, as instanced in the water dimer, is a sine qua non in modeling condensed phases of water.2 Hydrogen-bonding is also significant in its gas phase, and binary and higher clusters are a ubiquitous component of water vapor,3 to the tune that water dimers present in the Earth’s atmosphere are suspected of having an important role in the atmospheric radiative budget.4,5 Conventionally tabulated thermochemistry of pure gas-phase substances corresponds to hypothetical ideal gas properties under standard pressure p° = 1 bar.6−8 Thus, the equilibrium constant for the vaporization process H2O (cr,l) → H2O (g) obtained from the thermochemistry of water in the condensed phase, H2O (cr,l), and in the gas phase, H2O (g), tabulated in CODATA7 or Gurvich et al.,9 indicates that at 373.124 K, when real water vapor attains the equilibrium pressure of 1.01325 bar (1 atm), the equilibrium pressure of H2O (g) is only 0.9979− 0.9981 bar.10 The reason is, of course, that the thermodynamic species H2O (g) represents the ideal gas monomer, and in order to reproduce the total equilibrium pressure of real water vapor, one would need to add the partial pressures of the water dimer, trimer, etc. The virial equation, which is one of the successful approaches in describing the pV behavior of real © XXXX American Chemical Society
Special Issue: Curt Wittig Festschrift Received: March 31, 2013 Revised: May 31, 2013
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temperature experimental ΔdissH°T or ΔdissG°T. While nominally straightforward, such an evaluation is in practice rendered complex by non-negligible scatter in both experimental and theoretical determinations, providing ample opportunities for any tested partition function to connect at least several data points, thus making the evaluation process a rather subjective exercise. However, the Active Thermochemical Tables (ATcT)101,102 approach is uniquely suited for making such evaluations more objective. Briefly, ATcT analyzes and solves an underlying thermochemical network (TN), which contains the available experimental and theoretical determinations. ATcT uses the partition functions to convert all determinations from their original forms (energies, enthalpies, or Gibbs free energies at various temperatures) to reaction enthalpies at one common temperature,103 and then proceeds by testing the converted TN for consistency. The latter is an iterative procedure that identifies the most likely outliers and attempts to alleviate the associated inconsistencies by iteratively incrementing in small steps the initially proposed uncertainties of the corresponding determinations (which in turn lowers their statistical weight). Once the TN is self-consistent, ATcT solves the system by simultaneously finding the enthalpies of formation of all targeted chemical species and produces a list of offenders (measurements and computations that needed to have their initially proposed uncertainties augmented), as well as an abundance of other information. The dimer partition functions explicitly probed as described above are those of Chao et al.,99 Glazunov and Sabirzyanov,100 and Scribano et al.85 Added to these is an RRHO partition function that is very similar in spirit to that of Muñoz-Caro and Niño44 (who unfortunately did not explicitly tabulate their dimer functions), based on experimental fundamentals104−109 complemented by computed fundamentals of Muñoz-Caro and Niño,44 with the contribution of the hindered rotation of the acceptor H2O evaluated by state count based on solving their one-dimensional potential. In each case, the ATcT TN contained the theoretical and experimental data mentioned earlier, where most of the original experimental data were reanalyzed in terms of the second and third law to extract representative ΔdissH°T or ΔdissG°T (complete with estimated uncertainties) at the average temperature of the fitted data set. Because of the normal scatter in the data (as well as occasional inconsistencies between data), it is entirely expected that the ATcT analysis will nearly always find some (hopefully smaller) inconsistencies. However, irrespective of which of the four dimer partition functions is supplied in the process, the ATcT analysis always finds major irreconcilable inconsistencies between the experimental and theoretical data. That is, if additional weight is given to the theoretical data, most of the experimental data ends up as outliers and vice versa, leading to the conclusion that none of the proposed dimer partition functions is sufficiently accurate. Quite recently, an important new experimental measurement has become available, making it possible to further simplify the above analysis. Namely, by velocity map imaging of water molecules that are produced during state-to-state vibrational predissociation of cold water dimer, Reisler’s group110 has very accurately determined D0 of the dimer as 1105 ± 10 cm−1. The (H2O)2 result was subsequently verified by determining the bond dissociation energy of the deuterated (D2O)2 analogue.111 Their experimental value is in outstanding agreement with the theoretical prediction by Shank et al.93 that was obtained from their HBB2 potential energy surface, the De of which is in turn
