Volume Isotope Effects Research Articles

Mass-independent Sn isotope fractionation and radiogenic 115Sn in chondrites and terrestrial rocks

Tin has ten stable isotopes, providing the opportunity to investigate and discriminate nucleosynthetic isotope anomalies from mass-dependent and mass-independent isotope fractionation. Novel protocols for chemical separation (based on TBP-resin) and MC-ICP-MS analyses are reported here for high precision Sn isotope measurements on terrestrial rocks and chondrites. Relative to the Sn reference standard (NIST SRM 3161a), terrestrial basalts and chondrites show isotope patterns that are consistent with mass-dependent and mass-independent isotope fractionation processes as well as with 115Sn radiogenic ingrowth from 115In.Two different mass-independent isotope effects are identified, namely the nuclear volume (or nuclear field shift) and the magnetic isotope effect. The magnetic isotope effect dominates in the two measured ordinary chondrites, while repeated analyses of the carbonaceous chondrite Murchison (CM2) display a pattern consistent with a nuclear volume effect. Terrestrial basalts show patterns that are compatible with a mixture of nuclear volume and magnetic isotope effects. The ultimate origin of the isotope fractionation is unclear but a fractionation induced during sample preparation seems unlikely because different groups of chondrites show distinctly different patterns, hence pointing towards natural geo/cosmochemical processes. Only the carbonaceous chondrite Murchison (CM2) shows a Sn isotope pattern similar to what expected for nucleosynthetic variations. However, this pattern is better reproduced by nuclear volume effects. Thus, after considering mass-independent and mass-dependent effects, we find no evidence of residual nucleosynthetic anomalies, in agreement with observations for most other elements with similar half-mass condensation temperatures.Most chondrites show a deficit in 115Sn/120Sn (typically −150 to −200 ppm) relative to terrestrial samples, with the exception of one ordinary chondrite that displays an excess of about +250 ppm. The 115Sn/120Sn data correlate with In/Sn, being consistent with the β− decay of 115In over the age of the solar system. This represents the first evidence of the 115In-115Sn decay system in natural samples. The radiogenic 115Sn signature of the BSE derives from a suprachondritic In/SnBSE, which reflects preferential partitioning of Sn into the Earth’s core.

New Insights into the Volume Isotope Effect of Ice Ih from Polarizable Many-Body Potentials.

The anomalous volume isotope effect (VIE) of ice Ih is calculated and analyzed based on the quasi-harmonic approximation to account for nuclear quantum effects in the Helmholtz free energy. While a lot of recently developed polarizable many-body potential functions give a normal VIE contrary to experimental results, we find that one of them, MB-pol, yields the anomalous VIE in good agreement with the most recent high-resolution neutron diffraction measurements─better than DFT calculations. The short-range three-body terms in the MB-pol function, which are fitted to CCSD(T) calculations, are found to have a surprisingly large influence. A vibrational mode group decomposition of the zero-point pressure together with a hitherto unconsidered benchmark value for the intramolecular stretching modes of H2O ice Ih obtained from Raman spectroscopy data unveils the reason for the VIE: a delicate competition between the latter and the librations.

The microscopic origin of the anomalous isotopic properties of ice relies on the strong quantum anharmonic regime of atomic vibration.

Water ice is a unique material presenting intriguing physical properties, such as negative thermal expansion and anomalous volume isotope effect (VIE). They arise from the interplay between weak hydrogen bonds and nuclear quantum fluctuations, making theoretical calculations challenging. Here, we employ the stochastic self-consistent harmonic approximation to investigate how thermal and quantum fluctuations affect the physical properties of ice XI with ab initio accuracy. Regarding the anomalous VIE, our work reveals that quantum effects on hydrogen are so strong to be in a nonlinear regime: When progressively increasing the mass of hydrogen from protium to infinity (classical limit), the volume first expands and then contracts, with a maximum slightly above the mass of tritium. We observe an anharmonic renormalization of about 10% in the bending and stretching phonon frequencies probed in IR and Raman experiments. For the first time, we report an accurate comparison of the low-energy phonon dispersion with the experimental data, possible only thanks to high-level accuracy in the electronic correlation and nuclear quantum and thermal fluctuations, paving the way for the study of thermal transport in ice from first-principles and the simulation of ice under pressure.

