ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT velocity a t 3 megacycles per second has been measured a t various mole fractions in 16 binaries of varying degrees of nonideality in studies conducted in this laboratory. Earlier experimental studies of this kind have been reviewed by Wada (6). The free volume theory is useful for interpreting and correlating sonic velocity data for liquid solutions. The application of the simple free volume theory to solutions has been well presented by Prigogine and Bellemans (4). For the consideration of sonic velocity data, the theory is extended as follo~s. The internal pressure, Pi,is given by the free volume equation of state
James (S), and Furth (I) have been used for solids in recent years. They are described and compared with data of Bridgman and with computations made by the Fermi-Thomas method. These equations are used with the hydrodynamic theory to obtain expressions for the pressure, temperature, and entropy increase in various metals as a result of shock wave passage through the solids. For the Pack-Evans-James equation of state, which appears to be adequate for interpolating between the Bridgman data and the Fermi-Thomas computations, the difference between Hugoniot and adiabatic pressure to fourth order in the compreasion is
(3) where V and V Onow refer to the solution molar volume and incompressible molar volume, respectively. The ratio ( V0/V)1/3 is evaluated by means of Equation 1. By the use of the partial molar concept
v~,
where N A and N B are the mole fractions and VA, and V ~ are B the partial molar volumes and incompressible molar volumes of the components A and B, respectively. The two right-hand terms in Equation 4 may be individually determined by the wellknown slope-intercept method from a plot of Pi from Equation 3 versus mole fraction. The values of v&41’3 and V O B ~which /~ are thus obtained vary slightly with mole fraction in the direction predicted by Prigogine and Bellemans (4). The excess internal pressure for the nonideal solutions does not correlate directly with the heat of mixing. The relation between the behavior of the excess internal pressure and the heat of mixing may be useful in future .studies of the molecular internal degrees of freedom in solution, as implicitly suggested by Bondi (11.
Literaiure Cited (1) Bondi, A., J . Phys. Chem., 58, 929 (1954). (2) Collins, F. C., and Navidi, M. H., J . Chem. Phys., 22, 1254 (1954). (3) Collins, F. C., and Raffel, H., Ibid., 22, 1728 (1954). ngogine, I., and Bellemans, A., Trans. Faraday Soc., 49, 80 (4) (1953). (5) Wada, Y . , J . Phys. SOC.J a p a n , 4, 280 (1949). Supported in part by Wright Air Development Center of U. 9. Air Force under Contract No. AF33 (616b373.
ENTROPIC EQUATIONS OF STATE AND THEIR APPLICATION TO SHOCK WAVE PHENOMENA GEORGE E. DUVALL AND BRUNO J. ZWOLlNSKl Poulfer laborafories, Stanford Research Institute, Stanford, Calif.
T
H E propagation of a shock wave through an inviscid fluid is described by a set of jump conditions, representing conservation of mass, momentum, and energy, which are the same for all mediums, and by an equation of state and the specific heat characteristic of the particular medium. If P P / d V z is positive, only compressive shocks can exist and the entropy increases discontinuously in the shock transition The hydrodynamic theory of shock waves in solids is based on the assumption that the shear modulus is completely negligible a t very high pressures and that the transition from the undisturbed to the shocked state occurs discontinuously a t the shock front. The region behind the shock front is then treated as an ideal fluid with a hydrostatic equation of state. Three equations of state that bear the names of Murnaghan ( 2 ) , Pack-Evans1182
The increase in entropy acrose the shock front to the same approximation is
where a, p are parameters of the Pack-Evans-James equation, VO = l / p o is specific volume a t room temperature, To, and zero pressure, C, is specific heat, g( Po)= ( ~ P / ~ T and ) T ~p ,is density behind the shock front.
Liieraiure Cited (1) Furth, R., Proc. Roy. Boc. (London), A183, 87 (1944). (2) Murnaghan, F. D., “Finite Deformation of an Elastic Solid,” Wiley, New York, 1951. (3) Pack, D. C., Evans, W. M., and James, H. J., Proc. Phys. Soc., 60, 1 (1948).
A N INSTRUMENT TO STUDY RELAXATION RATES BEHIND SHOCK WAVES E. L. RESLER, JR., AND M. SCHEIBE lnsfitute for Fluid Dynamics and Applied Mathemafics, University of Maryland College Park, M d .
A
N INSTRUMENT is described which combines the schlieren
technique, a photomultiplier tube, and an oscilloscope to measure the density distribution behind shock waves (in gases) produced in a shock tube. The use of shock waves and of optical techniques to study the chemical kinetics of reactions has been reported (1, 2 ) . This instrument however, is simpler in many respects than those previously used and is capable of better space resolution and higher sensitivity while still recording the same information. Basically the instrument functions in the following manner. A photomultiplier tube intercepts all the light clearing the knife edge of a schlieren system, the light beam of which passes through a section of the shock tube. A property of the schlieren system is that light will escape the knife edge in an amount proportional to the density derivative a t each point in the field of the light beam. Since the photomultiplier tube gathers all the light passing the knife edge it integrates the density derivative over the light beam or puts out a signal proportional to the density difference a t the edges of the light beam. Thus the system has the sensitivity of a schlieren and photomultiplier combination and the space resolution is determined not by the beam width but by how well the edges of the light beam are defined or to what degree the light in the beam is parallel. Therefore, if in the shock tube a shock wave with a relaxation zone behind i t characterized by a density distribution passes through the light beam of the schlieren system and if the length of the relaxation zone behind the shock wave is less than the width of the schlieren light beam, then the recorded signal from the photomultiplier tube will be the density
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47,No. 6