at some known temperature T. The second group consists of a temperature-dependent partition function QT, from which one can obtain the enthalpy increment H°T − H°0, entropy S°T, and isobaric heat capacity C°p,T, and hence compute the enthalpy and Gibbs free energy of formation, ΔfH°T and ΔfG°T, at other desired temperatures. In the case of water dimer, virtually all available information belonging to the first group corresponds to (or can be converted to) determinations of the dimer dissociation energy, enthalpy, or free energy: (H 2O)2 (g) → 2H 2O(g)
(1)
A significant body of experimental literature is rooted on measuring and analyzing various properties of water vapor and reporting the enthalpy ΔdissH°T and/or Gibbs free energy ΔdissG° T of dimer dissociation at one or more finite temperature(s),3,15−36 generally in the 290−450 K range (and occasionally at higher temperatures). Not surprisingly, there is also a significant body of theoretical literature reporting the dimer bond dissociation energy, obtaining it either directly from electronic structure calculations at various levels of treatment that range from an early Hartree−Fock study37 to more accurate approaches,35,38−63 or by extracting this datum from fitted and adjusted dimer potential energy surfaces.40,49,52,54,55,64−95 Theoretical literature typically reports the dimer dissociation energy at 0 K, D0 = ΔdissH°0 = ΔdissG°0, though sometimes only the dimer binding energy −De is reported. The thermodynamical usefulness of the latter is limited because of the missing zero-point energy,96 without which the computed binding energy cannot be used to infer practical thermochemistry. Importantly, accurate determination of zero-point energies, at least in the case of state-of-the-art electronic structure methods, is often a crucial step limiting the overall accuracy of the theoretical computation.97 Computed D0 and experimental ΔdissH°T or ΔdissG°T at various temperatures can be easily interconverted and compared using the partition functions for the dimer and monomer. Since the partition function for the monomer is in principle known rather accurately,7,9 the accuracy of such conversions essentially depends on the quality of the dimer partition function. At least98 five, rather different, partition functions for water dimer have been proposed in the literature. Chao et al.99 have computed an rigid rotor harmonic oscillator (RRHO) partition function based on estimated vibrational frequencies and rotational constants, with additional corrections for the hindered rotation of the acceptor H2O around the hydrogen bond. Muñoz-Caro and Niño44 have explored several RRHO functions, based on computed vibrational harmonics, estimated fundamentals, and a computed potential energy function for the hindered rotation. Glazunov and Sabirzyanov100 have also devised an RRHO partition function, corrected for internal rotation as well as for centrifugal stretching, based on computed spectroscopic constants that were compared to available experimental constants. As opposed to these three RRHO functions, Goldman et al.82 have obtained their partition function by state counting, based on solving their VRT(ASP-W)III dimer potential for total angular momentum J = 0 and estimating the remaining rotational contributions. In a subsequent paper from the same group, Scribano et al.85 have further refined their earlier partition function by explicitly including solutions for higher angular momenta. The quality of the dimer partition function can be evaluated by testing how well it connects the theoretical D0 and the finiteB
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in excellent agreement with the very high-end electronic structure computation of Tschumper et al.48 Similarly, the experimental D0 for the deuterated analogue111 is in outstanding agreement with the theoretical prediction obtained from the same HBB2 surface.112 The experimental D0 was subsequently also confirmed by Leforestier et al.,95 who used their newest CCpol-8sf potential energy surface. This development reduces the scatter in the 0 K thermochemistry of the dimer and thus restricts a degree of freedom that was previously possible when examining proposed dimer partition functions: sliding the 0 K enthalpy of formation of the dimer up or down in order to accommodate this or that higher temperature determination. Figure 1 shows a representative subset of experimental data falling in the 290− 450 K range as individual points with error bars.113 The lines in the figure are ΔdissH°T (top panel) or ΔdissG°T (bottom panel) that are obtained by using each of the existing dimer partition functions discussed above, while anchoring their low-temperature end to the experimental D0. Clearly, none of the four dimer partition functions is satisfactory: they all miss nearly all of the experimental data points. This is particularly noticeable in the bottom panel (ΔdissG°T), where the experimental error bars are lower.
