p- V-T Studies on Molten Alkali Nitrates Part 2.-Internal Energies and Equation of State BY JOHN E. BANNARD Department of Metallurgy and Materials Science, University of Nottingham, Nottingham NG7 2RD Received 20th December, 1976 p-V-T data for the molten alkali nitrates to a temperature of 800 K and a pressure of 1400 bar were used to derive internal pressures and specific heats, and to derive the parameters of the van der Waals equation. p-V-T measurements on liquids have been restricted to studies at relatively low temperatures.' However for a study of liquids of simple internal structure one must look to the molten salts, particularly to the molten alkali halides.2 The melting points of these salts are usually in excess of 750°C although much lower temperatures may be used in a study of the nitrates.Part 1 presented the p-Y-T data for the alkali nitrates in the form of thermal pressure coefficients, and used this data to derive densities and compressibilities. The isobaric expansivity, %[ = ;(g)J and the isothermal compressibility, are related to the isochoric thermal pressure coefficient, yv, by Furthermore these so-called mechanical coefficients may be related to a number of other properties, e.g. to the internal pressure, n, where and to the heat capacities, Cp and Cv because T Va; C,-C, = - BT The individual values of Cp and Cv can be resolved either by independent calori- metric means or in terms of the corresponding adiabatic compressibility, BS, which is defined by the relationship 163164 MOLTEN ALKALI NITRATES and which is normally determined from measurements of the velocity of sound in the liquid, W, i.e.( p is the density). Evidently, CPICV = PTlPS (3) and from experimental values of /IT and Ps together with eqn (2), values of C3 and Cv can be found. The acoustical method is precise and has been used in the evaluation of heat ~apacities.~ The determination of isothermal compressibilities is more difficult and probably less precise (especially at high temperatures and pressures). The Equation of State interrelates p-V-T data. There are a large number of empirical equations such as the Virial Equation of State, where, using modern computers, any number of virial coefficients may be used to cause the equation to fit experimental data. The approach taken here is to examine how closely the measured thermodynamic values fit equations of the van der Waals type. THE EQUATION OF STATE A comparison of the yv values obtained earlier can be made by plotting V,/y, against molar volume, V,.It is seen from fig. 1 that the mechanical coefficients of a large range of liquids are comparable. If these plots are considered to be straight, the slope is [l/y, x V,/(Vm- VO)], a type of free-volume function where Yo wiil be the intercept on the molar volume axis. This relationship is based on an application of the van der Waals equation of state, e.g. (P+n)(V-b) = BRT (4) where n is the internal pressure, b the excluded volume and B is a constant equal to 1 for a monatomic fluid and 2 for a polyatomic fluid. Recalling that the internal pressure may be written as (a,/P,)T-p, then eqn (4) may be modified to where the excluded volume term is considered as a definition of the free volume, VF = V-b.According to this simple relation yv should be a linear function of V and at l/y, = 0, V = b. That this is very nearly so is evident from fig. 1 which illustrates the relation for the five alkali nitrates. The precision of the results is not sufficient to establish the linearity of the relationship; indeed for all the liquids there seems to be a slight systematic curvature. The dashed-arrows indicate the values of solid volume, V,, for the five nitrates. (Virtually no measurements have been made of cc, or /IT for ionic solids, but yv for solid NaNO, has been estimated at = 100 bar K-I). The slopes of the lines show some scatter but a reasonably constant B value of 2-3 indicates that the p-V-T properties of these melts are approximately expressed by a van der Waals equation for a polyatomic f l ~ i d .