J . Chem. SOC., Faraday Trans. 1, 1982, 78, 1039-1042 Theoretical Evaluation of the Velocity of Sound in Liquid Mercury at Elevated Pressures BY BRIJ R. CHATURVEDI, RAMESHWAR P. PANDEY AND JATA D. PANDEY* Department of Chemistry, University of Allahabad, Allahabad-2 1 1002, India Received 19th March, 1981 An attempt has been made to evaluate the velocity of sound in liquid mercury over a wide range of temperatures and pressures in the light of Flory ’s statistical theory. The agreement between theoretical and experimental values is satisfactory. The theoretical estimation of sound velocities in molecular liquids and binary liquid mixtures from Flory ’s statistical in conjunction with Auerbach’s relati~n,~ has been discussed recently by Pandey5p and Mishra.’? Pandey6 and Mishra8 have also extended Flory’s theory to enable the velocity of sound to be evaluated in pure molecular liquids at elevated pressures.Recently Pandey et aL9 have also successfully calculated the surface tension of molten salts using the Flory theory. It appears from the literature that no attempt has so far been made to examine the applicability of Flory’s theory to liquid metals. In the present paper such an attempt is made. The theoretical evaluation of sound velocities in the case of liquid metals is rare in the literature. Although KonyuchenkolO calculated sound velocities for a variety of molten metals and compared the results with experiment, he did not utilize Flory’s theory for this purpose. We have computed here the velocity of sound in liquid mercury over a wide range of temperatures and pressures in the light of Flory ’s theory. THEORETICAL According to Auerbach4 the velocity of sound, U, is expressed by the relation where c7 and p are the surface tension and density, respectively. According to Flory’s statistical theory the surface tension is expressed as c7 = c7* 6(V) (4 where c7* and 6( V ) are the characteristic surface tension and reduced surface tension, respectively.Patterson and Rastogi’ll in their extension of the corresponding states’ theory to the case of surface tension,l29l3 obtained the following relations for characteristic and reduced surface tensions : o* = ki p*$T*f (3) 10391040 VELOCITY OF SOUND I N LIQUID MERCURY Here k is Boltzmann's constant and M is the fractional reduction in the number of neighbours of a cell owing to migration from the bulk phase to the surface phase.Different values have been given for M.l1* 14-16 Mishra', and Pandey5? have suggested some modification to its previous values11v14-16 in order to obtain better results at elevated pressures. Recently Pandey et al. used an alternative value in the case of molten salts. This value9 differs from those used for molecular liquids and liquid mixtures. In the present case we have used 0.69 for M. Other symbols in the eqn (3) and (4) have their usual meanings. These equations have been derived in the light of the reduced equation of as described by Flory using the following relati~nsl-~v 6 v where a and BT are the thermal expansion coefficient and isothermal compressibility, respectively .17501 1700 1400- 1700 - I I " I ' ' I I I ' FIG. 2 4 6 8 10 12 13 0 p/108 Pa p , for various temperatures: (a) 294.9, (b) 313.5, (c) 325.9 K. 1.-Theoretical (0) and experimental (0) values of the velocity of sound as a function of pressure,B. R. CHATURVEDI, R. P. PANDEY A N D J. D . PANDEY 1041 TABLE 1 .-REDUCED VOLUME, CHARACTERISTIC PRESSURE, AND CALCULATED AND EXPERIMENTAL SOUND VELOCITIES IN LIQUID MERCURY Ucak Uexpt T/K p/108 Pa v p*/107 Pa /m s-' /m s-' A (%) 294.9 1 x 10-3 1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 2 3 4 5 6 7 8 9 10 1 1 12 13 313.5 1 x 10-3 325.9 1 x 10-3 average % error 1.0516 1.0503 1.0492 1.048 1 1.047 1 1.0462 1.0452 1.0444 1.0435 1.0429 1.042 1 1.0416 1.0407 1.0402 1.0546 1.0533 1.0520 1.0509 1.0498 1.0488 1.0478 1.0469 1.0462 1.0453 1.0441 1.0438 1.0429 1.0423 1.0566 1.0552 1.0539 1.0527 1.0516 1.0505 1.0495 1.0486 1.0476 1.0467 1.0461 1.0452 1.0446 1.0439 - 146.7 148.0 149.2 150.3 151.3 152.5 153.2 154.2 155.0 156.7 156.9 158.0 158.3 159.5 153.2 154.6 155.7 157.1 158.3 159.3 160.4 161.4 163.0 163.6 163.7 165.1 165.2 166.7 157.5 159.0 160.3 161.6 162.7 163.9 164.9 166.1 166.7 167.2 169.1 169.1 170.5 172.0 - 1465 1473 1489 1500 1510 1521 1530 1540 1548 1560 1565 1575 1581 1590 1491 1504 1516 1521 1540 1550 1560 1570 1581 1589 1595 1604 161 1 1622 1508 1521 