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AIM AND METHODS A closer look at Figure 1 reveals some additional insights on the existing partition functions for the dimer. Noting that the partition function for the monomer is the same in all cases, the behavior on the bottom panel, ΔdissG°T, gives some information about the absolute magnitudes of the tested dimer partition functions. Clearly, the dimer partition functions of Scribano et al.,85 Glazunov and Sabirzyanov,100 and Chao et al.99 have too small a magnitude, with the latter being significantly smaller than the former two, while the partition function that is similar to that used by Muñoz-Caro and Niño44 is too large. In addition, the fact that the latter partition function runs approximately parallel to the bulk of the data points, suggests that, at least in the temperature range 300−450 K, its magnitude is too large by a nearly constant factor, roughly estimated as ∼2.5. Similarly, the function of Glazunov and Sabirzyanov100 appears to be underestimated by roughly a factor of ∼2 in this temperature range, almost as if these authors have included a superfluous symmetry number, though that does not seem to be the case. However, the partition functions of Chao et al.99 and Scribano et al.85 appear to be underestimated by factors that are becoming progressively larger as the temperature increases. The behavior on the upper panel of Figure 1, ΔdissH°T, reveals something about the rate at which the partition functions are growing as the temperature increases, rather than their absolute values. The functions that are higher than the bulk of the data, Scribano et al.,85 Chao et al.,99 and the function similar to that of Muñoz-Caro and Niño,44 generally appear to grow too slowly, with the partition function of Scribano et al.85 departing most noticeably from the growth rate implied by the experimental data. Conversely, the partition function of Glazunov and Sabirzyanov100 appears to be growing somewhat too rapidly, at least at the higher end of the explored temperature range. In retrospect, it is perhaps not too surprising that the RRHO partition functions prove to be inadequate, irrespective of whether the vibrational contribution is evaluated by using harmonics or fundamentals.114 The dimer has six higher frequency vibrational modes that belong to the internal
Figure 1. Water dimer bond dissociation enthalpy (upper panel, ΔdissH°T) and Gibbs free energy (bottom panel, ΔdissG°T): comparison of existing dimer partition functions (lines) with experimental data (points). Points: black circle, Gebbie et al.;3 purple circle, DianovKlokov et al.;26 green circle, Cormier et al.;34 brown square, Bolander et al.;21 dark blue triangle, Fiadzomor et al.;36 purple triangle, Nakayama et al.;35 gray square and gray circles, Ptashnik et al.;32 yellow circle, Keyes;16 green triangle, Luck;22 dark blue circle, Curtiss et al.;24 green square, Keyes16 and Curtiss et al.;23−25 light blue circles, light blue triangle, and light blue square, Harvey and Lemmon;33 purple square, Vukalovich et al.;19 brown triangle, Lambert;18 yellow triangle, Pfeilsticker et al.30 Lines: purple, Scribano et al.;85 blue, Chao et al.;99 orange, Muñoz-Caro and Niño;44 green, Glazunov and Sabirzyanov.100
vibrations (two stretches and a bend) of the donor and the acceptor molecule. Though somewhat anharmonic, these are unlikely to be a prohibitive problem for the RRHO method. However, the remaining six soft vibrational modes, which correspond to relative motions of the acceptor and donor molecule, undergo complicated tunneling involving 8 equivalent global minima115 and a corresponding complement of shallower wells representing the other conformers. The soft modes are highly anharmonic and are coupled to the rotational levels, which are in turn affected by nuclear spin statistics.116 Thus, each rotation−vibration level is split by acceptor switching tunneling, interchange tunneling, and bifurcation tunneling, producing a complex structure of the resulting levels, C
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form expressions for rotational contributions have been obtained by integrating rotational levels to infinity, and those for vibrational contributions by finding the sum of an infinite series.117 The error introduced by assuming nondissociating oscillators should become progressively worse as the temperature increases, though this effect may be partially canceled out by the error arising from unaccounted anharmonic effects. Just as in the case of level counting, the temperature at which inherent overcounting by RRHO becomes significant depends on the energy at which the molecule in question dissociates. In view of the uncovered problems with RRHO and levelcounting partition functions, at least as implemented in the dimer partition functions that were proposed so far, the aim here is to explore a different approach. The process exploits the available equation of state of water vapor, which implicitly has some embedded information about the dimer properties. Wagner and Pruss118 have formulated the most recent equation of state for liquid water and water vapor, called IAPWS-95 (where IAPWS stands for International Association for the Properties of Water and Steam). This equation of