Mercury Isotope Fractionation during the Photochemical Reduction of Hg(II) Coordinated with Organic Ligands.

The photochemical reduction of Hg(II) is an important pathway in the environmental Hg cycle because it competes with Hg methylation and potentially limits the formation of bioaccumulative methylmercury. Hg stable isotope systematics have proven to be an effective tool for investigating the transport, transformation, and bioaccumulation of Hg. The dominant cause of mass independent isotope fractionation (MIF) of Hg in nature is the photochemical reduction of various species of Hg(II). However, it is difficult to fully interpret Hg stable isotope signatures due to the lack of mechanistic information about which Hg compounds are susceptible to MIF and why. This study investigates Hg isotope fractionation during the photochemical reduction of Hg(II) complexed to organic ligands, which are representative of the available binding sites in natural dissolved organic matter. The photochemical reduction of Hg(II) in the presence of cysteine resulted in both negative and positive MIF in residual Hg(II), where the sign depended on pH and dissolved oxygen level. In the presence of serine, either nuclear volume or magnetic isotope effects were observed depending on the wavelength of light and the extent of Hg(II) complexation by serine. In the presence of ethylenediamine, MIF was negative. Our Hg stable isotope results suggest that MDF and MIF are induced at different steps in the overall photochemical reduction reaction and that MIF does not depend on the rate-determining step but instead depends on photophysical aspects of the reaction such as intersystem crossing and hyperfine coupling. The behavior of Hg isotopes reported here will allow for a better understanding of the underlying reaction mechanisms controlling the Hg isotope signatures recorded in natural samples.

Selectively Deuterated Poly(ε-caprolactone)s: Synthesis and Isotope Effects on the Crystal Structures and Properties

Selective deuteration is an important tool for many analytical techniques including neutron scattering and spectroscopies. However, the availability of deuterated materials is limited because of the challenges in their synthesis. Here, we report the synthesis of partially and fully deuterated ε-caprolactone monomers and their corresponding polymers, poly(ε-caprolactone)s (PCLs), and the investigation of isotope effects on their crystalline structures and physical properties. Deuteration of PCLs leads to smaller crystal lattices and volumes compared to protiated PCLs by the amount proportional to the deuteration levels. The linear trend suggests that the volume isotope effect in PCL is primarily governed by the vibrations of C–D and C–H bonds. The large intrachain contraction of deuterated PCLs compared to that of polyethylene reported in the literature can be ascribed to the presence of polar ester groups in PCLs. Deuterated PCLs also display lower melting temperatures than protiated PCLs proportional to their deuteration levels because of weaker intermolecular interactions in deuterated polymers. FTIR spectroscopy, with support from density functional theory calculations, shows large red shifts of the stretching and bending frequencies of C–D versus C–H bonds as dictated by their relative reduced masses 13/7, and to a smaller degree for the C═O stretching frequency. This work is among the very few studies comparing the effects of partial versus full deuteration on the structures and properties in semicrystalline polymers. These results not only advance our understanding of isotope effects in polymeric materials but also provide an important avenue to design polymers with desirable properties.

Ice Ih anomalies: Thermal contraction, anomalous volume isotope effect, and pressure-induced amorphization.

