102 THE DETONATION PROCESS 



ternal energy of the products. The total energy change is of course 

 independent of the path by which the transition is assumed to take 

 place. It is convenient in calculation to evaluate the heat of reaction 

 at the initial temperature and pressure (essentially zero) , heat the prod- 

 ucts to the final temperature at constant volume which can be taken to 

 be infinite and compress the products at the final temperature to the 

 final volume. The expression of this process for M grams of explosive 

 is 



(3.22) J:N, Hi - Ho + RToZNi = C, • (T - T ) + ( a|)/^ 



In this equation, Ni is the number of moles of product species i having a 

 heat of formation Hi per mole, Ho is the heat of formation of M grams 

 of the explosive, Cv is the total mean heat capacity of all products over 

 the temperature range and M/p is the final volume. From the Wilson- 

 Kistiakowsky equation 



E{V, T) = Eo + NCriT - 300) + NRTaxe^' 

 and so 





^\ dV = NRTaxe^'' 



dV T 



where a: is a function of temperature and the final composition. 



Knowing the specific heats and heats of formation, Eq. (3.22) gives 

 an implicit solution for the final temperature which for a given com- 

 position can be solved by successive approximations. Use of the con- 

 ditions l^E = for the adiabatic conversion and A*S = for the sub- 

 sequent isentropic expansion, together with the thermochemical data, 

 thus permits evaluation of the quantities necessary for approximate 

 propagation theories. Numerical methods and auxiliary tables for 

 computation arc given in a report by Kirkwood, Brinkle}^. and Richard- 

 son (59, V). 



3.8. Boundary Conditions and Initial Motion of the Gas Sphere 



The propagation theory of Kirkwood and Bethe for the underwater 

 shock wave reduces the determination of shock wave parameters to an 

 evaluation of the "kinetic enthalpy" on the surface of the gas sphere at 

 a variable retarded time r after detonation is complete. As already dis- 

 cussed in section 3.5, Kirkwood and Bethe neglect the head of the 

 detonation wave and use an initial condition of adiabatic explosion at 



