Velocity of sound isotherms in liquid krypton and xenon

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1 Velocity of sound isotherms in liquid krypton and xenon C. C. LM, D. H. BOWMAN, AND RONALD A. AZZ Department of Physics, University of Waterloo, Waterloo, Otztario Received May 2, 1968 The velocity of sound was measured with a precision of 0.1 % in liquid krypton and xenon at pressures between the vapor pressure and about 65 atrn, from near their triple points to near their critical points. A corresponding states treatment of these measurements and previous results in argon showed that, with a suitable choice of relative molecular parameters (u,~), the W*(P*,T*) surfaces were coincident to within the experimental error, except for argon near the critical temperature. The relative values of the effective atomic radii o obtained from this analysis were somewhat lower than those obtained from other thermodynamic properties. Canadian Journal of Chemistry, 46, 3477 (1968) ntroduction Previous measurements (1,2) of the velocity of sound under saturated vapor pressure in argon, 6W krypton, and xenon have indicated that the principle of corresponding states is obeyed extremely well in these substances. t is of interest 'O to examine whether such similarities exist with respect to the pressure dependence of sound velocity in the region above the vapor pressure curve. Experimental Method 1 2 ~ lrn ~ The velocity of sound was measured by the resonance FG. 2. Liquid krypton velocity of sound isobars technique of Guptill et at. (3, employing a cylindrical W(m/s) versus T("K)., barium titanate transducer. Measurements were made at frequencies in the range 0.5 to 2.1 MHz. The apparatus Scale by the accuracy of the standard resistor used for and method of measurement have been described elsewhere (1, 2, 4). The velocity of sound was measured at comparison. During measurements along any one isoseveral pressures along each of a series of isotherms for therm, the temperature was held constant to _ OK. both liquids. The temperature of the liquid was measured A Texas nstruments pressure gauge model 141 was by means of a platinum resistance thermometer calibrated used to measure the pressure. The uncertainty in pressure by the National Physical Laboratories. The accuracy of measurements was due to the gauge (accuracy atm) temperature measurements was limited to and the unknown hydrostatic pressure head (f0.04 atm f "K maximum). with respect to the nternational Practical Temperature The velocity of sound was measured to a precision of better than 0.1 % over the entire range of temperature and pressure. Uncertainties in temperature and pressure KRYPTON SOTHERMS TEMPERlTURE 700 C: " produced a comparatively small uncertainty in the : A : velocity of sound except along the steeper portions of the L O c C isotherms near the critical temperatures W 4CC a a PRESSURE (mml LMe0 EECh_ E e g 9 '.o 20 o, 0 FG. 1. Liquid krypton velocity of sound isotherms W (mls) versus P (atm). 0, data points; A, values interpolated from W. results in ref. (2). 1 Results Ninetythree velocity of sound measurements were made along 11 isotherms in liquid krypton. The results are presented in Table and plotted in Fig. 1, together with values obtained by interpolation from previous measurements (2) under saturated vapor pressure. The velocity isotherms appear similar in shape to those presented earlier for argon (4), being almost linear at the low temperatures, and steeper and more curved at the

2 CANADAN JOURNAL OF CHEMSTRY. VOL. 46, 1968 TABLE The velocity of sound in liquid krypton "K "K "K "K P (atm) W (m/s) P (atm) W (m/s) P(atm) W(m/s) P(atm) W(rn/s) "K "K "K "K P (atm) W (m/s) P (atm) W (m/s) P (atm) W (m/s) P(atm) W(rn/s) "K "K "K P (atm) W (m/s) P (atm) W (m/s) P (atm) W (m/s) higher temperatures. nterpolated values of the velocity of sound at 10,20, 30,40,50, and 60 atrn are plotted as functions of temperature in Fig. 2. 7oo These velocity isobars are also similar in shape to 600 those for argon. n xenon, 150 velocity measurements were made along 13 isotherms. The results are presented in Table 1 and plotted in Fig. 3, together w with values obtained by interpolation from pre.oovious results (2). The velocity ofsound isobars are plotted in Fig. 4. The curves are similar in shape 200 to those for krypton and argon. Polynomials of degree four or less were sufficient to describe the isotherms. The coefficients of the equations : :, 2 PENM( SOTHERMS msce,~m..~.r,c...>. L 2,....,,o, El M i 0 'O',..,:,,z. :,%,,a d= r " f _ 7," r o :.,m, N. fk 0 >*,a" wy: / PRESSURE (a,rn) 0 20 X FG. 3. Liquid xenon velocity of sound isotherms W(m/s) versus P(atm). 0,. data points; A, values W = A, + ALP + AzP2 + A3P3 t A4P4 interpolated from W, results m ref. (2). ",,,,