state is a scientific equivalent of the algebraically simpler (and thus less accurate) IAPWS-IF97 formulation,119 which is geared toward industrial use (and replaces various previous editions of the socalled Steam Tables120−122). IAPWS-95 is essentially an accurate empirical description of the relative123 free energy of real water obtained by a least-squares fit of virtually all available experimental measurements of water properties, producing a mathematically rather involved form that includes well over a hundred fitted parameters. In conjunction with the IAPWS-95 formulation, Feistel and Wagner124 have developed an analogous equation of state for ice. The equations of state for liquid water and its vapor are functions of density and temperature and explicit in the (relative) Helmholtz free energy, while the equation of state for ice is a function of pressure and temperature and explicit in the (relative) Gibbs free energy. Other desired thermodynamic properties, such as relative enthalpy and entropy, isobaric and isochoric heat capacity, etc., can be then obtained by combining appropriate derivatives of the equations explicit in the free energy. The IAPWS-95 formulation for liquid and vapor is valid in the entire stable fluid region of water and that for ice between 0 K and the triple point of water. The central assumptions underlying the process of devising the partition function for water dimer are
the effect of which is highly unlikely to be correctly captured by an RRHO partition function, even if one were to make additional corrections9 for anharmonicities, rotation−vibration coupling, and centrifugal stretching. What is surprising, though, is that the partition function of Scribano et al.85 is also inadequate, though it was obtained by level counting. Assuming that the actual rotation−vibration levels are correctly reproduced by the solutions of the potential used by Scribano et al., at least part of the problem of their partition function is in abruptly terminating the count at the dissociation limit. For diatomics, it is standard practice, propagated by Gurvich et al.,9 to obtain the partition function by level counting (rather than by the RRHO approach), and terminating the count at the dissociation limit (which is increased for each total angular momentum J by adding the appropriate centrifugal barrier). Clearly, for a diatomic molecule, the levels in the dissociation continuum belong to the partition functions of the two constituent atoms rather than to the partition function of the diatomic. For polyatomics, the situation is less clear, not only because there is more than one dissociation limit but also because in general there will be rotation−vibration levels belonging to the same electronic state that are quasi-bound though they are nominally above the dissociation limit, and thus may have a long enough lifetime to warrant their inclusion in the partition function. Although very few partition functions for polyatomics have been so far obtained by level counting, the question of which rotation− vibration levels above the dissociation limit, if any, should be included, typically becomes a moot point if the first dissociation limit is sufficiently high, in which case the effect of including/ excluding certain high energy levels affects the partition function only at very high temperatures, at which point there are plenty of opportunities for other sources of error in the partition function to overshadow the counting problem. Contrary to such cases, the dissociation limit of water dimer is quite low and thus undercounting the rotation−vibration levels above the dissociation limit that should have been included will affect the partition function already at midrange temperatures. Scribano et al.85 did indeed discuss the problem of omitting the quasi-bound states of the dimer. Scribano et al. have also explored a version of the partition function that systematically included the quasi-bound states, but concluded that this leads to overcounting, and thus their final dimer partition function did not include any quasi-bound states. Figure 1 suggests that this partition function suffers from an undercount. Namely, an undercount will clearly result in an underestimation of the partition function magnitude. Furthermore, systemic undercount beyond a certain fixed cutoff energy will lead to a partition function that is progressively more and more underestimated as the temperature increases. Both of these effects appear to be consistent with the observed behavior of the dimer partition function of Scribano et al.,85 making the undercount a plausible explanation for its failure. Returning to the RRHO partition functions, one needs to note that a general underestimate or overestimate of their absolute magnitudes can have multiple sources, some of which relate to the particular choice of fundamentals for the soft modes and to various anharmonic effects that are unaccounted for. However, it may be worthwhile to point out that in the case of water dimer the RRHO approach may have an additional problem, which is the opposite of the just discussed problem of potentially terminating the level count too soon. Namely, RRHO inherently counts levels ad infinitum since the closed-
preal = p1 + K 2p12 + K3p13 + ...