~ Some exception would be expected for RbNO, because it is known that the solid undergoes a number of phase transitions to low density structures. As a consequence rubidium nitrate belongs to that rare group of solids which contract on fusion. But it is seen that none of the values of AVfUsi,, correspond to the value V - b, so that the relevance of the V, values in fig. 1 should not be overemphasised. The value of the intercept, b, together with the molar volume of the solid at 25°C and the molar volume of the liquid at the melting point are summarised in table 1.J .E . BANNARD 165 The approximate nature of the van der Waals equation can be illustrated by evaluating b from a rearrangement of eqn (3, i.e. and b values for the nitrates may be deduced from table 2. Their average values can be compared with the volume change on fusion tabulated in table 1, and the differences illustrate the difficulty of unambiguously defining Fg. b = V-BR/yy, Vm/cm3 FIG. l.-Vm/yv against molar volume for various liquids. TABLE 1 .-FREE-VOLUME FOR THE ALKALI NITRATES m.pt. Vo = b V V S AVtus /K /cm3 mol-1 /cmJ mol-1 /cmJ mol-1 /cm3 mol-1 B LiN03 525 29 38.7 29.0 6.84 2.1 NaN0, 580 32 44.6 37.6 4.32 3.1 KNO, 611 45 54.0 48 .O 1.73 2.2 cSNO3 679 53 69.2 52.9 7.48 2.9 RbN03 581 45 59.3 47.5 -0.14 3.2 TABLE 2.-PARAMETERS OF THE VAN DER WAALS EQUATION T/K LiN03 600 700 NaNO, 600 700 KN03 600 700 RbN03 600 700 CsN03 600 700 YV V Y-b /bar K-1 /cm3 mol-1 /cmJ 16.57 39.6 10.6 20.01 44.8 12.8 17.87 46.5 14.5 20.20 53.9 8.9 16.58 56.0 11.0 17.97 59.2 14.2 15.42 61.7 16.7 13.58 69.4 16.4 - - - - - - (Rlyv) /cm3 mol-1 4.96 4.1 1 4.59 4.10 4.99 4.61 5.36 6.10 - - (BR/rv)- /cm3 mol 1 10.4 12.7 14.2 9.0 11.0 14.7 17.1 - p+n/bar 9940 11990 12440 12130 11590 10780 10820 - - - 17.7 9500 p+n/RT 0.183 0.244 0.217 0.247 0.202 0.219 0.189 0.165 - -166 MOLTEN ALKALI NITRATES The other term of the van der Waals equation, the internal pressure term, can be calculated directly from the observed values of yv and p , i.e.from the relation n = T(dp/aT)v-p. The various parameters of the van der Waals equation are compared in table 2.SPECIFIC HEATS The mechanical coefficients are also related to the specific heats. Thus, the difference in the isobaric and isochoric heat capacities is related to the expansivity and compressibility by the thermodynamic relation given in eqn (2). The determina- tion of individual values of Cp and C, requires either that one be known from other sources or that another relation be used ; i.e. eqn (3), where ps is the adiabatic com- pressibility which is normally found from measurements of the velocity of sound in the system. Adiabatic compressibilities have been derived for Li, Na and K nitrates from the measurement of sonic velocities in these melts. The heat capacities obtained 1 bar 1000 bar 1 bar 1000 bar 1 bar 1000 bar TABLE 3 .-TEMPERATURE AND PRESSURE DEPENDENCE OF SPECIFIC HEATS temp./K /I(-1 /cm2 dyne-1 /cm2 dyne-1 Cp/Cv /J mol-1 K-1 /J mol-1 K-1 /J mol-1 K-1 104ap 1 0 1 2 ~ ~ 1012BP BTlPS = c p - c v CP c v LiN03 573 3.111 17.7(18.3) 18.3 1.034 11.88 360 348 673 3.210 21.0(20.9) 22.0 1.052 12.76 259 246 573 3.059 16.2 16.6 1.026 12.47 493 481 673 3.149 18.8 19.6 1.043 13.56 332 318 NaN03 573 3.719 15.6(15.6) 17.9 1.147 19.58 154 134 673 3.864 18.7(18.4) 21.3 1.139 21.67 177 155 773 4.018 22.6(22.1) 25.2 1.115 23.68 229 205 573 3.660 14.4 17.2 1.194 19.50 120 100 673 3.790 17.0 19.8 1.165 22.09 156 134 773 3.927 19.5 22.8 1.169 24.52 171 146 KN03 573 3.843 16.1(16.1) 18.3 1.143 24.69 197 172 673 3.997 19.3(19.1) 23.0 1.195 25.90 1 60 134 773 4.163 23.3(22.8) 30.4 1.300 25.48 109 84 573 3.781 14.8 16.8 1.136 25.52 214 188 673 3.915 17.6 20.4 1.159 27.36 199 172 773 4.057 20.4 25.4 1.245 28.24 141 113 by means of eqn (6) and (3) are compared at different temperatures and pressures in table 3.The values of BS in brackets are those derived from the ultrasonic data of Higgs and Litovitz;' they differ slightly from the other values taken from similar work by Bockris and Richards.