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Universal behavior of the enthalpy of vaporization: an empirical equation
Fluid Phase Equilibria 218 (2004) 205–207
Universal behavior of the enthalpy of vaporization: an empirical equation Ariel G. Meyra, Victor A. Kuz, Guillermo J. Zarragoicoechea∗ IFLYSIB (UNLP, CONICET, CICPBA), C.C. 565, 1900 La Plata, Argentina Received 22 August 2003; accepted 23 December 2003
Abstract Values of the enthalpy of vaporization from the critical to the triple point are correlated by an empirical equation. The equation contains parameters which characterize each substance: the critical and triple point temperatures and the enthalpy of vaporization at the triple point, and for all substances the same universal critical ratio. This work suggests that a wide class of fluids with the exception of quantal liquids shows an universal behavior along the coexistence curve. © 2004 Elsevier B.V. All rights reserved. Keywords: Enthalpy of vaporization; Vapor–liquid equilibria
1. Introduction There is a vast experimental knowledge of the liquid–vapour equilibrium behavior of pure substances. Empirical equations of vapor pressure are known since Dalton [1,2]. All these equations relate pressure with temperature, the coefficients must be evaluated for each substance by fitting the available experimental data and the applicability is limited to the temperature interval where the fitting is acceptable. There is not yet a general and universal vapor pressure equation that could be applied to any substance. The coexistence volumes are wrongly predicted by theoretically based equations of state and empirical equations are limited in the range of applicability [3]. The situation is different for enthalpies or entropies along the coexistence curve. Here there are correlations for organic and inorganic fluids and for metal and quantal liquids [4,5]. These quantities are of importance in processes where thermal energies of phase transition are required. The enthalpy of vaporization for any substance is zero at the critical point and has a maximum value at the triple point. It is a bounded function like all the functions along the coexistence curve. Theoretical analysis [6] predicts the enthalpy of vaporization to be proportional to (Tc −T)1/2 , for ∗ Corresponding author. Present address: Instituto de Fisica de Liquidos y Sistemas Biologicos, Universidad Nacional de la Plata, 59 No. 789, La Plata 1900, Argentina. Tel.: +54-221-425-4904; fax: +54-221-425-7317. E-mail address: [email protected] (G.J. Zarragoicoechea).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2003.12.011
points close to the critical point; also it must be mentioned that van der Waals equation gives enthalpies of vaporization too small except for those points close to the critical point [7]. van der Waals equation and experiment agree in showing that the enthalpy of vaporization decreases to zero at the critical point and the curves are of similar shape. Empirical equations like that of Watson [4] or Fish and Lielmezs [5] use the critical temperature and the boiling temperature to construct different dimensionless temperatures. Following this line of reasoning, in the present work we propose an universal empirical equation for the enthalpy of vaporization that correlates the available experimental data. Our equation is able to correlate the enthalpy of vaporization along the whole coexistence curve with less dispersion than that of Watson or Fish and Lielmezs.
2. An universal empirical equation for the enthalpy of the liquid–vapour phase transition Let us first represent the experimental data for the enthalpy of vaporization (EV) as a function of temperature for nitrogen, propylene [8], water, argon and methane [1], in dimensionless variables. The temperature T is always bounded between the critical temperature Tc and the triple temperature Tt , that is Tt ≤ T ≤ Tc . For any substance the EV is zero at the critical point and has a maximum at the triple temperature [1,8,9]. We scale H(T) with its maximum value
206
A.G. Meyra et al. / Fluid Phase Equilibria 218 (2004) 205–207 Table 1 Standard deviations of empirical equations for the enthalpy of vaporization, H/Ht Substance
N
This work
Watson [4]
Fish and Lielmezs [5]
Water Propylene Argon Nitrogen Methane
12 25 11 14 11
0.329 0.721 0.383 0.305 1.443
1.211 1.384 1.732 1.893 3.070
0.683 0.802 0.829 1.085 1.116
N is the number of data points [1,8].
From Eq. (3), by using the definitions (1) and (2) we can write the EV as 2 Tc − T Zc [(T −Tt )/(Tc −Tt )]+Zc (5) H(T) = Ht Tc − T t Fig. 1. Reduced enthalpy of vaporization as a function of dimensionless temperature. Points are experimental data [1,8], solid line is the plot of Eq. (3).
