Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX
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Modified Trouton’s Rule for the Estimation, Correlation, and Evaluation of Pure-Component Vapor Pressure Paul M. Mathias,* Garry Jacobs, and Jesus Cabrera Fluor Corporation, 3 Polaris Way, Aliso Viejo, California 92698, United States ABSTRACT: Insights from the venerable Trouton’s Rule have been used to guide the development of an applied-thermodynamic method for the estimation, correlation, and evaluation of pure-component vapor pressure. Trouton’s Rule very simply and succinctly states that the entropy of vaporization of fluids at their normal boiling point is a constant (≈10.5 times the gas constant). Detailed evaluation of the data for many families of chemical compounds reveals the subtle patterns of departures from the rule, and facilitates the development of a useful new correlation. Several examples are presented to demonstrate the value of the new correlation to estimate, correlate, extrapolate, and evaluate vapor-pressure data, and to understand the patterns of vapor-pressure behavior. The methodology provides a guide for the development of thermodynamic correlations, and the resulting correlations are expected to be useful for the practice of applied thermodynamics.
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INTRODUCTION The vapor pressure of pure fluids is an extremely important physical property, and extensive efforts in applied thermodynamics have been devoted to developing methods for the analysis, correlation and estimation of vapor pressure. Many of these approaches have used the Clausius−Clapeyron equation, which provides an approximate relationship between the temperature derivative of the vapor pressure and the enthalpy of vaporization.1 The Gibbs−Helmholtz equation provides a similar relationship between the temperature derivative of the saturated fugacity and the liquid enthalpy departure.2 Trouton’s Rule3,4 is an empirical and classical observation that relates the enthalpy of vaporization of fluids to the temperature of vapor−liquid phase change. In this paper, we combine the insights from the Clausius−Clapeyron equation, the Gibbs−Helmholtz equation, and Trouton’s Rule to develop a practical vapor-pressure correlation procedure that has a small number of parameters, and demonstrate its utility. Today, chemical technologists have access to a large number of electronic databases and correlations,5 and these include: NIST,6 NIST WebBook,7 DIPPR,8 PPDS,9 DETHERM ··· on the WEB,10 and the Korean Thermophysical Properties Data Bank.11 The databases provide a powerful way to evaluate and develop property models, as is demonstrated in this paper through the use of the DIPPR database.8 But databases must also be tested and evaluated, and an effective way to do this is visually and through intuitive relationships.12 Also, engineering property correlations are improved when based upon intuitive understanding in addition to rigorous theoretical and scientific relationships. Here we demonstrate the development and application of a robust and intuitive correlation based upon Trouton’s Rule. © XXXX American Chemical Society
While today we have access to extensive databases, many new compounds (e.g., pharmaceutical active ingredients, new chemicals, degradation compounds, etc.) have only minimal data, often just a single vapor pressure and a liquid density. Even if additional data are available, they must be evaluated and tested. Correlations with just a few adjustable parameters enable a useful technique to evaluate and extrapolate data, and we also demonstrate this capability of the modified Trouton’s Rule correlation presented in this paper. The quantitative analyses in this paper have been executed using Solver in Excel.
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THERMODYNAMICS
The goal of this work is to develop a useful correlation by exploiting the relationship between vapor pressure and various enthalpies−which must be unambiguously defined and objectively chosen. We begin by discussing exact and approximate relationships from thermodynamics and applying them to data for representative substances. The Clausius equation is an exact thermodynamic equation that relates the temperature derivative of the vapor pressure to the enthalpy of vaporization and the difference between the saturated vapor and liquid molar volumes. dP S ΔHVL = dT T ΔV VL
(1)
Special Issue: In Honor of Cor Peters Received: August 29, 2017 Accepted: December 5, 2017
A
DOI: 10.1021/acs.jced.7b00767 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
Article
where, PS is the vapor pressure at temperature T, ΔHVL is the enthalpy of vaporization at T, and ΔVVL is the difference between the vapor and liquid saturated molar volumes at T. The Clausius equation is valid between any two phases in equilibrium (e.g., liquid−solid and vapor−solid), and eq 1 only presents the vapor−liquid relationship. At relatively low temperatures (usually below the normal boiling point), the vapor molar volume is much larger than the liquid molar volume, and the vapor molar volume may be approximated by the ideal-gas equation. Under these approximations, eq 1 reduces to the simplified Clausius−Clapeyron equation that is presented in most thermodynamics textbooks. dln(P S) ΔHVL − CC =− d(1/T ) R
(2)
where, ΔHVL−CC is the Clausius−Clapeyron approximation to the enthalpy of vaporization. Mathias and O’Connell2 used the Gibbs−Helmholtz equation to derive the exact thermodynamic relationship between the saturated liquid enthalpy and other thermodynamic properties. ΔH L = R
d ln(f S ) V L dP S − d(1/T ) T d(1/T )
ΔHL ≡ f ‐term − P ‐term
Figure 1. Various enthalpies of propane calculated by a highly accurate equation of state.14 The double line is the enthalpy of vaporization (eq 1), the solid line in the negative of the liquid enthalpy departure (eq 3), the dashed line is the negative of the f-term (eq 4), the dotted line is the approximate enthalpy of vaporization (Clausius−Clapeyron equation, eq 2), and the heavy double line is the P-term (eq 4). The points are accepted enthalpy of vaporization experimental data from the DIPPR database.
