26 RADIATION BIOLOGY 



must follow a similar cycle, and with it the Boltzmann distribution of 

 energies among the various degrees of freedom. At high supersonic fre- 

 quencies, vibrational-translational interaction will be inadequate to estab- 

 lish thermal equilibrium with internal degrees of freedom, and the heat- 

 capacity (determined from velocity v) measurements by means of the 

 well-known formula 



(where Cp and Cv = heat capacity at constant pressure and volume, 

 respectively; P = pressure; and p = density) will not contain the con- 

 tribution from internal modes of motion. In the neighborhood of some 

 critical frequency dependent on the material, the physical state, and the 

 temperature, energy will be restored to external degrees of freedom just 

 180° out of phase with the compression wave. As a result, the wave 

 will be damped and retarded, thus producing the phenomena of absorp- 

 tion and dispersion of sound. Sound dispersion has been thoroughly 

 treated both experimentally and theoretically (reviewed by Richards, 

 1939). Some important theoretical treatments are due to Kneser (1931, 

 1933), Herzfeld and Rice (1928), Saxton (1938), and their various 

 coworkers. The energies involved are frequently small with respect to 

 those of the reaction potential barrier, so that chemical effects may not 

 be important. That is, the collision paths generally lie about the lower 

 equipotential lines, which are seldom distorted. It is to be expected that 

 the persistence of vibrational energy will consequently be considerable. 

 Nevertheless, as Eucken and Becker (1934) and Eucken (1935) pointed 

 out for sound dispersion, chemical affinity is the most important con- 

 sideration in these processes. Table 1-1 illustrates the importance of 

 affinity, which can probably best be explained as due to a general lowering 

 of the surfaces rather than to an increase in distortion. Like molecules 

 have considerable affinity for each other, but it is clear that trace sub- 

 stances with high dipole moments or high reactivity, such as water and 

 carbon dioxide, for instance, are frequently even more effective in pre- 

 venting sound dispersion. 



The number of collisions required for equilibrium can be most readily 

 calculated after the manner of Bethe and Teller (1940). Diatomic mole- 

 cules treated as harmonic oscillators are considered, since multiply vibrat- 

 ing molecules add little save complexity. The circular acoustic frequency 

 of maximum absorption of sound per wave length comax can be determined 

 from experiment. It is related to the relaxation time l/coo for the equi- 

 libration of internal degrees of freedom by the expression 



^0 ^ f Cp[Cp — R] \ (1-15) 



Wmax \Cp[Cp — R]/ 



