27Ji. MOTION OF THE GAS SPHERE 



outside the boundary, and the second term is the work done against 

 hydrostatic pressure. 



If the products of explosion behave as ideal gases with a constant 

 ratio of specific heats 7 and are further assumed to undergo adiabatic 

 changes, the pressure-volume relation is P{V/Wy = k, where W is the 

 mass of explosive products in grams and A; is a constant. The internal 

 energy E{a) is then given by 



J ;.(„) T - 1 7-1 \V(a)/ 



From the last expression, it is evident that E{a) decreases rapidly 

 with increasing volume (proportional to a^), and at sufficiently ex- 

 panded stages of the motion represents a negligible fraction of the 

 initial energy of the products. The values of radius a and correspond- 

 ing pressure Pa for which E(a) has a given value can be estimated from 

 a knowledge of the adiabatic law for the products and the initial energy. 

 For TNT, the calculations of Jones (described in section 3.5) give the 

 adiabatic relation, valid for P< 4,500 Ib./in.^ 



©'■ 



25 



= 7.8 



where P is in kilobars, W is in grams, and V is in cm.^ The total energy 

 released by 1 gram of TNT is about 1,060 cal./gm. (= 4.44 X 10^° 

 ergs./gm.) of which approximately half is emitted in the shock w^ave 

 (see section 4.8 for a detailed discussion). 



The fraction F of the remaining energy Y which is present as in- 

 ternal energy at any state of expansion is 



F = ^ = 0.166 P,;/^ = 0.42 



©'■■ 



where P is in Ib./in.^, W is in lb., a in ft., and Y is taken to be 440 cal./ 

 gm. This fraction is less than 25 per cent for P < 7.6 lb. /in.-. For a 

 300 pound charge the corresponding radius is 13.5 feet, rather less than 

 the maximum radius of about 20 feet for a charge detonated 50 feet 

 below the surface. The curve of bubble expansion has the same 

 qualitative manner of variation with time as shown in Fig. 8.1, which 

 leads to the estimate that the internal energy of the products is less than 

 25 per cent of the total for more than 70 per cent of the entire cycle. 

 The fact that the internal energy is relatively unimi)()rtant over 