* For lithium nitrate and for potassium nitrate the ratio pT/Ps rises with rise in temperature and falls with rise in pressure. However for sodium nitrate the ratio falls with rise in temperature and rises with rise in pressure. Subsequently the trend in specific heats is reversed in the case of NaNO,. The actual values obtained for the specific heats are of the magnitude expected for highly associated liquids, cf.the values quoted by Rowlinson for liquids of various kinds and those for the oxyanion melts quoted by L~msden.~ In general Cp and Cv converge as the triple point is approached, i.e. as ( p + z) AV becomes zero,J. E . BANNARD 167 and at higher temperatures Cp usually rises but sometimes only after a low temperature fall. However the apparent variations of Cp and Cv with temperature as given in table 3, may or may not be significant because the precision with which the absolute values of these quantities is known is really very low. The fact that they are large and similar in magnitude makes their evaluation from their ratio and difference a very uncertain procedure and the data derived here (and elsewhere) from com- pressibility measurements must be regarded with reserve.The two groups of ultrasonic workers ' 9 * disagree in their adiabatic compressibilities by up to 4 %, so an uncertainty in ps of +4 % is immediately introduced. This together with an uncertainty of up to +4.5 % in PT produces the large errors. Once, of course, a I 1 6h ' *6 40 390 39.5 Vm/cm3 temperatures in K. FIG. 2.-Change of internal energy with molar volume for LiN03. 0, 1 bar ; x , lo00 bar; single calorimetric value of Cp or Cv is known, then the other data could be evaluated with confidence. Assuming, however, that both the magnitude and the sense of the specific heats given in table 3 do have some significance, differences between the structures of the three melts are suggested. The minimum in Cp for water between the melting point and the boiling point has been explained in terms of the large change in the configurational contribution to the specific heat,lo which itself is a measure of the distortion and breaking of the cohesive bonds in the liquid. The maximum contribution to the specific heat from the " internal " energies, i.e.from rotational, vibrational and translational modes, can be roughly estimated. If the melt consists of fully-liberated, non-planar M-N03 species then the maximum contribution from these sources is 12R, i.e. approximately 100 J mol-1 K-l. The difference between168 MOLTEN ALKALI NITRATES this quantity and the values given in table 3 must be accounted for by the configura- tional specific heat, the specific heat arising from dynamic changes in structure. The values for LiN03 are considerably above 100 J mol-1 K-l at the temperatures calculated, although showing a very large temperature-dependence and a very large pressure dependence.This supports the anticipated highly-structured configuration for that melt. INTERNAL ENERGY The internal pressure of the liquid, n, may be obtained from the thermal pressure coefficients according to eqn (I), hence plots of yv T-p may be made as a function of molar volume and these are given in fig. 2 to 6. Values are also given for 1000 bar, and although it is immediately apparent that this range of pressure produces very 45 46 Vm/cm3 temperatures in K. FIG. 3.-Change of internal energy with molar volume for NaN03. 