Hirschfelder equation

Other forms of Hirschfelder equation are given in Ref 9d and Addnl Ref G... 

The reader is reminded that no experimental data are available in regions II and III and that the estimations of accuracy given for these two regions are based on the reliability of the values taken from [ ] and the Hirschfelder equation [ ]. 

Calculation of binary diffusion coefficients based on Eqs. (3.1.69),(3.1.71),(3.1.72) and (3.1.79) is limited because estimations of the collision cross-section of the molecules cr (and of the influence of the temperature on kinetic theory of gases. The so-called Hirschfelder equation is frequently presented in many textbooks and used in the literature, but values of parameters such as the collision diameters of the molecules and characteristic energies are needed. Instead, many authors have developed empirical relations. For non-polar gas pairs, D B.g is in good approximation (deviation <10%) given by the equation of Slattery and Bird (1958) ... 

Pack R T and Hirschfelder J O 1968 Separation of rotational coordinates from the W-electron diatomic Schrddinger equation J. Chem. Phys. 49 4009... 

The diffusion coefficient can be determined from the Hirschfelder-Bird-Spotz equation as follows ... 

The error in Runge-Kutta calculations depends on h, the step size. In systems of differential equations that are said to be stiff, the value of h must be quite small to attain acceptable accuracy. This slows the calculation intolerably. Stiffness in a set of differential equations arises in general when the time constants vary widely in magnitude for different steps. The complications of stiffness for problems in chemical kinetics were first recognized by Curtiss and Hirschfelder.27... 

The AIChE design manual recommends the Wilke and Chang (1955) equation for liquid diffusivities and the Wilke and Lee (1955) modification to the Hirschfelder, Bird and Spotz equation for gas diffusivities. 

At moderate pressures, the virial equation of state, truncated after the second virial coefficient, can be used to describe the vapor phase. As suggested by Hirschfelder, et. al. (1 3) the temperature dependence of the virial coefficients is expressed... 

In developing the equations governing the thermal and diffusional processes, Hirschfelder obtained a set of complicated nonlinear equations that could be solved only by numerical methods. In order to solve the set of equations, Hirschfelder had to postulate some heat sink for a boundary condition on the cold side. The need for this sink was dictated by the use of the Arrhenius expressions for the reaction rate. The complexity is that the Arrhenius expression requires a finite reaction rate even at x = —°°, where the temperature is that of the unbumed gas. 

In order to simplify the Hirschfelder solution, Friedman and Burke [8] modified the Arrhenius reaction rate equation so the rate was zero at T = T0, but their simplification also required numerical calculations. 

Then it became apparent that certain physical principles could be used to simplify the complete equations so they could be solved relatively easily. Such a simplification was first carried out by von Karman and Penner [9], Their approach was considered one of the more significant advances in laminar flame propagation, but it could not have been developed and verified if it were not for the extensive work of Hirschfelder and his collaborators. The major simplification that von Karman and Penner introduced is the fact that the eigenvalue solution of the equations is the same for all ignition temperatures, whether it be near T or not. More recently, asymptotic analyses have been developed that provide formulas with greater accuracy and further clarification of the wave structure. These developments are described in detail in three books [10-12],... 

A more detailed account of the Schrodinger equation can be found in physical chemistry textbooks such as those of Hinshelwood (1951) and Atkins (1978), or in more specialized texts such as that of Hirschfelder et al. (1954). An excellent review of the applications of quantum mechanics to geochemistry has recently been proposed by Tossell and Vaughan (1992). 

Boltzmann Equation of State and Its Modification by Hirschfelder Rosevere. See under Detonation (and Explosion), Equations of State... 

From the principle of corresponding states, Hirschfelder et al (Ref 4, p 236), derived another equation ... 

In conclusion, it may be said that van der Waals equations can only provide reasonably accurate representation over limited ranges of variation of the pressure and temperature. For this reason many attempts were made to produce a more satisfactory equation by modifying vanderWaals equations. Such modifications were made by Berthelot (See item b ), Callendar (See item c-2), Clausius (See item C3), Dieterici (See item d2), Hirschfelder et al (See item 113), Keyes (See item ki), Lees (See item I3), and Macleod (See item mi) Accdg to Dunkle (Ref 17), for high temps and moderate pressures which give rise to large values of V so that P55>a/v2, van der Waals equation reduces to Abel s equation (See item ai)... 