H(T = Tt ) = Ht , the EV at the triple point. The dimensionless temperature and the EV are scaled in such a way that temperature varies between zero at the triple point and one at the critical point while the dimensionless EV goes from 1 to 0, respectively: t=
T − Tt Tc − T t
(1)
and h(t) =
H(T) Ht
(2)
Using this dimensionless parameters to represent the experimental data, we see in Fig. 1 that all curves for the very different substances mentioned above reduce to an universal curve. We propose the following function to represent this curve: h(t) = (1 − t)Zc t+Zc 2
The values of the Tc , Tt , Ht depend on the substance, therefore for each substance we have a curve H(T). In Fig. 2 we represent Eq. (5) and the corresponding experimental data of nitrogen, propylene, water, argon and methane. There, it can be seen that the transformation does not introduce significant differences giving a very good correspondence with the experiments. Let us see now how Eq. (5) predicts the EV when we have incomplete experimental data. That is, we have the critical temperature, some experimental values of the latent heat but we do not have neither the triple temperature nor the latent heat at the triple point, as it is the case for mercury [9]. In this situation we assume the following approximation. Tt and Ht are replaced by Tmin and H(Tmin ), being Tmin the lowest experimental temperature available. Even within this approximation empirical equation (5) represents very good values of the EV, as it is shown in Fig. 3. It is clear that the applicability is now reduced to temperatures in the range Tmin –Tc .
(3)
where Zc = 0.292 is the universal critical ratio suggested by Guggenheim [3]. This value comes from the second virial coefficient for the Lennard–Jones potential [10], where ε = 0.292 (4) kTB where ε is the energy parameter in the Lennard–Jones potential and TB the temperature where the second virial coefficient is null (Boyle temperature). As we can see from Fig. 1, the function h(t) fits very well the dimensionless experimental data, with no adjustable parameters. It is important to mention that when t → 1, Eq. (3) behaves as Watson’s equation, but the standard deviations of our equation are less than that of the Watson or Fish and Lielmezs equations (see Table 1).
Fig. 2. Enthalpy of vaporization as a function of temperature. Points are experimental data [1,8], solid lines are the plots of Eq. (5) with the corresponding values of Tc , Tt , and Ht .
A.G. Meyra et al. / Fluid Phase Equilibria 218 (2004) 205–207
207
[4,5] give also good results but with greater dispersions than Eq. (5). Faizullin [14] has recently suggested a new correlation for non-associated liquids but in his equation it is necessary to know the liquid volume on the saturation line. As a future work it would be very interesting to find a thermodynamic foundation of the empirical equation presented here. List of symbols H enthalpy of vaporization T temperature Boyle temperature TB Tc critical temperature triple point temperature Tt Zc universal critical ratio Fig. 3. Enthalpy of vaporization of mercury. Points are experimental [9], and the solid line is the prediction of the empirical equation (5).
In view of the behavior of the function H(T) we went a little further and used Eq. (5) to predict the EV at the triple point of, e.g., butane and iron. The critical and triple point temperatures of butane are Tc = 425.16 K [11] and Tt = 134.86 K [12], respectively. By using Eq. (5) for the known data [11] T = 173.15 K and H(173.15 K) = 26.88 kJ/mol, we obtain Ht = 28.06 kJ/mol for EV at the triple point. Proceeding in the same way for the case of iron, we know an estimate of the critical temperature Tc = 10000 K [13], and from the periodic table of elements, the boiling temperature Tb = 3023 K, the EV at the boiling point Hb = 349 kJ/mol, and the temperature of fusion Tf = 1802.2 K. With a good approximation we supposed that Tt ∼ = Tf . By elimination of Ht in Eq. (5), applied at T = Tb , we obtained Ht = 366.6 kJ/mol, a reasonable value when compared with Hb . 3. Discussion In this paper we present an empirical equation for the enthalpy of vaporization of the liquid–vapour phase transition of a pure substance. Though the functional form of this equation is not usual for representing thermodynamic quantities, it has the outstanding characteristic of not having adjustable parameters. It has as inputs Tc , Tt , and Ht , which characterize the substances and the universal critical ratio Zc . In the dimensionless representation h(t) is a unique function for all the substances studied (Fig. 1). The equation does not represent any particular real fluid but all of them simultaneously. With dimensions, H(T) gives a curve for each substance (Figs. 2 and 3) with very good correlation. This equation is novel and of practical use and can be applied over the whole coexistence region. Empirical equations for the enthalpy of vaporization previously proposed
Greek letter ε energy parameter of the Lennard–Jones potential
Acknowledgements This work was partially supported by Universidad Nacional de La Plata and by the Agencia Nacional de Promoción Cient´ıfica y Tecnológica (ANPCyT) PICT 03-4517. GJZ is member of ‘Carrera del Investigador Cient´ıfico’ CICPBA.