Ice Ih displays several anomalous thermodynamic properties such as thermal contraction at low temperatures, an anomalous volume isotope effect (VIE) rendering the volume of D2O ice greater than that of H2O ice, and a pressure-induced transition to the high-density amorphous (HDA) phase. Furthermore, the anomalous VIE increases with temperature, despite its quantum-mechanical origin. Here, embedded-fragment ab initio second-order many-body perturbation (MP2) theory in the quasiharmonic approximation (QHA) is applied to the Gibbs energy of an infinite, proton-disordered crystal of ice Ih at wide ranges of temperatures and pressures. The quantum effect of nuclei moving in anharmonic potentials is taken into account from first principles without any empirical or nonsystematic approximation to either the electronic or vibrational Hamiltonian. MP2 predicts quantitatively correctly the thermal contraction at low temperatures, which is confirmed to originate from the volume-contracting hydrogen-bond bending modes (acoustic phonons). It qualitatively reproduces (but underestimates) the thermal expansion at higher temperatures, caused by the volume-expanding hydrogen-bond stretching (and to a lesser extent librational) modes. The anomalous VIE is found to be the result of subtle cancellations among closely competing isotope effects on volume from all modes. Consequently, even ab initio MP2 with the aug-cc-pVDZ and aug-cc-pVTZ basis sets has difficulty reproducing this anomaly, yielding qualitatively varied predictions of the sign of the VIE depending on such computational details as the choice of the embedding field. However, the temperature growth of the anomalous VIE is reproduced robustly and is ascribed to the librational modes. These solid-state MP2 calculations, as well as MP2 Born-Oppenheimer molecular dynamics, find a volume collapse and a loss of symmetry and long-range order in ice Ih upon pressure loading of 2.35 GPa or higher. Concomitantly, rapid softening of acoustic phonons is observed starting around 2 GPa. They constitute a computational detection of a mechanical instability in ice Ih and the resulting pressure-induced amorphization to HDA.

Nature of the Volume Isotope Effect in Ice.

The substitution of hydrogen (H) by deuterium (D) in ice Ih and in its H-ordered version, ice XI, produces an anomalous form of volume isotope effect (VIE), i.e., volume expansion. This VIE contrasts with the normal VIE (volume contraction) predicted in ice-VIII and in its H-disordered form, ice VII. Here we investigate the VIE in ice XI and in ice VIII using first principles quasiharmonic calculations. We conclude that normal and anomalous VIEs can be produced in ice VIII and ice XI in sequence by application of pressure (ice XI starting at negative pressures) followed by a third type-anomalous VIE with zero-point volume contraction. The latter should also contribute to the isotope effect in the ice VII → ice X transition. The predicted change between normal and anomalous VIE in ice VIII at 14.3 GPa and 300 K is well reproduced experimentally in ice VII using x-ray diffraction measurements. The present discussion of the VIE is general, and conclusions should be applicable to other solid phases of H(2)O, possibly to liquid water under pressure, and to other H-bonded materials.

Very large differences in intramolecular D-H partitioning in hydrated silicate melts synthesized at upper mantle pressures and temperatures

Hydrated (with D2O and H2O) sodium tetrasilicate glasses, quenched from melts at 1400 °C and 1.5 GPa, are studied using 1H, 2H, and 29Si solid-state nuclear magnetic resonance (NMR) spectroscopy. Whereas D2O and H2O depolymerize the silicate melt to similar degrees, protium and deuterium intramolecular partitioning between different molecular sites within the glasses is very different and exemplified by a strong preferential association of deuterons to sites with short O-D···O distances. This preference is independent of total water content and D/H ratio. Substantially different intramolecular D-H partitioning is also observed in a glass with a model hydrous basalt composition. Such large differences in isotope partitioning cannot result from classic equilibrium fractionation because of the high synthesis temperature. Potential kinetic isotope effects are excluded via a slow quench experiment. The apparent fractionation is likely governed by density/molar volume isotope effects, where deuterium prefers sites with smaller molar volume. Large differences in intramolecular site partitioning in melts could lead to significant differences in D-H partitioning between water-saturated melt and exsolved aqueous fluid (where D/HW,Melt ≠ D/HW,Fluid) during crystallization of Earth’s magma ocean, potentially controlling the D/H content of the Earth’s oceans.

Modeling nuclear volume isotope effects in crystals.