(2)
Mave /MH2O = (p1 + 2K 2p12 + 3K3p13 + ...)/preal
(3)
Here, preal denotes the total pressure of real vapor, p1 denotes the partial pressure due to the monomer, Kk is the equilibrium constant for the cluster containing k monomers, MH2O is the molecular mass of the monomer, and Mave is the average molecular mass of the real vapor. Equation 2 is simply the assumption that real water vapor contains monomers, dimers, etc., expressed in terms of Dalton’s law of partial pressures. Equation 3 reflects the fact that if one evaporates n molecules of liquid water, one does not necessarily end up with n independent particles in the vapor phase since some of the particles will be in the form of binary or higher clusters. This effectively increases the average relative molecular weight of the vapor. For a fixed temperature, the quantities on the left sides of eqs 2 and 3 can be computed from the equation of state. In D
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CODATA 2006 fundamental physical constants125 were used throughout. The relative molecular mass of the dimer was taken as 36.030536, that of water was 18.015268. These values result from the relative atomic masses of the stable isotopes of hydrogen and oxygen,126,127 and the isotopic composition of VSMOW (Vienna Standard Mean Ocean Water),128 which is now accepted as a standard for water. In eq 4, D0 was taken to be the value obtained by Reisler’s group.110 In principle, the required monomer partition function could have been used from CODATA7 or from Gurvich et al.,9 but for the present purposes that would have required extensive interpolation because their values are tabulated in 100 K increments. In addition, these partition functions for water have been partially superseded at first by Harris et al.,129 who recalculated the partition function for gas-phase water by counting accurately computed rotation−vibration levels, and subsequently by Vidler and Tennyson,130 who counted experimental levels. The combination of the fact that in the temperature range of interest here the effective differences between the earlier functions and those of Vidler and Tennyson are marginal and that the significance of relatively small changes may be compromised by their use of older values for the fundamental physical constants,131 in conjunction with the fact that we have an ongoing ATcT-related long-term effort of refurbishing as many partition functions as possible, provided an impetus to develop our own version of the partition function of the monomer by similar level counting of experimental values for the rotation−vibration levels of the water molecule,132 complemented by theoretical values.133 Missing high-energy levels in the list of experimental values were identified by keeping track of all relevant quantum numbers and filled in by a combination of techniques that included consulting the list of theoretical values, solving the rotational levels of the appropriate asymmetric rotor, matching the existing levels of the same total angular momentum, and applying sum rules.