This equation, modified by Hirschfelder Roseveare, was found to be suitable for moderatelyriiigh pressures. It is given in Ref 8e, p 262 in the following form ... 

In order to understand the reason for calling LJD equation as the one "based on intermolecular forces,t there is included a brief explanation based on the discussion given in the book of Hirschfelder, Curtis ... 

Since Lennard-Jones (6-12) potential has been widely used for calcn of properties of matter in the gaseous, liquid, and solid states, Hirschfelder et al (Ref 8e, pp 162ff) discuss it in detail. They show that the parameters o and ( of the potential function may be determined by analysis of the second virial coefficient of the LJD equation of state... 

The Boltzmann equation (See item bg) is also of the third degree, but Hirschfelder Roseveare s modification of Boltzmann equation (See under item bg) is of the fourth degree... 

Pure shock waves) 4) G.B. Kistiakowsky, p 951 in Kirk Othmer 5 (1950), pp given in the text (Not included in the 2nd edition) 5) Corner, Ballistics (1950), 100-01 (Corner Noble-Abel equations of state) 6) SAC MS, Ballistics (1951), 18 (Covolume and equation of state of propint gases) 7) Taylor(1952), 34 (Boltzmann and Hirschfelder Roseveare equation of state for the expln products) 69-72 (Rankine-Hugoniot equation of state) 87-98 (Abel, Boltzmann and other equations of state applicable to deton of condensed expls yielding only gaseous products) 114 (Equations of state applicable to deton of condensed expls whose products contain a condensed phase)... 

Rankine-Hugoniot equations 181-87 (Equations of state which include among others the following Jones Miller, Lennard-Jones Devonshire, Halford-Kistiakowsky-Wilson, Joffe its modification by Su Chang, Taylor, Kihara Hikita, Travers, Cook, Kistiakowsky-Wilson-Brinkley and Polytropic equations) 194 (Landau-Stanyukovich and Hirschfelder et al equations of state 11) J.F. Roth, Explosiv-stoffe 1958, 50 (Abel sche Zustandsgleichung fur die Detonation) 12) Cook (1958), 37 (General equation of state) 62-3 [Halford-Kistiakowsky-W ilson-Brinkley equation of state, (listed as K-H-W-B equation of state)] ... 

Hydrodynamic theory of deton and shock waves) 29c) J.O. Hirschfelder et al, OSRD 547(1942) (Thermochemistry and the equation of state of the propellant gases) 30) J. von Neumann, OSRD 549(1942) (Theory of deton waves)... 

Equation (56) states that the effect of a thermal gradient on the material transport bears a reciprocal relationship to the effect of a composition gradient upon the thermal transport. Examples of Land L are the coefficient of thermal diffusion (S19) and the coefficient of the Dufour effect (D6). The Onsager reciprocity relationships (Dl, 01, 02) are based upon certain linear approximations that have a firm physical foundation only when close to equilibrium. For this reason it is possible that under circumstances in which unusually high potential gradients are encountered the coupling between mutually related effects may be somewhat more complicated than that indicated by Eq. (56). Hirschfelder (BIO, HI) discussed many aspects of these cross linkings of transport phenomena. 

Cross section and potential. Collision cross sections are related to the intermolecular potential by well-known classical and quantum expressions (Hirschfelder et al, 1965 Maitland et al, 1981). Based on Newton s equation of motion the classical theory derives the expression for the scattering angle,... 

The gas-phase diffusivity of sodium in helium Dvl may then be evaluated from the experimental data by using Equation 15. A value of 1.96 cm.2/sec. was obtained, which compares favorably with 2.11 cm.2/ sec. estimated from an equation given by Hirschfelder, Curtiss, and Bird (8), using Lennard-Jones parameters given by Chapman (5). The close agreement obtained here seems to justify the assumption of a stagnant gas layer through which both sodium and cesium diffuse. 

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