Similarity law for Widom lines and coexistence lines

The coexistence line of a fluid separates liquid and gaseous states at subcritical pressures, ending at the critical point. Only recently, it became clear that the supercritical state space can likewise be divided into regions with liquidlike and gaslike properties, separated by an extension to the coexistence line. This crossover line is commonly referred to as the Widom line, and is characterized by large changes in density or enthalpy, manifesting as maxima in the thermodynamic response functions. Thus, a reliable representation of the coexistence line and the Widom line is important for sub- and supercritical applications that depend on an accurate prediction of fluid properties. While it is known for subcritical pressures that nondimensionalization with the respective species critical pressures $p_{\mathrm{cr}}$ and temperatures $T_{\mathrm{cr}}$ only collapses coexistence line data for simple fluids, this approach is used for Widom lines of all fluids. However, we show here that the Widom line does not adhere to the corresponding states principle, but instead to the extended corresponding states principle. We resolve this problem in two steps. First, we propose a Widom line functional based on the Clapeyron equation and derive an analytical, species specific expression for the only parameter from the Soave-Redlich-Kwong equation of state. This parameter is a function of the acentric factor $\omega$ and compares well with experimental data. Second, we introduce the scaled reduced pressure $p_{\mathrm{r}}^{*}$ to replace the previously used reduced pressure $p_{\mathrm{r}}=p/p_{\mathrm{cr}}$. We show that $p_{\mathrm{r}}^{*}$ is a function of the acentric factor only and can thus be readily determined from fluid property tables. It collapses both subcritical coexistence line and supercritical Widom line data over a wide range of species with acentric factors ranging from $-0.38$ (helium) to 0.34 (water), including alkanes up to n-hexane. By using $p_{\mathrm{r}}^{*}$, the extended corresponding states principle can be applied within corresponding states principle formalism. Furthermore, $p_{\mathrm{r}}^{*}$ provides a theoretical foundation to compare Widom lines of different fluids.

Figure 1

Thermodynamic state plane and supercritical state structure. The Widom line is an extension to the coexistence line at supercritical pressure, defined as the locus of maximum isobaric heat capacity. It is a marker of the crossover between supercritical liquidlike and gaslike states. The contour represents the density distribution, showing the sharp transition at subcritical pressure and a smooth crossover at supercritical pressure. The dashed line at $Z=pv/\left(RT\right)=0.95$ denotes the transition to an ideal gas. The reduced density ${\rho }_{\mathrm{r}}$ is $\rho /{\rho }_{\mathrm{cr}}, c_{p,\mathrm{r}}=c_p/c_{p,\mathrm{iG}},c_{p,\mathrm{iG}}=\gamma R/(\gamma -1), R$ is the gas constant, and $\gamma$ the isentropic exponent. Data for oxygen is from NIST [22].

Figure 2

Comparison of Widom line equations from Gorelli et al. [8], Arias-Zugasti et al. [24], and Banuti [16] with heat capacity peaks from NIST [22] for oxygen, nitrogen, water, and hydrogen. The relations of Gorelli et al. and Banuti are targeted towards capturing nitrogen and oxygen data (open symbols). None of the relations matches hydrogen or water (closed symbols) data; furthermore, data of different species do not collapse when plotted as functions of the reduced pressure and the reduced temperature.

Figure 3

Comparison of the slope of the Widom line $A_s$ as a function of the acentric factor $\omega$, obtained from the Soave-Redlich-Kwong equation of state (19) (line) and NIST reference data [22] (symbols).

Figure 4

Species specific Widom line from Eq. (9) and Table 1 for oxygen, hydrogen, and water.

Figure 5

Collapse of (a) coexistence lines and (b) Widom lines from NIST data when using $p_{\mathrm{r}}^{*}$ of Eq. (21) instead of $p_{\mathrm{r}}$. The $p_{\mathrm{r}}^{*}$ correlation is given by Eq. (22).