Higher Approximations

Diatomic Molecules, For diatomic molecules the correction terms can be evaluated explicitly, and in recent years have to an increasing extent been calculated from improved spectroscopic data now available. The 

The vibrational anharmonicity gives a first additional term in the vibrational energy of — xdtfv + he, [The second term is the coefficient of (v + i) and is shown by observed spectra to be negligibly small.] Here, Xe is the (first) anharmonicity constant and is the equilibrium wave-number, for small displacements. The energy of the vth vibrationally excited state is then given by 

The coupling of rotation and vibration gives an additional term in the expression for (e — So)/hc of — a(y + i)J(J + 1) where a is the rotational-vibrational coupling constant. The effect of centrifugal stretching is expressed by a further term of — DJ%J + 1) where D is the rotational stretching coefficient, which is related to the rotational constant for equilibrium separation, jBe. by Z) = ABljvl. If So is the rotational constant for the first vibrationally excited state, then Bq = Be — a/2. 

Defining the quantities u = hcvJkT, 8 = a/Bo, and y = D/ABeY = (Belve), then the additional correction to the partition functions can be expressed, after some simplification, by 

The resulting corrections to the thermodynamic properties then follow as functions of x, 8, y, and u, all of which are derivable from spectroscopy. Tabulated values of the functions involving u are available, which facilitate the calculation, an example of which is given in Table 2. 

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At the end of the paper, condensed tables of the higher approximations have been carried out with respect to atomic and molecular systems. For atoms, the tables are arranged after the number of electrons involved, which means that, e.g., N = 2 refers to the series of He-like ions H", He, Li+, Be2+, etc. For molecules, there is a table for H2 a table for other simple molecules (LiH, BeH+, H20, NH3, etc.) with all or almost all electrons treated, and finally a special table for the n electron systems in the two latter cases, the references to the best SCF data available are also contained for comparison. 

Craig, D. P., Proc. Roy. Soc. London) A200, 474, Configurational interaction in molecular orbital theory. A higher approximation in the non-empirical method." (i)... 

From this point of view the early K.B. theory was somewhat less general than the theory of Poincar6, in which higher approximations are obtained by taking more terms in the series Eq. (6-55). 

For NGF, the loss of capacity is much higher (approximately 68%) than the loss of graphite active mass (approximately 2.5%). Thus, for this type of graphite the corrosion mechanisms (4) and (5) with formation of surface groups may be considered predominant. 

The concentration of insulin present in soluble insulin preparations (i.e. fast-acting insulins), is much higher (approximately 1 x KT2 3 mol I ). At this concentration, the soluble insulin exists as a mixture of monomer, dimer, tetramer and zinc-insulin hexamer. These insulin complexes have to dissociate in order to be absorbed from the injection site into the blood, which slows down the onset of hormone action. 

The Chapman-Enskog theory was developed for dilute, monatomic gases for pure substances and for binary mixtures. The extension to multicomponent gas mixtures was performed by Curtiss and Hirschfelder (C12, Hll), who in addition have shown that the Chapman-Enskog results may also be obtained by means of an alternate variational method. Recently Kihara (K3) has shown how expressions for the higher approximations to the transport coefficients may be obtained, which are considerably simpler than those previously proposed by Chapman and Cowling these simpler formulas are particularly advantageous for calculating the coefficients of diffusion and thermal diffusion (M3, M4). 

O-th approximation under very general conditions. But this is, though necessary, not sufficient for practical purposes. In this and the following sections we shall give sufficient conditions for higher approximations to he valid. The f undamental conditions A,) plus C), or 1>) plus B0) plus 0), are always assumed in... 

These results are generalized to higher approximations as follows. Suppose that the quantities... 

Higher approximations can be treated in similar manner and lead to results corresponding to those of 9, but it will not be necessary to state them in detail. 

In these equations, T is the temperature, p is the pressure, X is the mole fraction of species k, m is the molecular mass, R is the universal gas constant, and / is the pure species viscosity. The T>jk are first order (in the Chapman-Enskog theory) binary diffusion coefficients, given by Eq. 12.113. It is actually inappropriate [103] to use a second-order or higher approximation [265] to the binary diffusion coefficients here. For this reason the Dixon-Lewis paper used the notation to emphasize that the first-order approxima-... 

Higher approximations give corrections to this Fokker-Planck equation, but it remains of the form of an M-equation. Thus the expansion in rc produces to all orders an approximation in the form of a Markov process, see XVI.5. 

This Ansatz is the essential step. The -expansion is not just one out of a plethora of approximation schemes, to be judged by comparison with experimental or numerical results 0. It is a systematic expansion in and is the basis for the existence of a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. It will appear that in the lowest approximation the noise is Gaussian, as is commonly postulated. In addition, however, it opens up the possibility of adding higher approximations. 

This is the final result. Higher approximations involving g3,g4,... do not change (4.7a) but merely add more terms to (4.7b). 

For the reader who felt uneasy with the various ad hoc approximations in the preceding section we now present a more systematic derivation of the same result. That also opens the door to higher approximations, although we shall not go through it. 

Kasimov and Zigmund (Kl2), 1963 General solution is given to equations of wavy film flow (with corrected continuity equation), using simple parabolic or higher approximation to velocity profile at given plane. It is shown that the film thickness and wave amplitude should increase in direction of flow. Effect of channel slope also considered. 

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