Equations of State for n-Hexadecane and n-Docosane

Equations of state for n-hexadecane (C16H34) and n-docosane (C22H46) have been developed as functions of the Helmholtz energy with independent variables of temperature and density. The equations were developed based on experimental values of density, speed of sound, isobaric heat capacity, and vapor pressure. With these equations, all thermodynamic properties of n-hexadecane and n-docosane can be calculated. For n-hexadecane, the uncertainty in vapor pressure is 0.5 %. The uncertainty of the saturated liquid density is 0.05 % from the triple point up to 400 K, and 0.2 % at higher temperatures. The uncertainty in densities is within 0.5 %. The speed of sound and isobaric heat capacity can be calculated within 0.25 %. The uncertainties of the properties calculated with the equation for n-docosane are 5 % for vapor pressure, 0.1 % for saturated liquid density, 1 % for density, 0.5 % and 1 % for speed of sound at atmospheric pressure and higher pressures, respectively, and within 3 % for heat capacity.


Introduction
n-Hexadecane (C 16 H 34 ) and n-docosane (C 22 H 46 ) are normal alkanes of interest in the petroleum industry for multiple applications. They are generally used as constituents in mixtures for fuel, especially for aviation. In this work, fundamental equations of state, in terms of the Helmholtz energy, are presented for both liquids. The equations of state are valid over the whole fluid region, and through them all thermodynamic properties can be calculated. Measurements of vapor pressure, density, speed of sound, and heat capacity are available in the literature for both states. The ancillary equations were developed by fitting calculated values of the saturation states (determined with the application of the Maxwell criteria applied to the equations of state) [87].

Ancillary Equations for n-hexadecane
The ancillary form for the vapor pressure

Equation of State
The form of the equation of state presented in this work is based on the Helmholtz energy as a function of density and temperature a(ρ,T), which is the most commonly used form for the calculation of thermodynamic properties of pure fluids and mixtures with low uncertainties. All thermodynamic properties can be estimated as derivatives of the Helmholtz energy, for example, pressure is calculated as The derivatives of the Helmholtz energy required to calculate other thermodynamic properties not reported are given elsewhere, e.g., [88].
The functional form for the reduced Helmholtz energy α, as a function of the dimensionless density and temperature, is where δ = ρ/ρ c , τ = T c /T, and R is the molar gas constant equal to 8.314462618 J·mol −1 ·K −1 [89].
The reduced Helmholtz energy α is the contribution from the ideal gas contribution α 0 , which represents the ideal gas properties, and the residual or real Helmholtz energy α r that accounts for the interactions between molecules.

Properties of the Ideal Gas
The ideal gas Helmholtz energy has the following form [90]: where the coefficients for n-hexadecane and n-docosane are given in Tables 1 and 2, respectively. In order to calculate thermodynamics properties, a model for the ideal gas isobaric heat capacity is necessary. The expression for the ideal gas isobaric heat capacity c p 0 used in this work that is required to derive the ideal gas Helmholtz energy is The u k coefficients contained in the Einstein functions used in this equation give the proper shape of the ideal gas heat capacity similar to that derived from statistical mechanical models.
The values of a k , v k , and u k are given in Table 1 for n-hexadecane and in Table 2 for n-docosane.

Properties of the Real Gas
The functional form often used till about the year 2000 for the residual Helmholtz energy equation was A form containing additional Gaussian bell-shaped terms is now typically used, including those for n-hexadecane and n-docosane in this work, and is expressed as where the coefficients and exponents are reported in Table 3 for n-hexadecane and in Table 4 for n-docosane and the values of k 1 , k 2 , and k 3 for the equations in this work are 5, 10, and 15. The Gaussian terms are useful in the determination of the fluid properties in the critical region.

Fitting Constraints
In order to develop the equations of state, several constraints were used to control the shape of the thermodynamic surface. In Table 5, a list of the main constraints for n-hexadecane is given. The table shows the properties for which a constraint was needed, the kind of constraint imposed, and the range over which it was applied.
Because the thermodynamic behavior of n-hexadecane and n-docosane is, for the most part, the same when viewed on a reduced basis (such as with τ and δ), the constraints used for n-hexadecane can be applied directly to the fitting of n-docosane after the temperature and density ranges of the constraints have been properly adjusted, and thus they are not reported here.
As an example, the rectilinear diameter (the average of the vapor and liquid saturated densities) was constrained to be linear by imposing zero curvature from 650 K

