Critical property isotope effects

Solutions in hand for the reference pairs, it is useful to write out empirical smoothing expressions for the rectilinear densities, reduced density differences, and reduced vapor pressures as functions of Tr and a, following which prediction of reduced liquid densities and vapor pressures is straightforward for systems where Tex and a (equivalently co) are known. If, in addition, the critical property IE s, ln(Tc /Tc), ln(PcVPc), and ln(pcVPc), are available from experiment, theory, or empirical correlation, one can calculate the molar density and vapor pressure IE s for 0.5 Tr 1, provided, for VPIE, that Aa/a is known or can be estimated. Thus to calculate liquid density IE s one uses the observed IE on Tc, ln(Tc /Tc), to find (Tr /Tr) at any temperature of interest, and employs the smoothing relations (or numerically solves Equation 13.1) to obtain (pR /pR). Since (MpIE)R = ln(pR /pR) = ln[(p /pc )/(p/pc)] it follows that ln(p7p)(MpIE)R- -ln(pcVpc). For VPIE s one proceeds similarly, substituting reduced temperatures, critical pressures and Aa/a into the smoothing equations to find ln(P /P)RED and thence ln(P /P), since ln(P /P) = I n( Pr /Pr) + In (Pc /Pc)- The approach outlined for molar density IE cannot be used to rationalize the vapor pressure IE without the introduction of isotope dependent system parameters Aa/a. 

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Reduced Equations of State Critical Property Isotope Effects... 

Table 13.1 Critical properties and critical property isotope effects for some isotopomer pairs...

Critical Property Isotope Effects 13.3.1 Experimental Data... 

Abstract In corresponding states (CS) theory the PVT properties of fluids are expressed in terms of the critical constants and one or more additional parameters. In this chapter the use of CS theory to correlate isotope effects on the physical properties of fluids is explored. 

Commonly encountered cubic equations of state are classical, and, of themselves, cannot rationalize IE s on PVT properties. Even so, the physical properties of iso-topomers are nearly the same, and it is likely in some sense they are in corresponding state when their reduced thermodynamic variables are the same that is the point explored in this chapter. By assuming that isotopomers are described by EOS s of identical form, the calculation of PVT isotope effects (i.e. the contribution of quantization) is reduced to a knowledge of critical property IE s (or for an extended EOS, to critical property IE s plus the acentric factor IE). One finds molar density IE s to be well described in terms of the critical property IE s alone (even though proper description of the parent molar densities themselves is impossible without the use of the acentric factor or equivalent), but rationalization of VPIE s requires the introduction of an IE on the acentric factor. 

Fig. 13.4 CS calculations for 3He/4He and H2/D2. Points are experimental, lines calculated. Heavy lines use observed critical property IE s and non-zero Aa/a (see text). Lighter lines employ correlated critical property IE s and non-zero Aa/a. The cross-hatched lines set Aa/a = 0. (a) (Upper) VPIE s. For 3He/4He and H2/D2 lines based on observed and correlated critical property IE s cannot be distinguished on the scale of the figure, (b) (Lower) molar density isotope effects. For both 3He/4He and H2/D2 cross-hatched lines assuming Aa/a = 0 nearly coincide with the heavy solid lines and are not plotted (Reprinted from Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M., Fluid Phase Equilib. 257, 35 (2007), copyright 2007, with permission from Elsevier)...

Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. Isotope effects on VLE properties of fluids and corresponding states critical point shifts on isotopic substitution. Fluid Phase Equilib. 257, 35 (2007). 

Symmetry factors, o, do not appear in eqn. (28) because the numbers of equivalent pathways have been allowed for in the definition of the kinetic isotope effect. F0(d is the critical energy of the decomposition involving the lighter isotope and F0(II) that of the decomposition involving the heavier isotope. The density of states, N(E), of the reactant ion is, of course, common to both decompositions and does not affect the intramolecular kinetic isotope effect. The intramolecular kinetic isotope effect is, therefore, dependent only upon transition state properties. 

For intramolecular isotope effects, as one starts from a unique precursor, the two processes correspond to the same internal energy distribution, so that useful information can be inferred even from ionabun-dances measured without internal energy selection (like metastable dissociations or field ionization kinetics). This is not the case for intermolecular effects, where there is no warranty that the two internal energy distributions are the same. Within the RRKM framework, the intramolecular kinetic isotope effect depends only on the transition state properties (critical energy, rotational constants and vibrational frequencies) and not on the reactant properties. 

It is critical when performing quantitative GC/MS procedures that appropriate internal standards are employed to account for variations in extraction efficiency, derivatization, injection volume, and matrix effects. For isotope dilution (ID) GC/MS analyses, it is crucial to select an appropriate internal standard. Ideally, the internal standard should have the same physical and chemical properties as the analyte of interest, but will be separated by mass. The best internal standards are nonradioactive stable isotopic analogs of the compounds of interest, differing by at least 3, and preferably by 4 or 5, atomic mass units. The only property that distinguishes the analyte from the internal standard in ID is a very small difference in mass, which is readily discerned by the mass spectrometer. Isotopic dilution procedures are among the most accurate and precise quantitative methods available to analytical chemists. It cannot be emphasized too strongly that internal standards of the same basic structure compensate for matrix effects in MS. Therefore, in the ID method, there is an absolute reference (i.e., the response factors of the analyte and the internal standard are considered to be identical Pickup and McPherson, 1976). 

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