92 THE DETONATION PROCESS 



rate with initial density, are used to determine the constants of equation 

 of state in the virial form 



^ = RT -\-bP + cP' + dP' 



N 



where N is the total number of moles of gaseous products per mole of 

 TNT, and, unless the composition is "frozen," is a function of P and T. 

 In an adiabatic expansion, the work done by the gases must equal 

 the sum of the energy liberated by the reaction and the loss of internal 

 energy, giving the relation 



(3.15) PdV = d{H - E) 



where // is the heat of formation of the products and E their total in- 

 ternal energy, both of which are functions of P and T. In order to 

 make further progress it is evidently necessary to introduce conditions 

 on the composition of the products. As the pressure and temperature 

 fall during expansion, the equilibrium composition will change but at 

 the same time the rates of reaction will decrease. The decreased rates 

 make it reasonable to suppose that equilibrium is not attained in later 

 stages of the expansion. Calculations based on the equilibrium as- 

 sumed by Jones for TNT (see, section 3.2) predict that at first the 

 amount of CO increases rapidly at the expense of CO2 and solid C, 

 reaching a maximum at about 1,800° K. At this point, the reaction 

 rates are enormously less than at 3,000° K., and Jones completes his 

 adiabatic calculation with the assumption of this fixed composition for 

 further expansion. As already mentioned, there is reason to question 

 the equilibrium results of Jones, and the final adiabatic relation may be 

 somewhat in error as a result. We shall see later, however, that the 

 exact form of this relation is not of paramount importance for most of 

 the calculations of pressures developed in underwater explosions. 



In order to integrate the differential eciuation (3.15) for the adi- 

 abatic expansion, a rather complicated set of equations is developed in 

 terms of the activity and composition variables. With the initial con- 

 ditions of pressure and composition assigned, the initial values of activ- 

 ity and temperature fix a starting point for numerical step by step 

 integration. The final P-V relation resulting from these calculations 

 is plotted in Fig. 3.2. On the logarithmic plot a perfect gas condition 

 would correspond to a straight line. It is seen that at the higher densi- 

 ties the pressures found are much higher, corresponding to the increased 

 effect of repulsive forces on the internal energy. The decrease in slope 

 at the smallest volumes is attributed to the smaller number of moles 

 (decreased concentration of CO) at the higher pressures. 