Fig. 1
Temperature as a function of density for n-hexadecane along several isobars: the straight line is the rectilinear diameter to the critical temperature; it results in the behavior shown in Fig. 1. In preliminary fits, the fourth virial coefficient was negative around 1500 K to 2500 K, thus it was forced to give positive values in the temperature range between 1300 K and 3000 K, as presented in Fig. 2.
The phase identification parameter (PIP) [91] required many constraints. An example is the constraint acting on the isobar at 1 MPa over the temperature range from 200 K to 420 K that forced the slope, curvature, and third and fourth derivatives to all be positive. A further constraint was used to obtain negative curvature Phase identification parameter (PIP) as a function of density for n-hexadecane along isotherms from 100 K to 1000 K of the isotherm at 1000 K between 1.08 mol·dm −3 and 1.85 mol·dm −3 (see Fig. 3). More details about the phase identification parameter and the fourth virial coefficient are reported in the following section.

Comparison to Experimental Data
The experimental data used for this work are summarized in Table 6 for n-hexadecane and in Table 7 for n-docosane. The experimental data were converted to kelvins (ITS-90) for temperatures, megapascals for pressures, and moles per cubic decimeter for densities.
In order to estimate the uncertainties of the equations of state, all available experimental data are considered. The uncertainties are estimates of the combined expanded uncertainties with a coverage factor of two. The accuracy of the equations of state were determined by statistical comparisons between the properties calculated from the equations of state and the experimental values. Tables 6 and 7 show the average absolute deviations (AAD) for any generic property X, as follows: where n is the number of data and ΔX is defined as

Comparison to n-hexadecane Experimental Data
In The deviations of saturated liquid densities for n-hexadecane are shown in Fig. 5. Most of the data have deviations lower than 0.1 % between 273 K and 373 K. In the temperature range between 303 K and 373 K, the data of Bolotnikov et al. [15], Prak et al. [63,64] show deviations lower than 0.05 %. The measurements of Plebanski et al. [62] were performed over a wide range of temperature (from about 300 K to 490 K), with a maximum deviation of 0.07 % from the equation and better than 0.02 % between 340 K and 420 K. All of the density data available are represented by the equation with deviations lower than about 1.5 % and most are within 0.3 %, as shown in Fig. 6. The data of Banipal et al. [10] are represented by the equation within 0.1 % for temperatures between 318 K and 373 K, and at pressures up to 10 MPa. At ambient pressure, the deviation is reduced to 0.05 %. The deviations of the data of Snyder and Winnick [70] are below 0.1 % at 298 K. Between 0.6 MPa and 28 MPa, the data are within 0.1 %. The measurements performed by Outcalt et al. [54] agree with the equation of state within 0.2 %, but the deviations are consistently negative.
The deviations in speed of sound data from the equation are shown in Fig. 7. The only available measurements at temperatures higher than 473 K are those in the work of Neruchev et al. [52]. The data show good agreement with the equation and are less than 0.5 % for temperatures up to 693 K. At pressures between 10 MPa and  For the isobaric heat capacity, there are fewer experimental data than those of the properties previously discussed. The measurements, shown in Fig. 8, are within 6 % from the equation. The work with the largest number of measurements is that of Banipal et al. [10]. They are well represented by the equation of state; the deviations are not higher than 0.3 % in the temperature range from 318 K to 373 K. At 298 K, the measurements of Benson et al. [12] deviate from the equation by less than 0.02 %. The deviations of their data increase to 0.5 % for higher temperatures (up to 358 K), and all of them are negative.

Comparison to n-docosane Experimental Data
For n-docosane, there are significantly fewer data points than for n-hexadecane. The vapor pressure is the most studied property for this fluid, covering a wide range of temperature (see Fig. 9). The data set of Morgan and Kobayashi [49] is best represented by the equation at high temperatures, between 453 K and 573 K, with an average deviation of 0.5 %. The data given in Sasse et al. [67] show different behaviors depending on the temperature. Between 393 K and 423 K, the data spread about the equation within 2 %. Below 393 K, they all show deviations between − 4 % and − 8 %. Above 423 K the data differ with the equation consistently around 2 %. The data of Chickos and Hanshaw [19] show deviations consistently around − 2 % at temperatures higher than 450 K, while the data and equation differ more as the temperature decreases.
All the experimental saturated liquid densities deviate from the equation of state within ± 0.2 %, as shown in Fig. 10. The measurements of Dutour et al. [23] deviate   Peters et al. [57] published the only pvT data reported for n-docosane, with temperatures from 323 K to 368 K and pressures up to 16 MPa. As shown in Fig. 11, the data show a systematic average deviation of − 1 % from the equation of state. The data at pressures near ambient differ from the saturated liquid density data by this amount. The data show a consistent offset from the equation at all pressures.
The measurements of Neruchev et al. [53] and Dutour et al. [23] are the only available data for n-docosane for the speed of sound as shown in Fig. 12. Between 353 K and 373 K the two sets overlap, and the deviations from the equation and For the isobaric heat capacity, just two sets of data are available: Atkinson et al. [6] and Durupt et al. [22]. The two data sets show substantial differences at the higher temperatures as shown in Fig. 13