particular, the quantity Mave/MH2O can be obtained from the hypothetical density of ideal gas and the actual density of water vapor corrected for the excluded volume. The latter was taken to be equal to the volume of the condensed phase at the same temperature, although variations between twice the volume of the condensed phase and zero have also been explored. For a fixed temperature, the equations of state for water vapor and the appropriate condensed phase were iteratively solved for the phase-equilibrium condition, i.e., the condition that the Gibbs free energies of the two phases are equal. This yielded the equilibrium pressure of water vapor above the condensed phase and the corresponding Mave/MH2O ratio, resulting in the first pair of data points for that temperature. Keeping the temperature fixed, an additional series of calculations at several pressures lower than the equilibrium pressure was performed, providing additional pairs of data points. For this, the pressures, starting at the equilibrium pressure, were decreased in a series of steps, each involving a factor of √2. This factor was found to be a reasonable compromise that produces a sufficient number of points before the pressure becomes so low that the behavior of the vapor is nearly that of an ideal gas and thus the useful information content on the clusters has become vanishingly small. It should be noted that the equation of state for the vapor allows limited extrapolation to vapor pressures higher than the equilibrium pressure. While this pressure range was also regularly explored, the resulting points were not used in the final fit because they would be technically outside the validity range of the equation of state. Using pairs of eqs 2 and 3, the entire set of point pairs that was obtained at one temperature was least-squares fitted, solving it for the monomer pressures corresponding to each point pair and the common equilibrium constants (as well as their uncertainties). The cutoff limit for the largest cluster size included in the fit was separately varied at each temperature, and the final cutoff size was based on the statistical significance of the corresponding equilibrium constant. The whole process was then repeated for a different temperature, covering the 200−500 K range in 5 or 10 K increments. The equilibrium constants for the dimer that were obtained as described above were used to extract the dimer partition function Q2 via the relationship Q 2 = K 2Q 12 exp[−D0 /(RT )]
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RESULTS AND DISCUSSION Figure 2 displays the same experimental points as Figure 1, but the line is now based on the present partition function of the dimer. Clearly, the line describes the experimental data quite accurately. It passes either through or extremely close to most data points (well within their error bars), the only exceptions being a few obvious outliers. At this point, one needs to stress the fact that the present partition function for the dimer was developed without directly relying on the data shown in the figure, and thus, the extremely good fit shown in Figure 2 is an unbiased attest of its fidelity. The only slight caveat to the claim of complete independence is that the IAPWS-95 equation of state of water was formulated by fitting a huge amount of experimental data available at the time, which included some (but not all) of the experimental data that resulted in the points shown in Figures 1 and 2. The most prominent data of Figure 2 that was not included in the IAPWS-95 formulation is that of Harvey and Lemmon.33 These authors, in fact, make the point of noting that their second virial coefficients differ slightly from the second virial coefficients of Wagner and Pruss.118 Thus, the fact that the present dimer partition function describes quite well the vast majority of the points, including those of Harvey and Lemmon, is significant. Table 1 is a JANAF-style tabulation of the thermochemical properties of the dimer and includes the isobaric heat capacity C°p,T, entropy S°T, reduced Gibbs energy function R ln Q°T,
(4)
where D0 is the 0 K dimer dissociation energy and Q1 is the monomer partition function. Once Q2 was obtained, the enthalpy increment H°T − H°0, entropy S°T, and isobaric heat capacity C°p,T, were obtained via standard relationships [H °T − H °0 ]/(RT ) = T ∂(ln QT)/∂T
(5)
S°T /R = [H °T − H °0 ]/(RT ) + ln QT
(6)
C °p,T /R = T 2∂ 2(ln QT)/∂T 2 + 2T ∂(ln QT)/∂T
(7)
The operation involved stripping ln Q2 of the translational contribution and least-squares fitting the residual with a ninth order polynomial in T. The fit provided the first and second derivative and slightly smoothed the residual. The fitted ln Q2 and its derivatives were used to compute the internal energy contribution to the enthalpy increment, entropy, and isobaric heat capacity. The final thermochemical functions of the dimer were obtained by restoring the appropriate translational contributions. E
