Molar Volume Isotope Effects

IE s on some of the other properties of water are shown in Table 5.9. Many properties (like the enthalpies of phase change, triple points, etc.) are closely related to VP and can be interpreted similarly. Molar volume isotope effects are interesting and are discussed in Chapters 12 and 13. In the low temperature liquids... 

Table 12.3 CDDR (continuous dilution differential refractometry) least squares parameters, molar volume isotope effects, and derived PIEs for some isotopomer solutions at 298.15K see Equation 12.15, AR/R = A + mv2 (Wieczorek, S. A., Urbanczyk, A. and Van Hook, W. A.,. /. Chem. Thermodyn. 28, 1009 (1996)) ...

Table 12.5 Molar volume isotope effects for some liquids, mostly at or below their normal boiling...

Fig. 12.9 Molar volume isotope effect for (H20/D20)ice left side) and (H2O/D2O) liquid (right side)... 

The ideal behavior of isotopic mixtures is expected if one assumes that the intermolecular forces between pairs of like molecules of each type and between unlike molecules are all the same, and further assumes the isotopic molecules to have the same size. Both assumptions are reasonable to a first approximation. Highly precise vapor pressure measurements on isotopic mixtures have shown, however, that even these mixtures exhibit small, but still significant, deviations from ideal behavior (Jancso et al. 1993, 1994). Theoretical analysis has demonstrated that the origin of nonideality is closely connected with the difference between the molar volumes of isotopic molecules ( molar volume isotope effect ). 

Experimental measurements of the molar volumes of benzene, toluene, cyclohexane, and methylcyclohexane and their deuterated analogs have shown that the lighter species have a greater volume by about 0.3% (Bartell and Roskos 1966). This difference in molar volumes molar volume isotope effect) arises from the fact that C-H bonds are longer than C-D bonds by about 0.005 A. Whereas in the above case, the hindered translational motion of the molecules as a whole plays a negUgible role, for small molecules (e.g., ethylene) the vibrations of the molecules give important contributions to the molar volume isotope effect (Menes et al. 1970). The volume dependence of the vibrational frequencies may also influence the molar volume isotope effect as was shown in the case of polyethylene (Lacks 1995). 

Studies of the molar volumes of perdeuteriated organic compounds might be expected to be informative about non-bonded intermolecular forces and their manifestations, and such studies might be considered to obviate the necessity of investigating steric isotope effects in reacting systems. The results from non-reacting systems could then be simply applied to the initial and transition states in order to account for a kinetic steric isotope effect. 

Abstract Isotope effects on the PVT properties of non-ideal gases and isotope effects on condensed phase physical properties such as vapor pressure, molar volume, heats of vaporization or solution, solubility, etc., are treated in some thermodynamic detail. Both pure component and mixture properties are considered. Numerous examples of condensed phase isotope effects are employed to illustrate theoretical and practical points of interest. 

A more careful analysis taking into account vapor nonideality through the second virial coefficient and the isotope effect on condensed phase molar volume yields Equation 5.16... 

An important attribute of Equation 5.16 is that the pressure exerted on both phases, Ptot, is common to both isotopomers. The important difference between Equations 5.16 and 5.9 is that the isotopic vapor pressure difference (P/ — P) does not enter the last two terms of Equation 5.16 as it does in Equation 5.9. Also isotope effects on the second virial coefficient AB/B = (B — B)/B and the condensed phase molar volume AV/V are significantly smaller than those on AP/P ln(P7P). Consequently the corrections in Equation 5.16 are considerably smaller than those in Equations 5.9 and 5.10, and can sooner be neglected. Thus to good approximation ln(a") is a direct measure of the logarithmic partition function ratio ln(Qv Q7QvQcO> provided the pressure is not too high, and assuming ideality for the condensed phase isotopomer solution. For nonideal solutions a modification to Equation 5.16 is necessary. 

Isotope Effects on Dipole Moments, Polarizability, NMR Shielding, and Molar Volume... 

Abstract Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. 

Using Equation 12.12 one obtains (AA/A — Av /v 2) = (Aao/ao). We see that precise refractive index differences measured over a reasonable range of wavelengths allow the recovery of the polarizability isotope effect (i.e. the isotope effect on the electric field induced dipole moment), provided the molar volume and its isotope effect are available. 

The Bartell mechanical model has also been used to estimate the isotope effect on molar volume due to the over all motion (i.e. hindered translation) of molecules interacting in a Lennard-Jones potential. For C6H6/C6D6 one finds AV/V 4 x 10-5, about two orders of magnitude smaller than the contribution of the internal modes (and experiment). We conclude that for all but very light molecules this contribution can be neglected. 

Bartell, L. S. and Roskos, R. R., Isotope effects on molar volume and surface tension simple theoretical model and experimental data for hydrocarbons. J. Chem. Phys. 44, 457 (1966)... 

Van Hook, W. A. and Wolfsberg, M., Comments on H/D isotope effects on polarizabilities. Correlation with virial coefficient, molar volume and electronic second moment isotope effects. Z Naturforsch. 49A, 563 (1994)... 

Table 13.1). In the solid P(CH4) > P(CD4) but the curves cross below the melting point and the vapor pressure IE for the liquids is inverse (Pd > Ph). For water and methane Tc > Tc, but for water Pc > Pc and for methane Pc < Pc- As always, the primes designate the lighter isotopomer. At LV coexistence pliq(D20) < Pliq(H20) at all temperatures (remember the p s are molar, not mass, densities). For methane pliq(CD4) < pLiq(CH4) only at high temperature. At lower temperatures Pliq(CH4) < pliq(CD4). The critical density of H20 is greater than D20, but for methane pc(CH4) < pc(CD4). Isotope effects are large in the hydrogen and helium systems and pLIQ/ < pLiQ and P > P across the liquid range. Pc < Pc and pc < pc for both pairs. Vapor pressure and molar volume IE s are discussed in the context of the statistical theory of isotope effects in condensed phases in Chapters 5 and 12, respectively. The CS treatment in this chapter offers an alternative description.

Molar volume and vapor pressure IE s may be calculated as f(TR) using critical parameters and their isotope effects (Table 13.1) assuming no IE on the system de-... 

Van Hook and Phillips (34) and Van Hook (37) have discussed the application of Equation 1 to gas-liquid and gas-solid chromatography respectively. First consider the corrective terms. The third term in Equation 1 corrects for the isotope effect on the partial molal volumes of the condensed phase. In the case of the two dimensional adsorbed film the term should be rewritten in terms of the surface tension and the molar coverage. In either event the correction is expected to be of the same order of magnitude as that for the pure liquids (where it reduces simply to the isotope effect on molar volumes). These corrections are negligibly small. They amount to only about 0.1% of the total isotope effect per D atom for representative hydrocarbons (35). Similarly, the fourth term which corrects for the isotope effect on the nonideality of the gas phase is readily shown to be negligibly small (31, 35) under normal chromatographic conditions. 

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