No abstract

In the study of chemical kinetics, electrical circuit theory, and problems of missile guidance a type of differential equation arises which is exceedingly difficult to solve by ordinary numerical procedures. A very satisfactory method of solution-of these equations is obtained by making use of a forward interpolation process. This scheme has the unusual property of singling out and approximating a particular solution of the differential equation to the exclusion of the manifold of other solutions. This behavior may be explained by a simple geometrical interpretation of the significance of the forward interpolation process. The differential equations to which this method applies are called "stiff."A typical example of a stiff equation is the equation representing the rate of formation of free radicals in a complex chemical reaction. The free radicals are created and destroyed so rapidly compared to the time scale for the over-all reaction that to a first approximation the rate of production is equal to the rate of depletion. This is the notion of the pseudo-stationary state. In some cases such as the fast reactions occurring in flames or detonations, this approximation is not sufficiently accurate. The method described in the present paper provides a means for obtaining solutions to equations of this type to any degree of accuracy.The nunmerical procedure described here can easily be extended to sets of simultaneous first-order differential equations. In any particular region, the differential equations can be uncoupled by introducing suitable linear combinations of the original dependent variables. Some of the uncoupled equations may be "stiff" in which case they can be integrated by the methods discussed here; whereas other uncoupled equations may be integrated by the more usual procedures.

Historically there have been two major formulations for the mass-flux relations in multicomponent diffusion: (1) a generalization of Fick's law in which the mass flux is written as a linear combination of concentration gradients and (2) a generalization of Maxwell's expression in which the concentration gradient is given as a linear combination of the mass fluxes. The thermodynamics of irreversible processes has made it possible to generalize these expressions to include thermal, pressure, and forced diffusion. Associated with these two formulations, there are various definitions for the multicomponent diffusivities. Here the interrelations among the variously defined diffusivities are given, some connections with molecular theories are made, and several neglected publications are cited.

Equations are derived for the viscosity, ordinary (pressure) diffusion, and thermal diffusion of multicomponent mixtures of gases. The ordinary diffusion velocities are expressed in terms of the usual diffusion coefficients for binary mixtures. The analysis is an extension of the work of Chapman and Cowling. It is shown that the Chapman-Cowling and Enskog procedure can be justified on the basis of a variational principle. This variational principle should be generally applicable to a large class of problems.

The free volume is calculated for non-interacting rigid sphere molecules taking into account the exact geometry imposed by the face-centered cubic packing. The size and shape are quite different from that of the inscribed spheres which correspond to the Lennard-Jones and Devonshire approximation. At high densities the equation of state obtained from the exact treatment agrees well with the Eyring or Lennard-Jones and Devonshire equation. When the specific volume is greater than twice the cube of the collision diameters, the molecules are no longer confined to cages formed by neighboring molecules. At these low densities the free volume concept is ambiguous. The equation of state depends upon the shape and orientation of the cells with respect to the lattice positions of the molecules. A particular choice is considered which leads to an equation of state that at low densities is accurate through the second virial coefficient. There are other shapes and orientations of the cells which would lead to other equations of state having this same property.