Wilson energy constants

Stage 1. The MeOH/H20/NaCl data are subjected to the correlation procedure described previously which gives values of the Wilson energy constants (Zi and Z2) and a new set of data for temperature and vapor composition that are internally consistent (see Table I). The small values of the standard deviation and the bias indicate good quality data in the salt effect field. For the analysis of serial correlation among the residuals we use the Durbin-Watson test (9). A run of positive or negative signs in the series of residuals is some indication that the model... 

The established method for calculating the vibrational frequencies of molecules is the Wilson GF method.27 In this method, the potential energy of a molecule is defined in terms of the force constants by a matrix F, and the kinetic energy, which depends on the geometry of the molecule, is defined by a matrix G. Using the methods of classical mechanics, the following equation may be derived. 

The energy levels of the vibrational modes can be predicted with a reasonable accuracy on the basis of the standard Wilson vibrational analysis (241,244) (called GF analysis). The vibrational motion of atoms in the polyatomic system is approximated by harmonic oscillations in a quadratic force field. Computations of the force constants are the subject of quantum chemistry. 

The two systems containing potassium nitrate and ammonium sulfate each have a region of partial miscibility. Adding a third constant to the Wilson equation allows correlation of such systems. The results are shown in Table II and Figure 2. (The broken line of Figure 2 is a feature of systems which show material instability. A graph of molar free energy... 

The coefficients of the B matrix have been developed by Wilson and Califano (Wilson, 1955 Califano, 1976). The corresponding formulas are included in all available normal coordinate packages. In the system of internal coordinates the force constants are defined as the second derivatives of the potential energy with respect to two coordinates. In order to obtain the Fx matrix, the Fr matrix with internal coordinates has to be transformed according to... 

In order to evaluate the thermodynamic functions of the process (5), it is necessary to know the interaction energy, equilibrium geometry and frequencies of the normal vibration modes of the bases and base pairs involved in equilibrium process. Interaction energies and geometries are evaluated using empirical potential or quantum chemically (see next section), and normal vibrational frequencies are determined by a Wilson FG analysis implemented in respective codes. Partition functions, computed from AMBER 4.1, HF/6-31G and MP2/6-31G (0.25) constants (see next section), are evaluated widiin the rigid rotor-harmonic oscillator-ideal gas approximations (RR-HO-IG). We have collected evidence [26] that the use of RR-HO-IG approximations yields reliable thermodynamic characteristics (comparable to experimental data) for ionic and moderately strong H-bonded complexes. We are, therefore,... 

Many such model force fields have been discussed in the literature. [For general discussions, see Herzberg (1945), Wilson et al. (1955), Woodward (1972), and Califano (1976). For discussions of the Urey-Bradley force field, see the review by Duncan (1975). For discussions of the entirely different consistent force field approach, see Lifson and Warshel (1968), Warshel et al. (1970), and Burkert and Allinger (1982).] We have chosen to use a simplified general valence force field (SGVFF), which has been defined as one which contains the minimum possible number of interaction constants compatible with a good fit of the spectra (Califano, 1976). Such a force field has been demonstrated to be very effective for hydrocarbons (Schachtschneider and Snyder, 1963). For this form the potential energy of Eq. (63) is written explicitly as... 

We shall not describe the internal coordinate system nor the symmetry coordinates, the symmetry force constants and the kinetic energy matrix, as they are detailed in McCullough s paper (/). Wilson s book, "Molecular vibrations , gives further explanations of the methods used (2). 

The relationship between the potential function K(R) and the observable spectroscopic parameters is summarized in Figure 2. The harmonic vibration frequencies are obtained as the eigenvalues of a secular determinant involving the quadratic force constants and the atomic masses and molecular geometry (the F and G matrices of Wilson s well-known formalism) by a calculational procedure discussed in detail by Wilson, Decius, and Cross.1 The eigenvectors determine the normal coordinates Q in terms of which the kinetic and quadratic potential energy terms are both diagonal (R = LQ). The various anharmonidty constants and vibration/rotation interaction constants are obtained in terms of the... 

Wilson [666, 667] and Bauman and Maron [47] show that it is possible to express the reaction rate as a function of film thickness, diffusion constant and solubility of oxygen in the film. When the thickness of the film is reduced to less than a certain value, the chemical reaction and not the diffusion becomes the controlling factor. The activation energy of the oxidation reaction amounts to 16—35 kcal mole-1, or even more, whereas the activation energy of the diffusion of gases in polymer films [39] is of the order of only 10 kcal mole-1. Control by diffusion is facilitated at higher temperatures by the decrease of oxygen solubility in the films. 

In this equation, ga is the interaction energy (analogous to Xj, of the Wilson equation). The parameter a arises from the fact that locally the composition of the solution may differ from the bulk concentration giving rise to nonrandom arrangement of molecules. It is usually set to a constant value between 0.2 and 0.3. Treating a as a constant, the NRTL model contains two adjustable parameters, T12 and T21 (notice that the individual values of the ga are not needed for the calculation). 

The calculations of the last two sections were made in order to interpret the experiments on the 1/v law of fission. These experiments were carried out toward the end of 1941 in Princeton by Creutz, Wilson and Kamm and were reported this year under C-109. The investigation was motivated by the surprisingly small ratio of thermal activity to epicadmium activity which appeared to indicate a deviation from the 1/v law for energies as low as. 5 eV. The suspicion of such a deviation (at least to any considerable extent) was dispelled by the experiments and their interpretation to be gone through below. Unfortunately, due to the constant pressure of other work, the experiments are less accurate than they would have been made had more time been available. 

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