References [1] J.S. Rowlinson, F.L. Swinton, Liquids and Liquid Mixtures, Butterworth Scientific, London, 1981, pp. 50–51. [2] D. Ambrose, in: M.L. McGlashan (Ed.), Chemical Thermodynamics, vol. 1, The Chemical Society, Burlington House, London, 1973. [3] E.A. Guggenheim, J. Chem. Phys. 13 (1945) 253–261. [4] K.M. Watson, Ind. Eng. Chem. 35 (1943) 398–406. [5] L.W. Fish, J. Lielmezs, Ind. Eng. Chem. Fundam. 14 (1975) 248–256. [6] L.D. Landau, E.M. Lifshitz, Fisica Estadistica, Reverte, Barcelona, 1969. [7] J.C. Slater, Introduction to Chemical Physics, McGraw-Hill, New York, 1939, p. 189. [8] S. Angus, K.M. de Reuck, B. Armstrong (Eds.), International Thermodynamic Tables of the Fluid State, vols. 6 and 7, Pergamon Press, Oxford, 1979. [9] R.C. Weast (Ed.), Handbook of Chemistry and Physics, The Chemical Rubber Co., Ohio, 1972–1973. [10] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964, p. 247. [11] American Society of Heating, Refrigerating and Air Conditionating (Ed.), Ashrae Handbook of Fundamentals, American Society of Heating, Refrigerating and Air Conditionating, Atlanta, 2001, p. 20.43. [12] H. Hausen (Ed.), Landolt-Börnstein, Zahlenwerte und funktionen, 4 Teil, Bandteil C, Berlin, 1980, p. 825. [13] A.R. Ubbelohde, The Molten State of Matter, Wiley, Chichester, 1978, p. 24. [14] M.Z. Faizullin, Fluid Phase Equilibria 211 (2003) 75–83.
Universal behavior of the enthalpy of vaporization: an empirical equation Ariel G. Meyra, Victor A. Kuz, Guillermo J. Zarragoicoechea∗ IFLYSIB (UNLP, CONICET, CICPBA), C.C. 565, 1900 La Plata, Argentina Received 22 August 2003; accepted 23 December 2003
Abstract Values of the enthalpy of vaporization from the critical to the triple point are correlated by an empirical equation. The equation contains parameters which characterize each substance: the critical and triple point temperatures and the enthalpy of vaporization at the triple point, and for all substances the same universal critical ratio. This work suggests that a wide class of fluids with the exception of quantal liquids shows an universal behavior along the coexistence curve. © 2004 Elsevier B.V. All rights reserved. Keywords: Enthalpy of vaporization; Vapor–liquid equilibria
1. Introduction There is a vast experimental knowledge of the liquid–vapour equilibrium behavior of pure substances. Empirical equations of vapor pressure are known since Dalton [1,2]. All these equations relate pressure with temperature, the coefficients must be evaluated for each substance by fitting the available experimental data and the applicability is limited to the temperature interval where the fitting is acceptable. There is not yet a general and universal vapor pressure equation that could be applied to any substance. The coexistence volumes are wrongly predicted by theoretically based equations of state and empirical equations are limited in the range of applicability [3]. The situation is different for enthalpies or entropies along the coexistence curve. Here there are correlations for organic and inorganic fluids and for metal and quantal liquids [4,5]. These quantities are of importance in processes where thermal energies of phase transition are required. The enthalpy of vaporization for any substance is zero at the critical point and has a maximum value at the triple point. It is a bounded function like all the functions along the coexistence curve. Theoretical analysis [6] predicts the enthalpy of vaporization to be proportional to (Tc −T)1/2 , for ∗ Corresponding author. Present address: Instituto de Fisica de Liquidos y Sistemas Biologicos, Universidad Nacional de la Plata, 59 No. 789, La Plata 1900, Argentina. Tel.: +54-221-425-4904; fax: +54-221-425-7317. E-mail address: [email protected] (G.J. Zarragoicoechea).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2003.12.011
points close to the critical point; also it must be mentioned that van der Waals equation gives enthalpies of vaporization too small except for those points close to the critical point [7]. van der Waals equation and experiment agree in showing that the enthalpy of vaporization decreases to zero at the critical point and the curves are of similar shape. Empirical equations like that of Watson [4] or Fish and Lielmezs [5] use the critical temperature and the boiling temperature to construct different dimensionless temperatures. Following this line of reasoning, in the present work we propose an universal empirical equation for the enthalpy of vaporization that correlates the available experimental data. Our equation is able to correlate the enthalpy of vaporization along the whole coexistence curve with less dispersion than that of Watson or Fish and Lielmezs.
