# Chemistry of Detonations - Kamlet M.J.

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PVJRT=1+X+0.625X2+0.2869X*+0.1928X4 (1)

5 For detailed references and a brief general discussion of other equations of state used in condensed-phase detonation problems, see S. J. Jacobs, ARS (Am. Rocket Soc. J.) 30, 151 (1960).

* J. O. Hirschfelder and W. E. Roseveare, J. Pbys. Chem. 43, 15 (1939).

4 J O. Hirschfelder, F. T. McClure, and C. F. Curtiss, “Thermochemistry and Equation of State of Propellant Gases,’’ OSRD Rept. 547 (1942).

1 J. Taylor, Detonation in Condensed Explosives (Oxford University Press, London, 1952).

with

x-t/r, b='?x?i, (2)

where bi is the molar covolume of the ith species, ë,- is the mole fraction of the ith species, Vt is the molar volume of the gas mixture, and P, T, and R are the pressure, temperature, and gas constant.

The bi are derived from the collision radii of the molecular species at high temperature and, as in the kinetic theory of gases at moderate pressure, are equal to four times the molecular volume multiplied by Avogadro’s number. Despite the use of diminished covolumes in the equation and despite the apparent theoretical basis of the model, the equation is oversimplified and the results of detonation calculations quite clearly show it to be inaccurate.

A quite different approach to the detonation product state has been to treat it as solidlike. Jones and Miller® performed equilibrium calculations on TNT with this idea in mind. They used an equation in which the volume was a virial expansion in the pressure. Other solidlike equations are cited in Ref. 2, but these have mostly been used for computing the state parameters with an assumed product state. The modified Kistia-kowsky-Wilson equation of interest to us here appears to be one of several possible compromises between the hard-sphere molecule approach and the solid state approach.

This approach, instigated by Kistiakowsky, Wilson, and Halford,7 may be said to have its roots in Eq. (1).. These authors modified an equation due to Becker

P=XT{l+Xt?)/V+f{V)t X=b/V (3)

by dropping the f(V) dependence, adding an adjustable constant /3 and making h a function of temperature. This “variable ñî volume” equation, as further modified by Fickett and Cowan,8 became

PVJRT^I+X#*, (4a)

X = KY,Xiki/VB{T+6)*. (4b)

Equation (4) is a variable covolume departure from the hard-sphere-molecule Eq. (1), for if /3=0.625, ê= 1, and a=0 the K-W equation would be identical with the Boltzmann equation to the third virial term, and the ki’s would be just the biS of the hard-sphere-molecule model. If /3 were 0.62S, with a about 0.2S to

0.5, one might consider the K-W equation to be a "soft-sphere” equation of state. In applying Eq. (4) to the calculation of detonation velocities it was quickly found, however, that jS could not be as large as 0.625,

e H. Jones and A. R. Miller, Proc. Roy. Soc. (London) A194, 480 (1948). ‘

1 G. B. Kistiakowsky and E. B. Wilson, Jr., “The Hydrodynamic Theory of Detonation and Shock Waves,” OSRD Rept. 114, 1941.

' R. D. Cowan and W. Fickett, J. Chem. Phye. 24, 932 (1956).

and in the earlier papers 0Ï this problem7,9>l° the values adopted were ê =1.0, /3=0.3, a=0.25, and 6=0.

With these parameters and the D~pî data for a number of explosives, kis were determined for the principal molecular species expected as detonation products from C-H-N-0 explosives. The values obtained came fairly close to agreeing with the biS of Eq. (1) if one defined bi as

(Ã-+4)" (5)

and assumed 7^4000° as typical of detonation temperatures found. Coihputations were made by estimating fixed detonation-product compositions as well as by the use of equilibrium calculations. The equilibrium calculations of Brinkley and Wilson9 tended, at that time, to favor an ÍãÎ-ÑÎ-ÑÎ* “arbitrary” method of estimating detonation-product compositions. Consequently, Snay and Christian15 tried both H1O-CO-CO2 and CO-HaO-COj arbitrary decomposition schemes to test the effects of changing composition on predicted detonation properties and covolume factors. By least squares they determined a best set of covolume factors for the above parameters. The results were not very much influenced by the decomposition assumption, but computed detonation pressures were lower than values found experimentally.

The next step toward better fitting of the K-W equation to detonation data was made by Cowan and Fickett,8 who established a substantially different set of parameters and covolume factors. More recent ad* justments by Mader11 have led to the parameter sets used most frequently today in the ruby code. Mader’s parameters were designed to give the best match with five experimental measurements considered to be highly accurate: the detonation pressure of RDX at 1.8 g/cc, the detonation velocities of RDX at 1.0 and 1.8 g/cc, and the detonation velocities of TNT at 1.0 and 1.64 g/cc. Fundamental difficulties in finding a single set of parameters to accommodate these five measurements led Mader to suggest dual sets of /3 and ê: an "RDX parameter set” to be used with compounds producing lesser amounts of solid carbon in the detonation, a “TNT parameter set” with explosives producing greater ^mounts of solid carbon (Table I).

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