0, 1 bar; x , 10o0 bar; little change in the derivative (aE/dY), for the alkali nitrates, the slopes of the isotherms between the 1 and 1000 bar points may be estimated and are given in table 4.The 6: 12 potential of Lennard-Jones and Devonshire for simple liquids leads to the well known energy/distance relationship ( E / r ) , and for any simple material, r will have a direct dependence on molar volume, V,, and the derivative i3E/i3Vm will take the form shown in fig. 7. It will be noted that the experimental plots of (yv T-p) given in fig. 2 to 6 take the form of the curve at positions A, B and C depending on the cation. There is a clear tendency for these isobaric plots to go through maxima. The isothermal slopes, between two pressure points, are given in table 4. Gibson et aZ.11-14 made a number of similar plots for a series of liquids ranging from water and ethylene-glycol to hydrocarbons, and obtained plotsJ .E. BANNARD 1 1 1 169 11.2 54 55 56 Vm/cm3 temperatures in K. FIG. 4.4hange of internal energy with molar volume for KNO+ 0, 1 bar; x , 10oO bar; t 0.m Vmlcm3 temperatures in K. Fro. 5.-Change of internal energy with molar volume for RbN03. 0, 1 bar; X , lo00 bar;170 MOLTEN ALKALI NITRATES of similar shape with, in some cases, maxima. Again, the simple model of a liquid describes the internal energy as a sum of the attractive potential energy and the repulsive : and E = EA-ER In the studied region - (a2ER/aV2)T increases more rapidly than (a2EA/a V2)= as the volume is diminished until at the maximum (dE,/aV), = -(aE,/aV,). Further decrease in volume increases the repulsive term causing (aE/aV), to fall.Of particular interest is the effect of temperature on the internal pressure, n. For most liquids n falls with rise in temperature which might be considered as a criterion of I I I 1 740 x - I I I 78ov\, 69 70 71 Vm/cm3 FIG. 6.-Change of internal energy with molar volume for CsN03. 0, 1 bar; x , lo00 bar; temperatures in K. " normality ".15 Two notable exceptions were the plots for water and ethylene glycol,14 where the n values rose with rise in temperature over the experimental range; there appears to be a tendency for this to be the case for the molten alkali nitrates although the experimental scatter is considerable. Gibson suggested that this is due to directional interaction, very strong in the case of water.For the effect of cation on the properties of a series of liquids like the alkali nitrates to be studied, acceptable corresponding states should be compared. However in broad terms the plots show that the internal pressure decreases markedly with increase in density (having passed through a maximum) which follows from the corresponding increase in the repulsive component of the intermolecular potential. Support for this is found in the data for LiN03 for which the total internal pressures are comparatively low and in which we suppose the repulsive contributions are the greatest. The internal pressure data for the salts just above the melting point isJ . E . BANNARD 171 given in table 5. A strict comparison on the basis of corresponding states is not easily achieved and here the simple expedient is followed of comparing the total internal pressure for the five salts at " corresponding " free volumes, i.e.at values of V, 25 % in excess of V,. As expected the increase in coulombic attraction with \ vm FIG. 7.-The Lennard-Jones potential for a liquid. decreasing cation radius systematically enhances the value of n from Csf to Na+, but thereafter coulombic repulsion becomes significant to the degree that n for lithium nitrate is reduced considerably. reveals that the alkali nitrates with the exception of lithium nitrate all have similar structures with the force constant K shown in An assessment of spectral data TABLE 4.