Isotope exchange equilibria

Kaplan and Thornton (1967) used three different sets of vibrational frequencies to estimate the zero-point energies of the reactants and products of the equilibrium, which provided three different isotope exchange equilibrium constants 1-163, 1-311 and 1-050. The value 1-311 is considered to be most reasonable, whereas the others are rejected as unrealistic for the case in hand. Calculations using the complete theory led to values that varied from 1-086 to 1-774 for different sets of valence-force constants for the compounds involved. 

This latter equilibrium is called an isotopic exchange equilibrium. Its equilibrium constant in terms of partition functions is from Equation 4.64... 

Finally in this section we turn our attention to the calculation of the isotope exchange equilibrium constant... 

Calculations of isotope effects and isotopic exchange equilibrium constants based on the Born-Oppenheimer (BO) and rigid-rotor-harmonic-oscillator (RRHO) approximations are generally considered adequate for most purposes. Even so, it may be necessary to consider corrections to these approximations when comparing the detailed theory with high precision high accuracy experimental data. 

In 1947 Harold Urey, the 1934 Nobel Laureate, recognized that the temperature dependence of the isotope exchange equilibrium between water and calcite (the principal mineral in marine limestones) could be employed as a paleo-thermometer. At 298.15 K the fractionation factor for calcite-water is 1.0286,... 

The value of the fractionation factor for any site will be determined by the shape of the potential well. If it is assumed that the potential well for the hydrogen-bonded proton in (2) is broader, with a lower force constant, than that for the proton in the monocarboxylic acid (Fig. 8), the value of the fractionation factor will be lower for the hydrogen-bonded proton than for the proton in the monocarboxylic acid. It follows that the equilibrium isotope effect on (2) will be less than unity. As a consequence, the isotope-exchange equilibrium will lie towards the left, and the heavier isotope (deuterium in this case) will fractionate into the monocarboxylic acid, where the bond has the larger force constant. 

The equilibrium constant for the isotope-exchange equilibrium can be expressed (6) in terms of the solvent isotope effects on the acid-dissociation constants and of the monocarboxylic acid and dicarboxylic acid monoanion, respectively. It follows that a lower value for the fractionation factor of the hydrogen-bonded proton means that the solvent isotope effect on the acid-dissociation constant will be lower for the dicarboxylic acid monoanion than for the monocarboxylic acid. 

A AZPE = AZPEii — AZPEd AZPEt) corresponds to the terms for the reactions of monodeuteriated aldehydes. Terms defined by IE = MMl x EXC x EXP (IE is the Isotopic exchange equilibrium, MMl is the mass moment of inertia term representing the rotational and translational partition function ratios, EXC is the vibrational excitation term and EXP is the exponential zero point energy). 

Often Ao is omitted, either deliberately or inadvertently. Although oo is small, it is not completely negligible for light molecules and affects their zero-point energies this must be taken into account when isotopic-exchange equilibrium constants are calculated see M. Wolfsberg, Advances in Chemistry Series, No. 89, p. 185 (1969). 

Isotopic Exchange/Equilibrium. Chemical steps are required at the outset of the procedure to insure isotopic exchange between the radionuclide to be analyzed (the radioanalyte) and the tracer or carrier that has been added. The carrier or tracer and the radioanalyte must be in the same oxidation state and chemical species in solution. This effort is not required for radionuclides that exist in only a single form, such as Group 1A (Li, Na, K, Rb) elements that are consistently in their +1 state in solution. Other elements (such as I or Ru) that have multiple oxidation states, and also can form stable complexes, will require steps to insure that the added carrier or tracer and the radioanalyte exchange before the analysis is started. 

Fig. 8 Potential energy wells for the isotope-exchange equilibrium in (3), L = H or D.

AICI3. In the isotope exchange equilibrium (12.9), the specific activity of (1) is given by... 

In order to investigate the validity of the gamma-bar method for these equilibrium cases, a single fit-producing y was calculated for each isotope-exchange equilibrium constant according to... 

None of the approximation methods investigated appears to be very reliable for application, in the room-temperature region, to the (nearunity) isotope-exchange equilibrium constants for the systems considered here. The inadequacy of the approximation methods in the majority of the cases studied arises, for the most part, from anomalies in the temperature dependences of the exact equilibrium constants. It has been shown (22) that whereas the logarithms of partition-function ratios are always smooth monotonic functions of temperature, plots of the logarithms of ratios of partition-function ratios—Le., isotope-exchange equilibrium constants—vs. T (or vs. log T) may exhibit maxima, minima, and inflection points. In addition, equilibrium constants may exhibit the... 

It is demonstrated that the formula for the vibrational energy states of a diatomic molecule should be written as En/hc = Go ioj n - - 1/2) — (OeX (n + 1/2) with non-zero Go. Go values are evaluated for some diatomic hydrides and the effect of Go on the theoretical calculation of isotopic exchange equilibrium constants is shown. 

When one takes into account anharmonicity in the theoretical calculation of isotopic exchange equilibrium constants, one usually employs only the anharmonic correction to the zero-point energy (—l/4[Pg.185]

The present discussion has been carried out within the framework of the Born-Oppenheimer approximation, as is usual in the discussion of thermodynamic isotope effects. It is conceivable, however, that correction terms to this approximation could make non-negligible contributions to calculated isotopic exchange equilibrium constants [see References 9 and JO]. 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

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