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The uncertainties in the tabulated values arise partly from the uncertainties in the IAPWS-95 formulation and partly from uncertainties related to the method used to produce the partition function. Wagner and Pruss118 state that the uncertainty in determining the equilibrium pressure of water vapor above the liquid from the IAPWS-95 formulation is ∼0.025% at temperatures above ∼370 K and less than that at lower temperatures, while the uncertainty in determining the density of the vapor is ∼0.03%. The IAPWS-95 equation of state of vapor is internally partitioned into the ideal gas part and a residual (real) correction. The former is locked into the older representation of the ideal gas partition function of water by Cooper,134 which was originally obtained by fitting the values tabulated by Wooley.135 As a consequence, the ideal gas part reproduces quite well the older isobaric heat capacities tabulated by JANAF,136,137 but less well those of Gurvich et al.9 or CODATA,7 or the more recent values of Vidler and Tennyson.130 Furthermore, the IAPWS-95 formulation uses an older value for the gas constant R. Wagner and Pruss118 discuss the latter problem and suggest that the specific properties (i.e., per unit mass) are still correct. However, one infers that the choice of an older ideal gas representation in IAPWS-95 as well as an outdated gas constant imply that at least at pressures lower than the equilibrium pressure, the water vapor properties are converging to a somewhat distorted set of values. The method adopted here to arrive at the dimer partition function introduces additional errors, arising from such things as the uncertainty introduced by the particular selection of the value for the excluded volume, potential overfitting or underfitting introduced by the particular selection of the cluster size cutoff, the uncertainty of the fit itself, the uncertainty in the D0 employed in the procedure, the additional uncertainty introduced by deriving the first and second derivatives of R ln Q°T from the polynomial fit, etc. Taking all of these potential sources of systematic errors into account, leads to an approximate estimate of the overall uncertainty of the reduced Gibbs energy function R ln Q°T of ±0.08 J K−1 mol−1 in the 265−500 K range, becoming progressively larger at lower temperatures and reaching ±0.15 J K−1 mol−1 at 200 K. The approximate uncertainty in the enthalpy increment H°T − H°0, which is higher since it relies on the derivative of ln Q°T, varies between ±0.15 and ±0.20 kJ mol−1 in the 200−295 K range, it is ±0.17 kJ mol−1 at 298.15 K, and gets progressively larger at higher temperatures, reaching ±0.59 kJ mol−1 at 500 K. The estimated uncertainty in the entropy S°T follows a similar pattern: from ±0.24 J K−1 mol−1 at 200 K, it diminishes to ±0.19−0.21 J K−1 mol−1 in the 220− 295 K range, it is ±0.19 J K−1 mol−1 at 298.15 K, and progressively grows from 300 K toward higher temperatures, reaching ±0.60 J K−1 mol−1 at 500 K. The isobaric heat capacity C°p,T relies both on the second and first derivatives of ln Q°T, which introduces a substantial error, making it by far the most inaccurate quantity in Table 1. The uncertainty in C°p,T is of the order of at least ±5−7 kJ/mol in the range of temperatures up to 350 K, (±6.5 J K−1 mol−1 at 298.15 K), growing toward higher temperatures to up to ±20 J K−1 mol−1 at 500 K. One notices that the estimated errors in the quantities given in Table 1 are generally larger toward the low and high ends of the explored temperature range. The reason for this is clear. Namely, the equilibrium water vapor pressure increases roughly exponentially as the temperature increases. The contribution of the dimer grows with the square of the monomer pressure, that of the trimer with the cube. At the low end of the temperature
Figure 2. Water dimer bond dissociation enthalpy (upper panel, ΔdissH°T) and Gibbs free energy (bottom panel, ΔdissG°T): comparison of the new dimer partition functions (line) with experimental data (points). Points: black circle, Gebbie et al.;3 purple circle, DianovKlokov et al.;26 green circle, Cormier et al.;34 brown square, Bolander et al.;21 dark blue triangle, Fiadzomor et al.;36 purple triangle, Nakayama et al.;35 gray square and gray circles, Ptashnik et al.;32 yellow circle, Keyes;16 green triangle, Luck;22 dark blue circle, Curtiss et al.;24 green square, Keyes16 and Curtiss et al.;23−25 light blue circles, light blue triangle, and light blue square, Harvey and Lemmon;33 purple square, Vukalovich et al.;19 brown triangle, Lambert;18 yellow triangle, Pfeilsticker et al.30 Red lines: present work.