-sLOPES OF ISOTHERMS (0-IOOO bar) temperature/K slopc/kbar cm-3 temperature/K slope/kbar cm-3 temperature/K slope/kbar cm-3 LiN03 NaN03 KN03 580 +0.30 660 +0.31 660 -0.18 600 + 0.20 680 + 0.30 680 - 0.26 620 +0.10 700 + 0.22 700 -0.33 640 + 0.08 720 +0.16 720 -0.38 660 0 740 +0.14 740 - 0.48 RbN03 CsN03 660 + 0.06 740 -0.11 680 - 0.02 760 -0.13 700 - 0.03 780 -0.17 720 - 0.08 740 - 0.08 760 -0.12172 MOLTEN ALKALI NITRATES fig.S(a) decreasing with the surface charge density of M+. The magnitude of this force constant implies a very definite association between the metal ion and the nitrate ion. Interactions of this type have been further suggested from heats of mixing of ternary halide + nitrate mixtures,l the excess enthalpies undoubtedly arising from departures from random mixing, and from n.m.r. studies.l* The force constant for Li was found to be lower than that for Na, in fact about the same as the K-N03 value.Preferential penetration of the small Lif into the bidentate position may well account for this, fig. 8(b). TABLE 5.-~”I’ERNAL PRESSURES FOR THE NITRATE MELTS density at m.pt. Vb m 125 %YO =i MIV; 1g cm-3 ;r/bar Volcm3 / a 3 / g cm-3 n at pb/bar LiN0, 1.781 9680 32.5 40.6 1.700 9940 NaNO, 1.91 5 11850 36 45 1.889 12100 KN03 1.872 12110 45 56.3 1.796 1 1490 RbN03 2.510 10740 49 61.3 2.406 10820 CSNO, 2.820 9530 57 71.3 2.734 9270 Ion association of this type seems to be important in nitrate melts and an increase in such association with increase in pressure, followed by loss of rotational freedom and increase in the amount of bidentate structure is probably the reason for the enhanced repulsive term.This also helps to explain why the entropy change for the fusion of lithium nitrate is so much higher than the corresponding chloride salt (see table 6). The low value of ASfusion for RbN03 indicates that the anion is:possibly freely rotating in the solid below the melting point. FIG. &-The structure of an M+ nitrate.J . E. BANNARD 173 TABLE 6.-THERMODYNAMIC FUNCTIONS RELATING TO THE FUSION OF ALKALI CHLORIDES AND NITRATES l9 LiCl NaCl KCl RbCl CSCl LiN03 NaN03 KN03 RbN03 CSNO~ m.pt./K 883 1073 1043 995 91 8 525 580 61 1 581 679 AHruaion /Id mol-1 19.92 27.99 26.53 23.72 18.74 25.61 14.73 11.72 4.64 14.10 A Vfwion A S ~ o l l / % /cm3 mol-1 /J mol-1 K-1 +26.2 5.8 22.55 f25.0 7.5 26.07 +17.3 7.3 25.44 +14.3 6.8 23.85 +10.0 5.6 22.05 +21.4 6.84 48.79 +10.7 4.32 25.52 +3.32 1.73 19.16 -0.23 -0.14 7.99 + 12.1 7.48 20.75 CONCLUSIONS The non-hard-sphere nature of the nitrate ion adds an element of complication to the interpretation of the mechanical coefficient data.In this respect a study of the technically more difficult halide melts would be useful. The significant results of this work on nitrate melts lie in the computed internal energy changes with density of the liquids, and the relative effects of different cations. However, for this analysis to be carried further, studies must be made over much greater ranges of T and p than those reported here. The author thanks the S.R.C. for financial support and Prof. G. J. Hills for making laboratory facilities available. J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworth, London, 1959). J. E. Bannard and A. F. M. Barton, J.C.S. Faraday I, 1978,74,153. K. F. Herzfeld and T. A. Litovitz, Adsorption and Dispersion of Sonic Waues (Academic Press, New York, 1959). R. N. Haward, Trans. Farahy SOC., 1966,62,828. B. Cleaver and J. F. Williams, J. Phys. Chem. Solids, 1968,29, 877. ’ R. W. Higgs and T. A. Litovitz, J. Acoust. SOC. Amer., 1960,32, 1108. a J. O’M. Bockris and N. E. Richards, Proc. Roy. SOC. A, 1957,241,44. J. Lumsden, Thermodynamics of Molten Salt Mixtures (Academic Press, 1966). ‘ G. Goldmann and K. Todheide, 2. Naturforsch., 1976,31a, 656. l o D. Eisenberg and W. Kauzmann, The Structure and Properties of Water (Oxford, 1969). l 1 R. E. 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