Density series generalized equation

Multiple sets of Burnett data were obtained for each isotherm—three sets for ethylene and two sets for helium. Each set consisted of data from a series of four consecutive expansions from the highest to the lowest pressure compatible with our optimum accuracy and precision. The initial pressure for each set was selected so as to intersperse the data from all of the sets over the entire pressure range of interest, 0.3 MPa to 3.7 MPa. Consistent with the extent of the nonideal behavior of the gas, the density-series generalized equation was applied to the ethylene data and the pressure-series generalized equation was applied to the helium data. The parameters in the resulting overdetermined sets of equations then were evaluated using the least-squares constraint. 

An even more effective numerical method for calculating the magnetic properties (25), (28), (31), and (34), based on formal annihilation of the paramagnetic contribution to the current density all over the molecular domain (CTOCD-PZ), has been outlined in a series of papers. Extensive numerical tests document the reliability, simplicity, and accuracy of this numerical procedure. To build up the CTCXID-PZ computational scheme it is expedient to define a set of generalized transformation functions theoretical methods which have been examined in detail by Coriani et al. Unlike the DZ procedure, the CTOCD-PZ equations cannot, in general, be solved to obtain closed form expressions. Rather the functions employed to cancel the transverse paramagnetic contribution to current density are evaluated pointwise via conditions imposed on first- and second-order current densities (compare for equations 113 and 115) ... 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

When a series of stirred-tanks is used as a chemical reactor, and the reactants are fed at a constant rate, eventually the system reaches a steady state such that the concentrations in the individual tanks, although different, do not vary with time. When the general material balance of equation 1.19 is applied, the accumulation term is therefore zero. Considering first of all the most general case in which the mass density of the mixture is not necessarily constant, the material balance on the reactant A is made on the basis of FA moles of A per unit time fed to the first tank. Then a material balance for the rth tank of volume V (Fig. 1.17) is, in the steady state ... 

First I will discuss Fourier series and the Fourier transform in general terms. I will emphasize the form of these equations and the information they contain, in the hope of helping you to interpret the equations — that is, to translate the equations into words and visual images. Then I will present the specific types of Fourier series that represent structure factors and electron density and show how the Fourier transform inter con verts them. 

The most general of the equations of state is the virial equation, which is also the most fundamental since it has a direct theoretical connection to the intermolecular potential function. The virial equation of state expresses the deviation from ideality as a series expansion in density and, in terms of molar volume, can be written... 

These equations are a part of the more general Eq. (40). They describe the dispersion of the quadratic term of the power series of the scattering function. For spherical structures like the ferritin molecule p is zero, as there is no change of the centre of mass at different solvent densities. Furthermore, the contributions to q and q" are small compared to q. The radius of gyration exhibits the dispersion of f ... 

The density of phosgene vapour under standard reference conditions was measured to be 4.526 [742] or 4.525 kg m 3 [1281]. Using the value of the standard molar volume, Vnj j, the density of the gas at 0 C and atmospheric pressure was calculated to be 4.413 kg m 3 Phosgene vapour is thus, unexpectedly, far removed from ideality. An attempt has been made to generalize the Benedict-Wee-Rubin equation of state using three polar parameters as part of a study of a large series of polar substances, which includes COClj as one of the examples [1518]. 

A statistical theory of turbulence which is applicable to continuous movements and which satisfies the equations of motion was introduced by Taylor [159, 160, 161] and [162, 163], and further developed by von Karman [178, 179]. Most of the fundamental ideas and concepts of the statistical turbulence theory were presented in the series of papers published by Taylor in 1935. The two-point correlation function is a central mathematical tool in this theory. Considering the statistics of continuous random functions the complexity of the probability density functions needed in a generalized flow situations was found not tractable in practice. An idealized flow based on the assumption of... 

The dielectric constant eo of a pure fluid in total equilibrium is in general a function of the density p0 and temperature T0 that is, eo = e(p0, To). This is called the dielectric equation of state. Clearly on the local level, there are small fluctuations in the local density and temperature so that we can write p(r, t) = p0 + 5p(r, t) and T(t, t)=T0 + ST(t, t). Thus if we assume local equilibrium, the dielectric equation of state can be used to determine the local value of the dielectric constant. Accordingly we write e(r, t) = e(p0 + p r, f) To + ST(t, t)). Since these fluctuations are expected to be quite small, this can be expanded in a (rapidly convergent) power series in these fluctuations. The local dielectric fluctuation <5c(r, t) = e(r, t) — e(p0, T0) is then to first order in the fluctuations dp, ST,... 

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