(3)
critical temperature, the magnitude of the liquid enthalpy departure decreases strongly, and this is because of the sharp change in the P-term. Figure 1 also presents enthalpy of vaporization data from DIPPR,8 and it is clear that the equation of state provides excellent agreement with experimental data. Most nonpolar and polar substances show behavior qualitatively equivalent to that of propane in Figure 1. For example, Figure 2 presents analogous calculations for water, and it is
(4)
where ΔHL is the saturated-liquid enthalpy departure, VL is the saturated liquid molar volume, f is the fugacity of the component (equal in the vapor and liquid phases), and the superscript s refers to the saturated condition (i.e., vapor and liquid in equilibrium). Mathias and O’Connell called the two terms on the right side of eq 3 the f-term and the P-term, and we retain these names. Equation 3 is the pure-component version of the general multicomponent equation, and Mathias13 showed that it can be used to derive the Clausius eq (eq 1). An approximation of eq 3 where the P-term is negligible (liquid molar volume is small and the temperature derivative of PS is also small) and f (the fugacity) is approximately equal to the pressure gives, d ln(P S) ΔHL ≈ R d(1/T )
(5)
Equations 2 and 5 indicate that the enthalpy of vaporization and the negative of the saturated liquid enthalpy departure have the same limiting value at low temperatures if the saturated liquid molar volume is small relative to the saturated vapor volume and the saturated vapor follows the ideal-gas law. We now present applications of these equations to representative chemicals to qualitatively and semi-quantitatively demonstrate the relative values. Figure 1 presents various enthalpies of propane calculated by a highly accurate equation of state.14 Four of these different enthalpies (excluding the P-term) have effectively >the same absolute value at temperatures below about 200 K. At the normal boiling temperature (231.1 K) the approximate enthalpy of vaporization (from the Clausius− Clapeyron equation, eq 2) has a higher magnitude than the other three enthalpies, but the differences are still quite small, and the differences among the four enthalpies increase as the temperature is further raised. At the critical temperature (369.8 K), the exact enthalpy of vaporization goes to zero while the other three enthalpies remain finite. At temperatures approaching the
Figure 2. Various enthalpies of water calculated by a highly accurate equation of state.14 The double line is the enthalpy of vaporization (eq 1), the solid line in the negative of the liquid enthalpy departure (eq 3), the dashed line is the negative of the f-term (eq 4), the dotted line is the approximate enthalpy of vaporization (Clausius−Clapeyron equation, eq 2), and the heavy double line is the P-term (eq 4).
clear that the charts for these two quite different substances are qualitatively equivalent; in particular, eqs 1, 2, and 3 give quantitatively equivalent results at low temperatures. Figure 3 presents various enthalpies of acetic acid, which, it should be noted, exhibits strong vapor-phase association, calculated using B
DOI: 10.1021/acs.jced.7b00767 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
Article
Figure 4. Comparison between the enthalpy of vaporization (DIPPR correlation, eq 1) and the approximate enthalpy of vaporization based upon the Clausius−Clapeyron eq (eq 2) for several families of compounds. The solid, dashed, and dotted lines indicate parity, +10% and +30%.
Figure 3. Various enthalpies of acetic acid calculated using a dimerization model15 and DIPPR correlations.8 The legend entries are similar to those in Figure 2. No P-term is shown. The points are accepted enthalpy of vaporization experimental data from the DIPPR database.
of ΔHVL. In the remainder of this paper we focus on the relationship between vapor pressure and ΔHVL‑CC.
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the vapor-phase association model of Togeas15 and DIPPR correlations. The vapor pressure, saturated liquid density, and enthalpy of vaporization were taken from DIPPR correlations and the vapor-phase fugacity was calculated using the dimerization model presented by Togeas.15 Figure 3 is radically different than Figure 1 and Figure 2 since the results from eqs 1 and 3 are similar, but eq 2 gives significantly different results even at low temperatures. The f-term provides a reasonable approximation to the liquid enthalpy departure at low temperatures because the contribution of the P-term is small, but both these enthalpies are considerably larger in magnitude than the approximate enthalpy of vaporization (eq 2) and the exact enthalpy of vaporization (eq 1). The enthalpy of vaporization based upon the Clausius−Clapeyron eq (eq 2) is larger than the exact enthalpy of vaporization by 70−80%, and the latter agrees well with experimental data. It should be noted that the vapor-phase fugacity calculated by association is not accurate at high pressures and hence the quantities calculated from eqs 3 and 4 are only qualitatively correct at high temperatures, say above the normal boiling point (393.1 K). Figure 3 is representative of the complex behavior displayed by substances exhibiting vaporphase association. The goal of this paper is to develop a vapor−pressure correlation by exploiting Trouton’s Rule, which postulates a relationship between the normal boiling point and the enthalpy of vaporization at the normal boiling point. We chose to use the enthalpy of vaporization based upon the temperature derivative of vapor pressure (eq 2 or eq 5) since this provides a simple, direct relationship. Figure 4 shows the relationship between ΔHVL and ΔHVL‑CC for compounds from several families, both calculated at their respective normal boiling points. It appears that there is a systematic relationship between ΔHVL and ΔHVL‑CC. At low values of ΔHVL (
