INTERNAL PRESSURE AS A FUNCTION OF PRESSURE

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1 Molecular and Quantum Acoustics vol. 27, (2006) 327 INTERNAL PRESSURE AS A FUNCTION OF PRESSURE Edward ZORĘBSKI University of Silesia, Institute of Chemistry, Szkolna 9, Katowice, POLAND emz@ich.us.edu.pl Internal pressure as a function of pressure was estimated from literature data. Mostly, the data on isothermal compressibility and isobaric coefficient of thermal expansion obtained from acoustic method were used. The results shown that internal pressure can reaches a maximum for moderate pressures and then decreases more and more rapidly. Keywords: internal pressure, pressure effect, isothermal compressibility, isobaric coefficient of thermal expansion, speed of ultrasound 1. INTRODUCTION As has been pointed out previously [1], relatively little has been published on internal pressure (also known as the cohesion pressure or energy-volume coefficient) of liquids, although, the internal pressure P in is a quantity that is well defined thermodynamically (see section 2, Eq.(1)) and can be used to describe the macroscopic resultant of molecular interaction. The effect of pressure on internal pressure has received still littler attention (the reports are rare and available data are in addition often scatter). In this article, a short presentation of this effect (based on literature data) will be made. Generally, the significance of the internal pressure and its correlation with the solubility parameter (widely used for instance in paints, pharmaceutics, polymers, and petroleum industry) has been discussed among others in review articles by Barton [2] and Dack [3]. The same problem in relation to the pressure influence was investigated by Renuncio et al. [4] and recently by Verdier and Andersen [5]. However, in both cases only very limited pressure ranges were examined, up to 100 MPa and 30 MPa, respectively. The water is of particular interest, because of the peculiar change in the internal pressure of water with the increase in the temperature T and pressure p [3,6-8]. First of all, in the region up to ca. 45 MPa and up to ca. 277 K internal pressure is negative [3,6-8]. Precisely, internal pressure is negative for matrix (p,t) above diagonal p-t about 45 MPa and 277 K [7]. In other

2 328 Zorębski E. words, at this line the change in internal pressure for water, negative to positive, with increase in temperature (isotherms) and pressure (isobars) are observed [3,6-8]. At the same time the P in values shows generally that at low temperatures P int increases, but at high temperatures P int decreases with increase in pressure. Consequently, at some temperature, the internal pressure of water is rather insensitive to pressure (up to 100 MPa) [7,8]; this temperature is reported as ca. 313 K [7] or ca. 318 K [8]. Simultaneously, in the investigated temperature and pressure range an extreme of the internal pressure is not observed. The results reported by Melent eva et al. [9] (an acoustic piezometer was used for speed of sound and density measurements) for 1-chlorohexane in the temperature range from ( to ) K and pressures from saturation up to 200 MPa are of interest too, because of the high chemical activity of halogen-substituted n-alkanes. Unfortunately the scatter of the obtained P int values is too large for detailed analysis (this lies probably mainly in relative large uncertainty of ±0.3 % for density measurements). Generally, taking into account declared uncertainty (±6 % [9]), the values of the internal pressure for 1-chlorohexane appear to be practically insensitive to the pressure for isotherms from ( to ) K and slightly increase with increasing pressure for isotherms from ( to ) K. Generally however, the internal pressure is known to reach a maximum with respect to pressure at constant temperature, i.e. ( P int / p) T = 0 for some p max. Such behaviour shown for instance diethyl ether [10] as well as some 1-alkanols [11]. In Figure 1, the internal pressure for diethyl ether at the temperature K is plotted against the pressure (data are taken from Ref. [10] and smoothed by fitting into an equation given in section 3; see Eq.(3) and Table A1). As can be seen, the internal pressure first increases slightly, then reaches maximum (at ca. 155 MPa), and then decreases slowly but as p exceeds ca. 500 MPa the internal pressure begins to decrease rapidly. At ca. 725 MPa the internal pressure shows inversion and goes to large negative values as the liquid is further compressed. Further discussion about this behaviour is given in section THEORETICAL ASPECTS OF CALCULATION The internal pressure P int may be evaluated from direct measurements of the thermal pressure coefficient, defined below by the Eqs (1) and (2), and the use of an apparatus called a constant-volume thermometer. This type of the apparatus has been described in the twenties of the last century [12] and subsequently used by several other workers [13,14]. These techniques so called the direct are rather rarely used. Mostly, the internal pressure P int in liquids is evaluated from direct or indirect measurements of the isothermal compressibility κ T