2. An universal empirical equation for the enthalpy of the liquid–vapour phase transition Let us first represent the experimental data for the enthalpy of vaporization (EV) as a function of temperature for nitrogen, propylene [8], water, argon and methane [1], in dimensionless variables. The temperature T is always bounded between the critical temperature Tc and the triple temperature Tt , that is Tt ≤ T ≤ Tc . For any substance the EV is zero at the critical point and has a maximum at the triple temperature [1,8,9]. We scale H(T) with its maximum value
206
A.G. Meyra et al. / Fluid Phase Equilibria 218 (2004) 205–207 Table 1 Standard deviations of empirical equations for the enthalpy of vaporization, H/Ht Substance
N
This work
Watson [4]
Fish and Lielmezs [5]
Water Propylene Argon Nitrogen Methane
12 25 11 14 11
0.329 0.721 0.383 0.305 1.443
1.211 1.384 1.732 1.893 3.070
0.683 0.802 0.829 1.085 1.116
N is the number of data points [1,8].
From Eq. (3), by using the definitions (1) and (2) we can write the EV as 2 Tc − T Zc [(T −Tt )/(Tc −Tt )]+Zc (5) H(T) = Ht Tc − T t Fig. 1. Reduced enthalpy of vaporization as a function of dimensionless temperature. Points are experimental data [1,8], solid line is the plot of Eq. (3).
H(T = Tt ) = Ht , the EV at the triple point. The dimensionless temperature and the EV are scaled in such a way that temperature varies between zero at the triple point and one at the critical point while the dimensionless EV goes from 1 to 0, respectively: t=
T − Tt Tc − T t
(1)
and h(t) =
H(T) Ht
(2)
Using this dimensionless parameters to represent the experimental data, we see in Fig. 1 that all curves for the very different substances mentioned above reduce to an universal curve. We propose the following function to represent this curve: h(t) = (1 − t)Zc t+Zc 2
The values of the Tc , Tt , Ht depend on the substance, therefore for each substance we have a curve H(T). In Fig. 2 we represent Eq. (5) and the corresponding experimental data of nitrogen, propylene, water, argon and methane. There, it can be seen that the transformation does not introduce significant differences giving a very good correspondence with the experiments. Let us see now how Eq. (5) predicts the EV when we have incomplete experimental data. That is, we have the critical temperature, some experimental values of the latent heat but we do not have neither the triple temperature nor the latent heat at the triple point, as it is the case for mercury [9]. In this situation we assume the following approximation. Tt and Ht are replaced by Tmin and H(Tmin ), being Tmin the lowest experimental temperature available. Even within this approximation empirical equation (5) represents very good values of the EV, as it is shown in Fig. 3. It is clear that the applicability is now reduced to temperatures in the range Tmin –Tc .
(3)
where Zc = 0.292 is the universal critical ratio suggested by Guggenheim [3]. This value comes from the second virial coefficient for the Lennard–Jones potential [10], where ε = 0.292 (4) kTB where ε is the energy parameter in the Lennard–Jones potential and TB the temperature where the second virial coefficient is null (Boyle temperature). As we can see from Fig. 1, the function h(t) fits very well the dimensionless experimental data, with no adjustable parameters. It is important to mention that when t → 1, Eq. (3) behaves as Watson’s equation, but the standard deviations of our equation are less than that of the Watson or Fish and Lielmezs equations (see Table 1).
Fig. 2. Enthalpy of vaporization as a function of temperature. Points are experimental data [1,8], solid lines are the plots of Eq. (5) with the corresponding values of Tc , Tt , and Ht .