Mass-independent isotope fractionations driven by differences in volumes and shapes of nuclei (the field shift effect) are known in several elements and are likely to be found in more. All-electron relativistic electronic structure calculations can predict this effect but at present are computationally intensive and limited to modeling small gas phase molecules and clusters. Density functional theory, using the projector augmented wave method (DFT-PAW), has advantages in greater speed and compatibility with a three-dimensional periodic boundary condition while preserving information about the effects of chemistry on electron densities within nuclei. These electron density variations determine the volume component of the field shift effect. In this study, DFT-PAW calculations are calibrated against all-electron, relativistic Dirac-Hartree-Fock, and coupled-cluster with single, double (triple) excitation methods for estimating nuclear volume isotope effects. DFT-PAW calculations accurately reproduce changes in electron densities within nuclei in typical molecules, when PAW datasets constructed with finite nuclei are used. Nuclear volume contributions to vapor-crystal isotope fractionation are calculated for elemental cadmium and mercury, showing good agreement with experiments. The nuclear-volume component of mercury and cadmium isotope fractionations between atomic vapor and montroydite (HgO), cinnabar (HgS), calomel (Hg2Cl2), monteponite (CdO), and the CdS polymorphs hawleyite and greenockite are calculated, indicating preferential incorporation of neutron-rich isotopes in more oxidized, ionically bonded phases. Finally, field shift energies are related to Mössbauer isomer shifts, and equilibrium mass-independent fractionations for several tin-bearing crystals are calculated from (119)Sn spectra. Isomer shift data should simplify calculations of mass-independent isotope fractionations in other elements with Mössbauer isotopes, such as platinum and uranium.

Molar volumes of LiI in H 2O and D 2O solutions; Structural hydration interactions

According to a recent study of the H 2O and D 2O molar volume isotope effect (MVIE) of the alkali metal chloride solutions, neither the standard nor the excess MVIE of the LiCl corresponds to the usual hydrophilic hydration characteristics of the inorganic ions above room temperatures. This phenomenon can not be rationalized by electrostriction, with the collapse of the “loose” tetrahedral (“ice-like”) water structure due to the electrostatic (ion + dipole) interaction. It seemed possible that this unique hydration behaviour of the Li + would be stronger and could reveal further structural information with a less hydrophilic anion than the chloride. Therefore we have determined the MVIE of the LiI as a function of temperature and concentration. The densities of normal and heavy water solutions of LiI have been measured with six-figure precision at T = (288.15, 298.15, and 308.15) K from (0.03 to 4) molal, m, using a vibrating-tube densitometer. The solvent isotope effect on the apparent molar volume, as well as on the solute and solvent partial molar volumes, was evaluated. As expected, with the rationalization of the MVIE of LiI instead of the geometric structural differences of the isotopic solvents, the energetic contributions have to be considered at all the temperatures investigated. At infinite dilution, a high degree of compensation between the reversed influences of the Li + and I − on the activities of the isotopic solvents determines the MVIE. By increasing concentration, the highly asymmetric energetic interactions of the Li + and the I − with the solvent apparently result in a “mutual salting-out” effect. At a concentration ≈0.7 m, a uniquely abrupt structural rearrangement results in a “solvent-separated ion-pair” solution structure.