Virial Coefficients
One of the most important validation tests in the development of equations of state comes from the analysis of its virial coefficients.
In Fig. 2, a plot of the second, third, and fourth virial coefficients is given as a function of temperature. The third virial coefficient C for n-hexadecane has a maximum value of 1.849 dm 6 ·mol −2 at 486.3 K, and is always positive above a temperature of 367.14 K. The maximum value of the fourth virial coefficient D is 1.120 dm 9 ·mol −3 at 489.3 K, has a minimum positive value at 1304 K, and a second maximum around 7536 K (for any temperature between 1304 K and above). These values and the shape of the function were achieved through the use of the constraints reported in Table 5 and explained in the section "Fitting constraints."

Extrapolation Behavior
The REFPROP software [92] was used to generate diagrams for inspection of the extrapolation behavior. One of the most important parameters used to verify the correct behavior of the equations is the phase identification parameter (PIP) defined in the work of Venkatarathnam and Oellrich [90]; this is an extremely sensitive property from which small inconsistencies can be seen that are not visible with other properties.
In Fig. 14, the PIP for n-hexadecane as a function of temperature is shown, along isobars and at saturation states. The plot shows positive curvature in the PIP over most of the liquid region, except for the region between 150 K and about 200 K. At very low temperatures (below 20 K), the curvature is still negative. This behavior of the PIP is also present in the plots for R-1234ze(E) reported in the work of Thol and Lemmon [93] and for R-245ca reported in Zhou and Lemmon [94].
Similar behavior is observed in the plot of the PIP versus temperature for n-docosane, shown in Fig. 15. The curvature is almost always positive over the liquid region, although it becomes slightly negative between 175 K and 205 K. The PIP for n-docosane, even at temperatures lower than 15 K, has positive curvature.
The validation of the equations can also be done by checking the behavior of other properties such as density, speed of sound, heat capacity, and the ideal curves. All of these properties show the expected trends as explained in other publications on equations of state, such as the work for the equation of state of propane [88] or R-125 [95].
The ideal curves are curves along which one property of a real fluid is equal to the hypothetical ideal gas. This definition can refer to any property, but usually the ideal curves of the compressibility factor, Z(T, ρ), and its derivatives are only considered, as follows: Ideal curve: Boyle curve: Joule-Thomson curve: Joule inversion curve: The ideal curves are used to judge the behavior of the equation of state. Even if the curves do not provide numerical information, reasonable shapes of the curves, such as the plots for propane [88], indicate correct extrapolation behavior of the   Figures 16 and 17 show the plots of the ideal curves obtained for n-hexadecane and n-docosane, which have the expected shape, giving confidence in the proper extrapolation behavior at high temperatures and pressures.

Conclusions
Fundamental equations of state for n-hexadecane and n-docosane were developed and presented here. The equations can be used to calculate all the thermodynamic properties of these alkanes over the entire fluid region.
For n-hexadecane, the uncertainty in vapor pressure is 0.5 %. For saturated liquid density, the uncertainty is 0.05 % for temperatures up to 400 K and increases to 0.2 % at higher temperatures. The estimated uncertainty in densities is 0.1 % from the triple point to 450 K for pressures below 50 MPa. Outside this range, the uncertainty is 0.5 %. The speed of sound has an uncertainty of 0.25 %. The uncertainty in isobaric heat capacity is estimated to be 0.25 %.
For n-docosane, the uncertainty in vapor pressure is about 5 % and the uncertainty in saturated liquid density is 0.1 %. At pressures up to 20 MPa, the uncertainty in density is about 1 %; no estimation can be provided at higher pressures. For speed of sound, the uncertainty is less than 0.5 % at ambient pressure and increases to 1 % at higher pressures. The uncertainty in heat capacity is 3 %.

Supplementary Information
The files containing the parameters of the equations of state for n-hexadecane and n-docosane are available for the use in REFPROP [92], TREND [96], and Cool-Prop [97].