enthalpy increment H°T − H°0, enthalpy of formation ΔfH°T, and Gibbs free energy of formation ΔfG°T in the 200−500 K range with 20 K increments. A more detailed version (with 5 or 10 K increments) is provided as Table S1 in the Supporting Information. We should note here that the tabulated partition function corresponds to a practical partition function that is standard in chemical applications. Practical functions ignore the overall nuclear spin contribution to the entropy as well as the isotope mixing component, which, in any stoichiometrically balanced chemical reaction, cancel out across the involved chemical species. In occasional applications where an absolute partition function is needed, the corresponding properties can be obtained by simply adding to the tabulated values of S°T and R ln Q°T, the term labeled S°nucl and given in the table header. F
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The Journal of Physical Chemistry A
Article
Table 1. Thermochemical Functions for Water Dimer Mr = 36.030536 p° = 1 bar S°nucl = 23.405 J K−1 mol−1 u95%(ΔfH°298) = ±0.052 kJ/mol
water dimer (H2O)2 (g) T (K)
Cp°T (J K−1 mol−1)
S°T (J K−1 mol−1)
R ln Q°T (J K−1 mol−1)
H°T − H°0 (kJ mol−1)
ΔfH°T (kJ mol−1)
ΔfG°T (kJ mol−1)
0 200 220 240 260 273.153 273.16 280 298.15 300 320 340 360 372.756 373.124 380 400 420 440 460 480 500
0.000 43.026 55.621 66.618 73.873 76.266 76.267 76.810 76.355 76.186 73.533 70.452 67.995 66.859 66.830 66.305 64.662 61.958 57.497 51.871 47.404 47.480
0.000 274.844 279.536 284.868 290.516 294.227 294.230 296.123 300.946 301.418 306.257 310.621 314.574 316.921 316.987 318.203 321.564 324.660 327.446 329.879 331.982 333.898
0.000 219.637 224.863 229.638 234.103 236.909 236.910 238.334 242.000 242.365 246.210 249.872 253.359 255.494 255.554 256.677 259.839 261.364 262.853 264.307 265.727 267.113
0.000 11.041 12.028 13.255 14.667 15.656 15.657 16.181 17.575 17.716 19.215 20.655 22.038 22.897 22.921 23.380 24.690 25.959 27.156 28.251 29.239 30.177
−491.075 −497.232 −497.934 −498.411 −498.717 −498.863 −498.864 −498.932 −499.115 −499.135 −499.381 −499.693 −500.067 −500.331 −500.339 −500.487 −500.944 −501.447 −502.026 −502.712 −503.510 −504.361
−491.075 −465.739 −462.553 −459.314 −456.043 −453.880 −453.879 −452.752 −449.753 −449.446 −446.126 −442.788 −439.430 −437.277 −437.215 −436.050 −432.647 −429.220 −425.768 −422.287 −418.773 −415.225
corresponding 0 K value is −491.075 kJ mol−1. The nominal uncertainty of the latter would be ±0.179 kJ mol−1, on account of the uncertainty of the corresponding enthalpy increment for the dimer discussed above. We shall return to that uncertainty in a moment. The 298.15 K enthalpy of formation of the dimer is the result of the current ATcT treatment of the data that were mentioned in the Introduction (and some of which is plotted in Figures 1 and 2), together with additional third law values in the 200− 500 K range for the equilibria 2H2O (cr,l) → (H2O)2 (g) and 2H2O (g) → (H2O)2 (g) extracted during the present analysis of the IAPWS-95 formulation. As noted in the Introduction, the ATcT treatment exploits the knowledge content of all relevant determinations in the underlying TN. An analysis based on decomposition of the associated variances indicates, somewhat surprisingly, that the ATcT value for the enthalpy of formation of the dimer is derived primarily from the experimental data in the 200−500 K range, such as the just mentioned IAPWS-95 based equilibria, and relies only minimally (