Evaluation of individual ionic partial molar volumes in aqueous solutions

In a recent H 2O–D 2O molar volume isotope effect study on the alkali metal halide solutions, individual ionic standard partial molar volume contributions ( V ¯ ion ∘ ( H 2 O ) - V ¯ ion ∘ ( D 2 O ) = Δ V ¯ ion ∘ ( H–D ) ) have been derived, dependent only on the chosen (Pauling) ionic radii, r ion. It has been concluded that the isotope effect of the cations (except for Li +) and that of the chloride anion exhibit a linear function of 1/ r ion, with Δ V ¯ ion ∘ ( H − D ) vanishing when 1/ r ion approaches zero. On the other hand, the directly measurable hydrodynamic “cosphere” volumes of the ions, V ¯ cosph ∘ , derived from the kinetic ionic (Stokes) radii, show a similar pattern, here also the cationic and anionic r ion relations cross or meet at V ¯ cosph ∘ ( Cs + ) ≅ V ¯ cosph ∘ ( Cl - ) . Because both Δ V ¯ ion ∘ ( H–D ) and V ¯ cosph ∘ originate in the hydration volumes of the ions, ( V ¯ hydr ∘ = V ¯ ion ∘ - V ¯ intr ∘ ) determined by electrostriction, this r ion pattern of the V ¯ hydr ∘ (ion) provides an “extra-thermodynamic” assumption to evaluate the individual cationic and anionic molar volume contributions in normal water solution. As a basis for the evaluation, this r ion pattern is applied to the conventional cationic and anionic V ¯ hydr ∘ (ion) values (related to V ¯ ion ∘ ( H + ,aq . ) = 0 cm 3 · mol - 1 ) , calculated from literature V ¯ ion ∘ data, assuming that the V ¯ intr ∘ (ion) = V ¯ cryst ∘ (ion). To adjust the V ¯ hydr ∘ ( Cl - ) to the V ¯ hydr ∘ ( Na + , K + , Rb + , Cs + ) versus r ion relation at T = 298.15 K, the conventional anionic V ¯ hydr ∘ (ion) data have to be increased, the cations have to be decreased by 4.2 cm 3 · mol −1. Similar adjustments at five different temperatures between T = (273 and 323) K, result in V ¯ ion ∘ ( H + ,aq . ) = - ( 4.2 ± 0.2 ) cm 3 · mol - 1 at T = 298.15 K, with an increment of −0.02 cm 3 · K −1 · mol −1. Related to these parameters, relations of V ¯ ion ∘ versus temperature for the alkali metal and halide ions were calculated. The cationic and anionic V ¯ hydr ∘ (ion) values, when related to V ¯ ion ∘ ( H + ,aq . ) = - 4.2 cm 3 · mol - 1 , show strong r ion correlation with V ¯ cosph ∘ , their ratio versus r ion offers a linear equation to compare the influence of the ions on the solvent structure on a well defined “quantitative” scale. The value of the derived V ¯ ion ∘ ( H + ,aq . ) , while being consistent with that widely accepted ca. − (5 ± 2) cm 3 · mol −1, has a higher precision with a reasonable basis of the applied “extra-thermodynamic” assumption. It can serve to verify other partitioning procedures used in non-aqueous solvents.

Novel Corresponding-States Principle Approach for the Equation of State of Isotopologues: H218O as an Example

The corresponding-states principle (CSP) has been considered for the development of the equations of state (EOS) of minor isotopologues that are usually unknown. We demonstrate that, for isotopologues of a given molecular fluid, a general extended multi-parameter corresponding-states EOS can be reduced to the three-parameter EOS, utilizing the critical parameters (temperature and density) and Pitzer's acentric factor as correlation parameters. Appropriate general CSP mathematical formalism and equations for constructing the EOS of minor isotopologues are described in detail. The formalism and equations were applied to isotopologues of water and demonstrated that the isotopic effect on the critical parameters and the acentric factor of H(2)(18)O can be successfully calculated from the EOS of H2O and experimental data on the isotope effects (liquid-vapor isotope fractionation factor and molar volume isotope effect). We have also shown that the experimental data on the vapor pressure isotope effect (VPIE) for 18O-substituted water are inconsistent within the framework of thermodynamics with the liquid-vapor oxygen isotope fractionation factor. The novel approach of CSP to isotopologues developed in this study creates a new opportunity for constructing the EOS of minor isotopologues for many other molecular fluids.

Pressure effects on the reduced partition function ratio for hydrogen isotopes in water