3 Molecular and Quantum Acoustics vol. 27, (2006) 329 and isobaric coefficient of thermal expansion α p. The respective well-known relations are given below. P = ( U / V ) = T ( p / T ) p int T V (1) where U is the internal energy, V is the molar volume, and ( p / T ) = α V p κ 1 T, and therefore P int 1 = T α κ p (2) p T 1 Term ( p / T ) = α κ is frequently called the isochoric thermal pressure coefficient and V p T denoted β V. It should be noted that for a perfect gas, the internal pressure is zero, but for imperfect gases and liquids it becomes appreciable, and is frequently much greater than the pressure p. Thus, for normal liquids below their boiling points, precisely at low pressures, where p << T ( p / T ) V = T α p κ 1 T, the second term in Eqs (1) and (2) can be entirely neglected without significant error. However, at high pressures this second term absolutely must be included P int / MPa Fig.1. Internal pressure P int vs. pressure p for diethyl ether at T = K; line calculated from Eq.(3). Data taken from Ref. [10]. The convenient and well-established method for the determination of the thermodynamic properties of liquids, especially under elevated pressures (direct measurements are here extremely difficult), is acoustic one [15-17]. This method based on the measurements of the speed of sound, and in consequence offer an indirect way to obtain information on related thermodynamic properties like density, heat capacity, compressibility, and last of all internal pressure. Thus, the P int is calculated here from indirect evaluated κ T and α p values. It must be emphasised that the greatest uncertainty is related to thermal expansion coefficient, i.e. the thermal expansion coefficient introduces the greatest uncertainty into internal pressure calculated by acoustic method; it refers generally to all indirect methods that

4 330 Zorębski E. are based on Eq.(2). Apart from experimental uncertainties, the greatest difficulty in the calculation of α p from ρ(t) p data lies in the obtaining of the ( ρ / T ) p, and precisely, in very often subjective determination of the ρ(t) p fitting equation, that in the next step is analytically differentiated. To avoid this problem, Cerderiña et al. [18] recommend numerical differentiation. It should be pointed out here that also the correlation based on the Tait-type relationship is successfully used for evaluation of the κ T (from the measurements of density as a function of temperature and pressure), and next the P int [4,5,19-22]. It seems that commercial available densimeters (which can be used to perform measurements in a relatively broad range of temperature and pressure) make easy suchlike investigations, the rather that the calculations are rather easy. Moreover, it is also interesting to note that the internal pressure can also be evaluated from other equations of state [23,24]. In both cases, mostly pure non-polar liquids were investigated. According to authors, results show fairly good agreement [23] and a close agreement (better than ±6%) [24] between the computed and experimental values. 3. RESULTS AND DISCUSSION By the use of Eq.(2), the P int was calculated from literature data on isothermal compressibility and isobaric coefficient of thermal expansion for: 1-hexanol [25], acetone [22], benzene [17], cyclohexane [17], n-decanol [5], and ethanol [5]. For benzene and cyclohexane, the internal pressures were calculated from data on isothermal compressibility and isobaric coefficient of thermal expansion obtained by acoustic method [17], whereas for acetone calculations based on the density measurements and Tait correlation [22]. On the other hand, for n-decane and ethanol [5], the isobaric coefficient of thermal expansion determined by microcalorimetry and isothermal compressibility determined by density measurements (Tait correlation) were used. In turn, in the case of liquid 1-hexanol [25], the isobaric thermal expansion measurements in a pressure-controlled scanning calorimeter, the specific volume isotherm at T = K (derived from isothermal compressibility measurements performed in function of pressure and from the density at atmospheric pressure) as well as the Tait correlation were used to evaluation of the respective values. Unfortunately, it appears that the results for acetone are unclear (a consistency of the obtained P int values is poor, especially along isobars), and finally, the pressure dependence of internal pressure in the temperature range from ( to ) K and up to 60 MPa is not shown. Nevertheless, it can be concluded that the P int generally increases with increasing pressure and for isotherms between ( and ) K a weak maximum is observed.

5 Molecular and Quantum Acoustics vol. 27, (2006) 331 For the rest, the results are given in graphic form in Figs 2-5. The presented isotherms of P int (p) are obtained by fitting into a polynomial in form of m P = a i p / int i= 1 i 1 ( 100) where a i are coefficients obtained by a least squares method and the upper limit of the summation m is equal 4 or 3. The coefficients of the isotherms with evident maximum are summarised in Table A1 (Appendix). The standard deviations for these isotherms are smaller than ±0.5 MPa for benzene and ±1.5 MPa for 1-hexanol. For diethyl ether, the standard deviation is found to be ±25 MPa. Figure 2 shows the isotherms of pressure dependence of internal pressure for 1-hexanol at selected temperatures below the normal boiling point ( K) and at pressures from the saturated vapour pressure up to 400 MPa. The isotherms have maxima at the pressures between ca. 60 MPa (for T = K) and 100 MPa (for T = K) (3) T = K P int / MPa T = K Fig.2. Internal pressure P int vs. pressure p for 1-hexanol in the temperature range ( to ) K with 25 K step; lines calculated from Eq.(3). Data for calculations were taken from Ref. [25]. In turn, based on the data published by Sun et al. [17], the internal pressure for benzene and cyclohexane in the temperature range from (283 to 323) K and up to the freezing pressures has been calculated. In both cases, the internal pressure decreases with increasing temperature, as is shown in Figs 3 and 4. Simultaneously, it appears that the pressure dependence of the internal pressure has maximum for benzene whereas respective dependence for cyclohexane shows no extreme in the investigated temperature and pressure range. In the case of benzene, the observed maximum of the P int shifts towards higher pressures with increasing temperature (Fig.3 and Table A1).