A.G. Meyra et al. / Fluid Phase Equilibria 218 (2004) 205–207
207
[4,5] give also good results but with greater dispersions than Eq. (5). Faizullin [14] has recently suggested a new correlation for non-associated liquids but in his equation it is necessary to know the liquid volume on the saturation line. As a future work it would be very interesting to find a thermodynamic foundation of the empirical equation presented here. List of symbols H enthalpy of vaporization T temperature Boyle temperature TB Tc critical temperature triple point temperature Tt Zc universal critical ratio Fig. 3. Enthalpy of vaporization of mercury. Points are experimental [9], and the solid line is the prediction of the empirical equation (5).
In view of the behavior of the function H(T) we went a little further and used Eq. (5) to predict the EV at the triple point of, e.g., butane and iron. The critical and triple point temperatures of butane are Tc = 425.16 K [11] and Tt = 134.86 K [12], respectively. By using Eq. (5) for the known data [11] T = 173.15 K and H(173.15 K) = 26.88 kJ/mol, we obtain Ht = 28.06 kJ/mol for EV at the triple point. Proceeding in the same way for the case of iron, we know an estimate of the critical temperature Tc = 10000 K [13], and from the periodic table of elements, the boiling temperature Tb = 3023 K, the EV at the boiling point Hb = 349 kJ/mol, and the temperature of fusion Tf = 1802.2 K. With a good approximation we supposed that Tt ∼ = Tf . By elimination of Ht in Eq. (5), applied at T = Tb , we obtained Ht = 366.6 kJ/mol, a reasonable value when compared with Hb . 3. Discussion In this paper we present an empirical equation for the enthalpy of vaporization of the liquid–vapour phase transition of a pure substance. Though the functional form of this equation is not usual for representing thermodynamic quantities, it has the outstanding characteristic of not having adjustable parameters. It has as inputs Tc , Tt , and Ht , which characterize the substances and the universal critical ratio Zc . In the dimensionless representation h(t) is a unique function for all the substances studied (Fig. 1). The equation does not represent any particular real fluid but all of them simultaneously. With dimensions, H(T) gives a curve for each substance (Figs. 2 and 3) with very good correlation. This equation is novel and of practical use and can be applied over the whole coexistence region. Empirical equations for the enthalpy of vaporization previously proposed
Greek letter ε energy parameter of the Lennard–Jones potential
Acknowledgements This work was partially supported by Universidad Nacional de La Plata and by the Agencia Nacional de Promoción Cient´ıfica y Tecnológica (ANPCyT) PICT 03-4517. GJZ is member of ‘Carrera del Investigador Cient´ıfico’ CICPBA.
References [1] J.S. Rowlinson, F.L. Swinton, Liquids and Liquid Mixtures, Butterworth Scientific, London, 1981, pp. 50–51. [2] D. Ambrose, in: M.L. McGlashan (Ed.), Chemical Thermodynamics, vol. 1, The Chemical Society, Burlington House, London, 1973. [3] E.A. Guggenheim, J. Chem. Phys. 13 (1945) 253–261. [4] K.M. Watson, Ind. Eng. Chem. 35 (1943) 398–406. [5] L.W. Fish, J. Lielmezs, Ind. Eng. Chem. Fundam. 14 (1975) 248–256. [6] L.D. Landau, E.M. Lifshitz, Fisica Estadistica, Reverte, Barcelona, 1969. [7] J.C. Slater, Introduction to Chemical Physics, McGraw-Hill, New York, 1939, p. 189. [8] S. Angus, K.M. de Reuck, B. Armstrong (Eds.), International Thermodynamic Tables of the Fluid State, vols. 6 and 7, Pergamon Press, Oxford, 1979. [9] R.C. Weast (Ed.), Handbook of Chemistry and Physics, The Chemical Rubber Co., Ohio, 1972–1973. [10] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964, p. 247. [11] American Society of Heating, Refrigerating and Air Conditionating (Ed.), Ashrae Handbook of Fundamentals, American Society of Heating, Refrigerating and Air Conditionating, Atlanta, 2001, p. 20.43. [12] H. Hausen (Ed.), Landolt-Börnstein, Zahlenwerte und funktionen, 4 Teil, Bandteil C, Berlin, 1980, p. 825. [13] A.R. Ubbelohde, The Molten State of Matter, Wiley, Chichester, 1978, p. 24. [14] M.Z. Faizullin, Fluid Phase Equilibria 211 (2003) 75–83.