The higher order distributions are defined analogously. 1 2 contains through the identity [Pg.692]

Little is known about higher order critical points. Tetracritical points, at least imsynnnetrical ones, require four components. However for tetracritical points, the crossover dimension

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe

One problem with this treatment is that it neglects higher-order terms depending on higher moments of i/au that become undefined for slowly decaying interaction potentials (see Problem III-9). [Pg.62]

In general terms, if the reaction to the desired product has a higher order than the byproduct reaction, use a batch or plug-flow reactor. If the reaction to the desired product has a lower order than the byproduct reaction, use a continuous well-mixed reactor. [Pg.30]

Here we have neglected derivatives of the local velocity of third and higher orders. Equation (A3.1.23) has the fonn of the phenomenological Newton s law of friction [Pg.675]

A number of improvements to the Bom approximation are possible, including higher order Born approximations (obtained by inserting lower order approximations to i jJ into equation (A3.11.40). then the result into (A3.11.41) and (A3.11.42)), and the distorted wave Bom approximation (obtained by replacing the free particle approximation for the solution to a Sclirodinger equation that includes part of the interaction potential). For chemical physics [Pg.968]

The virial equation for the pressure is also modified by the tliree-body and higher-order temrs, and is given in general by [Pg.475]

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy [Pg.472]

Figure 2.3 Choice of reactor type for mixed parallel and series reactions when the parallel reaction has a higher order than the primary reaction. |

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. [Pg.188]

In principle, one can do beder by allowing for R-dependence to U and T. If we allow them to vary linearly with R, then we have Gordon s method [42]. However, the higher order evaluation in this case leads to a much more cumbersome theory that is often less efficient even though larger steps can be used. [Pg.985]

Multiple reactions. For multiple reactions in which the byproduct is formed in parallel, the selectivity may increase or decrease as conversion increases. If the byproduct reaction is a higher order than the primary reaction, selectivity increases for increasing reactor conversion. In this case, the same initial setting as single reactions should be used. If the byproduct reaction of the parallel system is a [Pg.63]

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. [Pg.423]

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. [Pg.465]

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. [Pg.643]

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