6 332 Zorębski E T = K 380 P int / MPa T = K Fig.3. Internal pressure P int vs. pressure p for benzene in the temperature range ( to ) K with 5 K step; lines calculated from Eq.(3). Data for calculations were taken from Ref. [17] T = K P int / MPa T = K Fig.4. Internal pressure P int vs. pressure p for cyclohexane in the temperature range ( to ) K with 5 K step; lines calculated from Eq.(3). Data for calculations were taken from Ref. [17]. Finally, the pressure effect on P int for n-decane and ethanol at T = K is seen in Fig.5. In the studied pressure range, an increase in P int with pressure is observed. As seen, this positive effect is evidently smaller for n-decane. Such small positive effects of pressure on P int (at T = K) were reported by Compostizo et al. [20] for carbon disulfide and carbon tetrachloride too (see Fig.6). Lugo et al. [19] reported similar results in the case of dimethyl carbonate and octane. They found that the P int increases as the pressure rises along the isotherms, and decreases with the temperature along isobars in the temperature range from ( to ) K and pressures up to 25 MPa. However, it must be emphasised that the pressure range in the above reports is very limited.

7 Molecular and Quantum Acoustics vol. 27, (2006) P int / MPa Fig.5. Internal pressure P int vs. pressure p for n-decane, - - and ethanol,- - at T = K; lines calculated from Eq.(3). Data for calculations were taken from Ref. [5] P int / MPa Fig.6. Internal pressure P int vs. pressure p for carbon tetrachloride, - - and carbon disulfide, - - at T = K; lines calculated from Eq.(3). Data taken from Ref. [20]. The most significant effect of the increase in pressure is the decrease in volume, i.e., a decrease in the molecular distance and free volume. This leads to variation of the molecular interactions. It is known that the internal pressure can be express as sum of an attractive and repulsive contribution. The attractive contributions are connected with the attractive forces that mainly comprise hydrogen bonding, dipole-dipole, multipolar, and dispersion interactions. Repulsive forces, acting over very small intermolecular distances, play a minor role in the cohesion process under normal conditions. But as the pressure increases, the repulsive forces increases also. At the same time change in the attractive forces are more complicated. In consequence, the internal pressure can both increase and decrease, as well as reaches a maximum as the pressure increases. And in some cases, the internal pressure can be even insensitive to pressure.

8 334 Zorębski E. As noted in the introduction, the P int for diethyl ether at some compression begins to decrease rapidly (see Fig.1). Further compression causes such decrease in P int that it becomes highly negative. In other words, the decrease in the molecular distance causes that the repulsive forces completely dominate the attractive forces. Because molar volumes reflect changes in the molecular distance (as an pressure is applied or as the liquid contract on cooling), Barton [26] has discussed the relationship between P int and molar volumes. He has shown that each liquid may be considered to have a P int (V) curve of the general form with maxima detected at moderate pressures and temperatures. At the same time only a part of the above-mentioned curve is experimentally accessible and the position of this part of curve is determined by the liquid properties. Generally, the explanation of the above-mentioned pressure behaviour of the P int is however difficult because as it has been noted in the introduction, the data sets are scarce. In addition, different experimental techniques as well as calculation methods were used. In consequence, the scatter among the various sets of data is considerable. For instance, for carbon disulfide Renuncio et al. [4] report that the P int has tendency to decrease with increasing pressure while the reverse is reported by Compostizo et al. [20]. In this work, an overall uncertainty in the calculated P int values is estimated roughly to be about 1-2 % or less. It should be emphasised that in the literature can be also found a totally unreliable values of P int that result from wrong evaluation by using the mean molecular radii (see, previous paper [1]). Some confusion also exist with relation to the sign of internal pressure, i.e., minus sign (groundless, in the present writer opinion) is noted for internal pressure by some authors [27]. 4. CONCLUSIONS In this work, the pressure effect on the internal pressure for chosen liquids is studied. It seems that the internal pressure shows indeed in some cases a maximum with respect to pressure at constant temperature. Generally, however, both the present contribution and available literature reports are not sufficient to yield any final conclusion. In the present writer opinion, first of all the systematic study of various class of liquids in broad pressure and temperature ranges is needed. Such study by means of the acoustic method was started recently in our laboratory for associated liquids. The results will be presented shortly. REFERENCES 1. E.Zorębski, Mol. Quant. Acoust., 26, , (2005). 2. A.F.M.Barton, Chem.Rev., 75(6), (1975).