We have developed a simple, yet accurate theoretical method for calculating the reduced isotope partition function ratio (RIPFR) for hydrogen of water at elevated pressures. This approach requires only accurate equations of state (EOS) for pure isotopic end-members (H 2O and D 2O), which are available in the literature. The effect of pressure or density on the RIPFR of water was calculated relative to that of ideal-gas water at infinitely low pressure for the temperature range from 0 to 527 °C. For gaseous and low-pressure (ca. ⩽15 MPa) supercritical phases of water, the RIPFR increases slightly (1–1.3‰) with pressure or density in a fashion similar to those of many other geologic materials. However, in liquid and high-pressure (>20 MPa) supercritical phases, the RIPFR of water decreases (0.5–6‰) with increasing pressure (or density) to 100 MPa. This rather unique phenomenon is ascribed to the inverse molar volume isotope effects (MVIE) of liquid and high-density supercritical waters, V (D 2O) > V (H 2O), while other substances including minerals show the normal MVIE. These theoretical predictions were experimentally confirmed by Horita et al. [Horita, J., Cole, D.R., Polyakov, V.B., Driesner, T., 2002. Experimental and theoretical study of pressure effects on hydrogen isotope fractionation in the system brucite–water at elevated temperatures. Geochim. Cosmochim. Acta 66, 3769 – 3788.] for the system brucite–water. Although the P–T ranges for the EOS of normal and heavy waters are rather limited, our modeling indicates that the RIPFR of water continues to decrease with pressure above 100 MPa. The method developed here can be applied to any other geologic fluids, if accurate EOS for their isotopic end-members is available. These results have important implications for the interpretation of high-pressure isotopic partitioning in the Earth, the outer planets, and their moons.

Chromatographic observation of boron isotopic fractionation between kaolin clay and boron bearing solution: A high pressure experiment

In order to determine the boron isotopic fractionation factor between boron adsorbed onto kaolin clay and boron in solution (S), as a first attempt, a break-through column chromatography at a high pressure (12.0 MPa) was performed. A boric acid standard solution (9.25 mmol/l, NIST SRM 951) was fed into a stainless steel column packed with boron-free kaolin clay with the flow rate of 0.8 ml/h. The S value obtained at 293 K was 1.0023, indicating that the lighter isotope (10B) was preferentially enriched into the clay. The magnitude of the S was apparently smaller than the previously reported values obtained by the batch method with seawater boron at atmospheric pressure. Besides the effect of pH and the influence of coexisting ions that would lead to a larger S value, as an aspect, dependence of pressure on the boron isotopic fractionation was a possible candidate reducing the S value. Assuming an occurrence of the molecular volume isotope effect (MVIE) on boron species, the S value would decrease with increasing pressure.

Evidence for nonideality in the fundamental liquid mixture (36Ar+40Ar)

Positive excess molar Gibbs energies were detected in liquid mixtures of (36Ar+40Ar) using a high-accuracy double differential manometric technique. The vapor pressure differences between a mixture of (36Ar+40Ar) and Ar40 and between Ar36 and Ar40 were measured simultaneously with the vapor pressure of Ar40. Four different mixtures were selected and 80 temperatures were chosen in the 85–96 K temperature range. The results show that the vapor pressure of an equimolar liquid mixture is greater than what would be expected if the solution were strictly ideal, by approximately 2.5 Pa in the 85–96 K temperature range. The calculated excess Gibbs energy for the equimolar mixture at 85 K is G1/2E=(0.021±0.004) J mol−1. For the excess molar enthalpy a value of H1/2E=(0.12±0.04) J mol−1 was found. The corresponding deviations from Raoult’s law are therefore extremely small (3 to 4 orders of magnitude smaller than those typically found in binary mixtures of nonisotopic constituents). The evidence of nonideal behavior in such a fundamental and simple system is relevant for the interpretation of the properties of liquid mixtures of more general importance. The pitfalls of the determination of activity coefficients through the comparison between vapor pressure isotope effect data and liquid–vapor isotope fractionation factors are discussed in some detail. The results were used to test three theoretical models, namely, the vdW-1 fluid theory, the 1cLJ perturbation theory, and the statistical theory of isotope effects in mixtures. While the first two proved to be inadequate, the isotope effect theory agrees reasonably well with experiment. The excess Gibbs energy of monatomic isotopic mixtures are related with the molar volume isotope effect and with the mean Laplacian of the potential energy in the liquid phase.