1534 1547 1558 1569 1579 1589 1598 1606 1617 1623 1632 1642 - 1450 1472 1493 1512 1531 1550 1567 1585 1601 1618 1633 1649 1664 1679 1442 1464 1485 1505 1524 1543 1561 1578 1595 1612 1628 1643 1659 1674 1436 1454 1465 1500 1520 1538 1556 1574 1591 1608 1624 1639 1655 1670 - - 1.0 - 0.4 0.2 0.8 1.4 1.9 2.4 2.8 3.3 3.6 4.2 4.5 5.0 5.3 - 3.5 - 2.8 -2.1 - 1.0 -1.0 - 0.5 0.1 0.5 0.9 1.4 2.0 2.4 2.9 3.1 - 5.0 - 4.6 - 4.8 -3.1 - 2.5 - 2.0 - 1.5 - 1.0 - 0.4 0.1 0.4 1 .o 1.4 1.7 2.11042 VELOCITY OF SOUND I N LIQUID MERCURY RESULTS AND DISCUSSION The values of reduced volume, v, and characteristic pressure, p*, as obtained through eqn (5) and (6), respectively, in the present case of liquid mercury over a wide range of temperatures and pressures are enlisted in table 1. The essential data required for the calculations have been taken from the 1iterat~re.l~ The values of reduced surface tension and charscteristic surface tension have been obtained with the help of these paraFeters, viz.V , p* and T*, vide eqn (3) and (2), respectively. These values of a* and 6 ( V ) have been utilized to obtain the values of surface tension which are in turn employed to predict the sound velocity through eqn (1). Although eqn (1) is empirical in nature its validity is well justified,le as it gives a maximum error of only & 4 % for theoretical sound-velocity results if all the experimental data are precise. Since in the present case the experimental surface tension data for liquid mercury at elevated pressures are not available in the literature, only a comparison of the experimental sound velocity with those predicted theoretically is possible. The predicted values of the sound velocities so obtained, along with the experimental values,17 are recorded in table 1.Both the theoretical and experimental values of the sound velocity are also represented graphically in fig. 1 as a function of pressure at all three temperatures. An inspection of the last three columns of table 1 and of fig. 1 reveals that the percentage deviation between theoretical and experimental values lies between 0.4 and 5.3, with an average of 2.1. Thus the agreement between experimental and theoretical results appears to be satisfactory. The results of the present calculation show that the rate of change of sound velocity with temperature is positive, whereas experimentally it is negative. Our main aim is to show the general applicability of the theory in a qualitative way. It appears that theory cannot explain the reverse trend of sound velocity with temperature, although the pressure variation of the sound velocity predicted theoretically is similar to those observed experimentally.All the reduced and characteristic parameters have been computed from the reduced equation of state as given by Flory,2y which is assumed to hold universally but which avoids the unrealistic parameterization of the intermolecular energy as stipulated by the theorem of corresponding states. One may therefore conclude that Flory’s theory can also be applied in the case of liquid metals over a wide range of temperatures and pressures. B. R. C. and R. P. P. are grateful to the Indian Council of Scientific and Industrial Research, New Delhi, for financial assistance. P. J. Flory, R. A. Orwall and A.Vrij, J. Am. Chem. SOC., 1964, 86, 3507. P. J. Flory, J. Am. Chem. SOC., 1955, 87, 1833. A. Abe and P. J. Flory, J . Am. Chem. SOC., 1965, 87, 1838. N. Auerbach, Experientia, 1948, 4, 473. J. D. Pandey, J. Chem. SOC., Faraday Trans. I , 1980, 76, 1215. J. D. Pandey, J . Chem. SOC., Faraday Trans. I , 1979, 75, 2160. R. L. Mishra, Acoustics Lett., 1979, 3, 1. R. L. Mishra, J. Chem. Phys., 1980, 73, 5301. J. D. Pandey, B. R. Chaturvedi and R. P. Pandey, J. Phys. Chem., 1981,85, 1750. lo G. V. Konyuchenko, Zzv. Vyssh. Uchelon. Zaved., Fiz., 1972, 15, 145. l1 D. Patterson and A. K. Rastogi, J. Phys. Chem., 1970, 74, 1067. l2 I. Prigogine and L. Saraga, J. Chim. Phys., 1952,49, 399. l3 R. Defay, I. Prigogine, A. Bellemans and D. H. Everett, Surface Tension and Adsorption (Longmans, l4 M. D. Croucher and M. L. Hair, J. Phys. Chem., 1977, 81, 163. Is D. V. S. Jain, S. Singh and R. K. Wadi, Trans. Faraday SOC., 1974, 70, 961. l6 R. L. Mishra and J. D. Pandey, Chem. Scr., 1977, 11, 117. l7 L. A. Davis and R. B. Gordon, J. Chem. Phys., 1967,46, 2650. London, 1966), chap. XI. R. L. Mishra, D. Phil. Thesis (University of Allahabad, 1977). (PAPER 1/447)