3 LM ET AL.: VELOCTY OF SOUND N KRYPTON AND XENON TABLE 1 The velocity of sound in liquid xenon O K "K "K OK P (atrn) W (m/s) P (atrn) W (rn/s) P (atm) W (rn/s) P (atm) W (m/s) "K "K O K "K P (atm) W (rn/s) P (atrn) W (rn/s) P (atm) W (rn/s) P (atrn) W (rn/s) OK "K P (a trn) W (rn/s) P (atrn) W (rn/s) P (atrn) OK W (m/s) P (atrn) "K W (rn/s) P (atrn) "K W (rn!s) are given for each isotherm of krypton and the region studied) with which to compare our xenon in Table 111, together with the standard data. Due to lack of density data in this region, error of estimate in m/s. No equation is given for we did not determine any other thermodynamic the highest temperature isotherm of each liquid quantities from the velocity data. because of the small pressure ranges obtainable. Apart from the data under saturated vapor pressure (1, 2), no velocity of sound measure Corresponding States Treatment ments could be found for krypton or xenon (in For reasons discussed in previous papers (1,2),

4 3480 CANADAN JOURNAL OF CHEMSTRY. VOL. 46, 1968 TABLE 111 Coefficients in the equation W = A. + A,P + A2P2 + A3P3 + A4P4, and standard error of estimate, 0, for krypton and xenon (W is in m/s and P is in atm) A z A 3 A 4 T ("K) Ao A, ( x (x (X ci Krypton Xenon , , FG. 4. Liquid xenon velocity of sound isobars W (mls) versus T ("K). one would expect the corresponding states principle to be valid to a good approximation in the heavier inert gas liquids: argon, krypton, and xenon. According to the quantum molecular formulation of this principle (5), the reduced velocity of sound W* is a universal function of the reduced pressure P'" the reduced temperature T*, and the de Boer quantum parameter A*. That is, W" WW*(P*,T:",A*) where W* = W/(N&/M)'12; P* = P/(&/03); T* = T/(&/k); and A * = h/ [o(rn~)'/~ 1. N is Avogadro's number, M is the molecular or atomic mass, k is Boltzmann's constant, h is Planck's constant, and E and o are the energy and length parameters in the interatomic potential function which, as a condition for the validity of the corresponding states principle, must be of the form Y = ~+(r/o), where + is a universal function. n a previous analysis, we compared the velocity of sound W, under saturated vapor pressure in liquid argon, krypton, and xenon. There, we chose as a reference, the value OK for &/k in argon, as found by Whalley and Schneider (6) from second virial coefficient data, assuming a LennardJones (612) interatomic potential. We then obtained coincidence of the W"(T") curves by adjusting &/k to the values 165.4, and 228.7, OK for krypton and xenon respectively. These values are close to those obtained from an analysis of vapor pressure data by Boato and Casanova (7). The data presented here allow us to extend the comparison of the three liquids over a region of the equation of state surface not studied to any great extent. n addition, the introduction of the

5 LM ET AL.: VELOCTY OF SOUND N KRYTON AND XENON TABLE V W* values for Ar, Kr, and Xe for various T* at constant P* W* (~/k (OK), ; (&/< (OK), ; (&/k(ok), ; T* o (A), 3.41) o(&, 3.613) 0 (A), 3.944) pressure as an independent variable allows us to compare, using the corresponding states principle, the length parameter o as well as the energy Darameter E. The procedure involved in comparing the present data can be visualized by considering a surface in a three dimensional rectangular coordinate system whose coordinates are pressure P, temperature T, and the velocity of sound W. The task is then to choose appropriate reduction factors for P, T, and W so that the reduced W"(P",T*) surfaces for the three liquids are coincident. t is noted here that the above procedure is a first approximation only, since it neglects possible differences due to quantum effects and manybody effects. When the reduction factors are expressed as above in terms of the molecular parameters E and o, then a single choice of relative E values determines the reduction of both T and W, assuming the molecular mass is known. f one assumes the relative E values given above are valid TABLE V Comparison of condensed state LennardJones (612) length parameter ratios Kr/Ar Xe/Ar Boato and Casanova (7) Horton and Leech (8) Levelt (9) 1.16 Leadbetter and Thomas (10) 1.17 This paper 1.06, 1.15, in this region, then one should need to adjust only the o ratios in order to obtain coincidence. f coincidence cannot be obtained over the whole surface in this manner, the reason may be an improper choice of effective relative E values, quantum effects, or actual lack of correspondence among the liquids. Using the above ~ / k values, we adjusted o ratios until the portions of the W'F(P':',T*) surfaces at low T* (T* = was chosen because it represented the lowest temperature isotherm in argon) were coincident. The adjusted o ratios so found were o(kr)/o(ar) = 1.06, and o(xe)/o(ar) = 1.15,. The coincidence was well within experimental error over the whole pressure range studied at this value of T*:. Our value for o(kr)/o(ar) = 1.06, compares favorably with those derived by Boato and Casanova (7) and Horton and Leech (8) (see Table V). n the case of o(xe)/o(ar), our value of 1.15, again agrees quite well with those derived by Boato and Casanova and by Horton and Leech. t also agrees quite well with the value derived by Levelt (9) (see Table V). However, it should be noted that Levelt's value was derived by requiring accurate correspondence of experimental compressibility data at high densities much in the same way as ours was derived by requiring correspondence of the velocity. Leadbetter and Thomas (10) have, on the basis of their density and surface tension data, suggested a slightly higher value for o(xe)/