Origins of molar volume isotope effects in hydrocarbon systems

Lattice dynamics calculations are carried out on the hydrogenated and deuterated forms of crystalline polyethylene to elucidate the origins of the molar volume isotope effect. These calculations focus on the consequences of molecular packing on the vibrational free energy, rather than the differences between the C–H and C–D bond lengths arising from differences in the vibrational amplitudes which has been the standard theory for the molar volume isotope effects. It is found that a significant molar volume isotope effect occurs even in the absence of C–H and C–D bond length differences, and is due to differences in intermolecular separations (i.e., H...H or D...D nonbonded distances) rather than intramolecular bond lengths. This effect arises primarily from the volume dependence of the C–H/D stretching frequencies. The total molar volume isotope effect will include both the intermolecular effects described here, plus the intramolecular C–H/D bond length effects described previously. These results show that the standard theory for the molar volume isotope effect, based on the differences in the C–H and C–D bond lengths, is not complete.

Comments on H/D Isotope Effects on Polarizabilities of Small Molecules. Correlation with Virial Coefficient, Molar Volume, and Electronic Second Moment Isotope Effects

Abstract Literature data on H/D isotope effects on polarizability are reviewed. An argument based on perturbation theory correlates the isotopic differences with electronic transition moments via vibrational averaging. The contribution of the second moment isotope effect is especially important. The approach is first illustrated with a calculation on H2/D2 and then employed to correlate differential refractive index data on about twenty other H/D isotopic pairs. A table of bond polarizability isotope effects is presented which permits the estimation of polarizability isotope effects for various kinds of H/D substituted compounds. Coupling these results with known isotope effects on vibrational amplitude permits calculation of vapor phase virial coefficient isotope effects, while combina­tion with refractive index data on liquids yields reasonable values for the sum of the isotope effects on molar volume and electronic second moment. However the quantitative correlation of the polarizability isotope effect with isotope effects on vapor pressure or thermodynamic activity is less direct.

Some Measurements of H/D Polarizability Isotope Effects Using Differential Refractometry

Refractive index differences between the H and D isomers of some common molecules in the liquid phase were measured between 404.7 and 690.0 nm. The data are combined with information on molar volume isotope effects to yield values for H/D isotope effects on the static polarizability, the vibrational contribution to the static and frequency dependent parts of the polarizability, and the H/D isotope effect on the second moment of the electronic charge distribution. The present results suffice to demonstrate the practicability of this technique to measure the components of the polarizability listed above. However for accurate resolution of the vibrational and second moment contributions, refractive index data of still greater precision will be required.

Excess free energy in solutions of isotopic isomers. I. Monatomic species. II. Polyatomic species

The excess free energy Aex for solutions of isotopes is discussed. We consider the volume dependence of the free energy and integrate over the concentration weighted molar volume isotope effect, but under the constraint of constant g(r*). The result is significantly different than results from integration along the isothermal equilibrium path employed by previous authors. Both internal and external modes can make important contributions to Aex. This is exemplified by treating solutions of 36Ar/40Ar, C6H6/C6D6, H216O/H218O, and H2O/HDO.

Isotope effects in condensed phases, the benzene example. Influence of anharmonicity; harmonic and anharmonic potential surfaces and their isotope independence. Molar volume effects in isotopic benzenes

Application of the harmonic oscillator cell model (HOCM) for condensed phase isotope effects to the benzene/deuterobenzene system using temperature dependent force fields, which yield good calculated values for vapor pressure isotope effects (VPIE) and reasonable agreement with spectroscopically observed frequency shifts on condensation, gives calculated values of isotope effect on energies of vaporization which are not in agreement with experiment. Pseudoharmonic corrections are inadequate to restore agreement with experiment. The VPIE in benzene is dominated by the contributions of the CH/CD stretching vibrations and consistency is achieved by expanding the model to include anharmonic corrections for these frequencies. In this fashion a combination of thermodynamic measurements of free energies (by vapor pressure) and energies (from the temperature coefficient of the vapor pressure or from calorimetry) can be employed to yield shifts in anharmonic vibrational constants on condensation. In the second part of the paper the molar volume isotope effect (MVIE) and the effects of volume, pressure, and temperature on the internal energy and its isotope effects (as monitored by the expansivity, compressibility, heat capacity, and their derivatives) are interpreted. The low-lying lattice modes make important contributions to these properties.