9 Molecular and Quantum Acoustics vol. 27, (2006) M.R.J.Dack, Chem.Soc.Rev., 4(2), (1975). 4. J.A.R.Renuncio, G.J.F.Breedveld, J.M.Prausnitz, J.Phys.Chem., 81(4), (1977). 5. S.Verdier, S.I.Andersen, Fluid Phase Equilib., 231, (2005). 6. J.V.Leyendekkers, J.Phys.Chem., 87, (1983). 7. M.J.Blandmer, J.Burgess, A.W.Hakin, J.Chem.Soc., Faraday Trans.1, 83(6), (1987). 8. R.E.Gibson, O.H.Loeffler, J.Am.Chem.Soc., 63, (1941). 9. V.V.Melent ev, M.F.Bolotnikov, Y.A.Neruchev, J.Chem.Eng.Data, 51, (2006). 10. W.J.Moore, Physical Chemistry, 3rd ed., Longmans, London A.Dudek, Termodynamiczne i akustyczne właściwości alkoholi pierwszorzędowych w obszarze podwyższonego ciśnienia, Msc Thesis, Silesian University, Katowice W.Westwater, H.W.Frantz, J.H.Hildebrand, Phys.Rev., 31, (1928). 13. U.Bianchi, G.Agabio, A.Turturro, J.Phys.Chem., 69(12), , (1965). 14. G.A.Few, M.Rigby, J.Phys.Chem., 79(15), (1975). 15. W.Marczak, M.Dzida, S.Ernst, High Temp. High Press., 32, (2000). 16. J.L.Daridon, A.Lagrabette, B.Lagourette, J.Chem.Thermodynamics, 30, (1998). 17. T.F.Sun, P.J.Kortbeek, N.J.Trappeniers, S.N.Biswas, Phys.Chem.Liq., 16, (1987). 18. C.A.Cerderiña, C.A.Tovar, D.Gonzales-Salgado, E.Carballo, L.Romani, Phys.Chem. Chem. Phys., 3, (2001). 19. L.Lugo, M.J.P.Comuñas, E.R.López, J.Fernández, Fluid Phase Equlib., 186, (2001). 20. A.Compostizo, A.C.Colin, M.R.Vigil, R.G.Rubio, M.Diaz Peña, J.Phys.Chem., 92, (1988). 21. G.Watson, C.K.Zéberg-Mikkelsen, A.Baylaucq, C.Boned, J.Chem.Eng.Data, 51, (2006). 22. R.Gomes de Azevedo, J.Szydłowski, P.F.Pires, J.M.S.S.Esperança, H.J.R.Guedes,. L.P.N.Rebelo, J.Chem.Thermodynamics, 36, (2004). 23. N.Pant, C.V.Chaturvedi, G.D.Chaturvedi, Z.Phys.Chem., 264, (1983). 24. E.K.Goharshadi, F.Nazari, Fluid Phase Equilib., , (2001). 25. S.I.Randzio, J-P.E.Grolier, J.R.Quint, Fluid Phase Equilib., 110, (1995). 26. A.F.M.Barton, J.Chem.Educ., 48(3), (1971). 27. V.N.Kartsev, M.N.Rodnikova, S.N.Shtykov, J.Struct.Chem., 45(1), (2004); and references therein.

10 336 Zorębski E. APPENDIX Table A1. Coefficients a i of Eq.(3) (i.e., pressure dependence of internal pressure) together with estimated p max at which ( P int / p) T = 0 T / K a 1 a 2 a 3 a 4 p max / MPa Diethyl eter a Benzene b Hexanol c a For p up to 1100 MPa; Ref. [10] b At given temperatures for p up to 50, 70, 90, 110, 130, 150, and 170 MPa, respectively; Ref.[17]. c For p up to 400 MPa; Ref. [25].

Speeds of sound and isothermal compressibility of ternary liquid systems: Application of Flory s statistical theory and hard sphere models

PRAMANA c Indian Academy of Sciences Vol. 70, No. 4 journal of April 2008 physics pp. 731 738 Speeds of sound and isothermal compressibility of ternary liquid systems: Application of Flory s statistical