6 3482 CANADAN JOURNAL OF CHEMSTRY. VOL. 46, 1968 o(ar). Choosing as a reference the value of 3.41 A for o(ar) as determined from second virial coefficient data by Whalley and Schneider (6), we therefore obtained 0 values of 3.41, 3.61,, and 3.94, A for argon, krypton, and xenon respectively. Using these values of 0, we determined W* for the other higher values of T* at two values of P* corresponding to the 30 and 60 atm regions. The values are given in Table V. As T* increases, the W* values for the three liquids deviate somewhat from each other. The differences between krypton and xenon are small, being less than 0.2% for all but the highest temperature. The W* values for argon, however, are lower by as much as 1.4% at T* = and 2.5% at T* = The uncertainty in W* due to experimental error in the velocity data is 0.1%. nterpolation of W along the isotherms and isobars may have resulted in as much as 0.2% error. For the temperatures considered in Table V, pressure measurement error could account for an uncertainty in W* of only 0.04%. Error in temperature, due mainly to possible inaccuracy of the nternational Temperature Scale used (see ref. 11 for a discussion of this) resulted in an uncertainty in Wand W* of 0.06% or less. Thus uncertainty in W.* could account for the relatively small differences between krypton and xenon in Table V, but not between these two and argon. The lower W* values for argon at higher T* are similar in sign and magnitude to those observed under saturated vapor pressure near the critical point (2). Quantum effects were suggested in ref. 2 as a possible reason for these differences. More recently, it has been suggested that the observed differences may be due in part to threebody forces1. The possible error in 0 due to uncertainty in the pressure and the W* values, and to the estimated uncertainty of 0.04 OK in the adjusted 'J. A. Barker. Private communication. elk values, is 0.01 A. Thus, from this analysis, the relative 0 values are 3.41, 3.61, , and 3.94, A for argon, krypton, and xenon respectively. These are smaller than those obtained by Boato and Casanova (7): 3.41, 3.66, and 3.97 A. The reason for this is not clear at present. No definite statement can be made regarding the direction of adjustment of the 0 ratios necessary to take into account quantum differences among the three liquids. t is possible that part of the discrepancy is due to quantum effects which were not considered here but were considered by Boato and Casanova. Theoretical calculations of W* as a function of T* and P" have been carried out by David and Hamann (12), using the cell model for a classical LennardJonesDevonshire liquid. Their values are much higher than those presented here. They do show, however, a similar decrease of W* with increasing T* at constant P*, and an increase in (a w*lap*),. with T*. Acknowledgment The research was supported by a grant from the National Research Council of Canada. 1. C. C. LM and R. A. Azrz. Can. J. Phys. 45, 1275 (1967). 2. R. A. Azrz, D. H. BOWMAN, and C. C. LM. Can. J. Chem. 45,2079 (1967). 3. E. W. GUPTLL, C. K. HOYT, and D. K. ROBNSON. Can. J. Phys. 33, 397 (1955). 4. D. H. BOWMAN, C. C. LM, and R. A. Azrz. Can. J. Chem. 46,1175 (1968). 5. J. DE BOER. Physica, 14,139 (1948). 6. E. WHALLEY and W. G. SCHNEDER. J. Chem. Phys. 23,1644 (1955). 7. G. BOATO and G. CASANOVA. Physica, 27, 571 (1961). 8. G. K. HORTON and J. W. LEECH. Proc. Phys. Soc. London, 82,816 (1963). 9. J. M. H. LEVELT. Physica, 26,361 (1960). 10. A. J. LEADBETER and H. E. THOMAS. Trans. Faraday Soc. 61, 10 (1965). 11. C. R. BARBER and A. HORSFORD. Metrologia, 1, 12 (1965). 12. H. G. DAVD and S. D. HAMANN. Australian J. Chem. 14,l (1961).

Refractive index and Lorentz-Lorenz function of xenon liquid and vapour

$Refractive index and Lorentz-Lorenz function of xenon liquid and vapour$ J. PHYS. B (PROC. PHYS. SOC.), 1968, SER. 2. VOL. 1. PRINTED IN GREAT BRITAIN Refractive index and LorentzLorenz function of xenon liquid and vapour D. H. GARSIDE, H. V. M0LGAARD and B. L. SMITH School