Appendix 1. Thermodynamic potentials of humid air
A1.1 Basic remarks
In the description of geophysical thermodynamic properties, the International Thermodynamic Equation of Seawater (TEOS-10) is the first international standard that is rigorously based on thermodynamic potentials in an axiomatic way (Feistel et al. 2016a, b; Feistel 2018). All required properties can be defined mathematically in terms of only three empirical functions, plus some general constants such as molar masses or the gas constant. This approach permits an analysis of internal relations between quantities that otherwise are often formulated from apparently independent experimental data. Here we express several fundamental quantities of atmospheric physics in terms of the TEOS-10 thermodynamic potentials.
A1.2 General thermodynamic relations
Thermodynamic properties of humid air, including its saturation states, can be derived from three analytical empirical functions, the so-called thermodynamic potentials of humid air, of liquid water and of ambient ice Ih. In the framework of TEOS-10, these functionsFootnote 2 are, respectively,
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(i) the specific Helmholtz energy, fAV(A,T,ϱAV), of humid air as a function of the dry-air mass fraction, A, the temperature, T, on the International Temperature Scale 1990 (ITS-90), and the mass density, ϱAV;
-
(ii) the specific Helmholtz energy, fW(T,ϱW), of liquid water as a function of the ITS-90 temperature, T, and the mass density of liquid water, ϱW (Wagner and Pruß 2002; IAPWS R6-95 2016) (known as “IAPWS-95”);
-
(iii) the specific Gibbs energy, gIh(T,p), of hexagonal ice I as a function of the ITS-90 temperature T, and the pressure, p (Feistel and Wagner 2006; IAPWS R10-06 2009).
In combination with air–water cross-virial coefficients (see Hyland and Wexler 1983a, 1983a, Harvey and Huang 2007, Feistel et al. 2010a) this set of thermodynamic potentials is used as the primary standard for pure water (liquid, vapor, and solid), seawater and humid air from which all other properties are derived by mathematical operations, i.e., without the need for additional empirical functions. For meteorological applications, the TEOS-10 Helmholtz function of humid air, fAV, may be approximated with sufficient accuracy by virial coefficients (Feistel et al. 2015a, Eq. (3) and references therein):
$$ f_{\text{AV}} \left( A,T,{\varrho}_{\text{AV}} \right) =(1-A) f_{\mathrm{V}} \left( T, {\varrho}_{\mathrm{V}} \right) + A f_{\mathrm{A}} \left( T,{\varrho}_{\mathrm{A}} \right) + f_{\text{mix}} \left( A,T,{\varrho}_{\text{AV}} \right) . $$
(A1.1)
The partial mass densities of the vapor and the air, respectively, are ϱV = (1 − A)ϱAV and ϱA = AϱAV, their related molar densities are \(\widetilde {{\varrho }}_{\mathrm {V}}={\varrho }_{\mathrm {V}}/M_{\mathrm {W}}\) and \(\widetilde {{\varrho }}_{\mathrm {A}}={\varrho }_{\mathrm {A}}/M_{\mathrm {A}}\), their molar masses are MW and MA, their specific gas constants are RW = R/MW and RA = R/MA, and R is the molar gas constant. According to Eq. (A1.1), the function \(f_{\text {AV}} \left (A,T,{\varrho }_{\text {AV}} \right )\) is composed of a dry-air part given by the specific Helmholtz energy for dry air, \(f_{\mathrm {A}} \left (T,{\varrho }_{\mathrm {A}} \right )\) Lemmon et al. (2000), of a water-vapor part given by the specific Helmholtz energy for water vapor, \(f_{\mathrm {V}} \left (T,{\varrho }_{\mathrm {V}} \right )\), which is defined by the thermal equation of state of fluid water (Wagner and Pruß 2002, IAPWS-95), and of a part describing the contribution of air–vapor interactions to the mixture properties, \(f_{\text {mix}} \left (A,T,{\varrho }_{\text {AV}} \right )\) (Feistel et al. 2015a, Eqs. (14), (15), (16), and references therein):
$$ \begin{array}{@{}rcl@{}} f_{\mathrm{V}}(T,{\varrho}_{\mathrm{V}}) & = & f_{\text{V,0}}(T) \\ & & + R_{\mathrm{W}} T \left[ \ln \frac{\widetilde{{\varrho}}_{\mathrm{V}}}{\widetilde{{\varrho}}_{0}} + B_{\text{WW}}(T) \widetilde{{\varrho}}_{\mathrm{V}} + \frac{1}{2} C_{\text{WWW}}(T) \left( \widetilde{{\varrho}}_{\mathrm{V}} \right)^{2} \right] , \end{array} $$
(A1.2)
$$ \begin{array}{@{}rcl@{}} f_{\mathrm{A}}(T,{\varrho}_{\mathrm{A}}) & = & f_{\text{A,0}}(T) \\ & & + R_{\mathrm{A}} T \left[ \ln \frac{\widetilde{{\varrho}}_{\mathrm{A}}}{\widetilde{{\varrho}}_{0}} + B_{\text{AA}}(T) \widetilde{{\varrho}}_{\mathrm{A}} + \frac{1}{2} C_{\text{AAA}}(T) \left( \widetilde{{\varrho}}_{\mathrm{A}} \right)^{2} \right] , \end{array} $$
(A1.3)
$$ \begin{array}{@{}rcl@{}} f_{\text{mix}} \left( A,T,{\varrho}_{\text{AV}} \right) & = & \widetilde{{\varrho}}_{\mathrm{A}} \widetilde{{\varrho}}_{\mathrm{V}} \frac{RT}{{\varrho}_{\text{AV}}} \\ & & \times \left[ 2 B_{\text{AW}}(T) + \frac{3}{2} C_{\text{AWW}}(T) \widetilde{{\varrho}}_{\mathrm{V}} + \frac{3}{2} C_{\text{AAW}}(T) \widetilde{{\varrho}}_{\mathrm{A}} \right] . \end{array} $$
(A1.4)
The quantity \(\widetilde {{\varrho }}_{0}\) is an arbitrary reference molar density, such as \(\widetilde {{\varrho }}_{0}=1 \text {mol m}^{-3}\), introduced only to render the argument of the logarithm unitless. The functions fV,0(T) and fA,0(T) are relatively complicated mathematical expressions related to ideal-gas heat capacities; these functions are not relevant for the following derivations, and explicitly reporting them is refrained from here.
In Eqs. (A1.2)–(A1.4), BWW(T) and CWWW(T) denote the second and third molar virial coefficients (VCs) for water–water interactions, BAA(T) and CAAA(T) the second and third molar virial coefficients for air–air interactions, and BAW(T), CAWW(T), CAAW(T) the second and third molar cross-virial coefficients for air–water interactions. The second virial coefficients are given in units of the molar volume (m3mol− 1), and the third ones in units of the square of the molar volume (m6mol− 2). These seven functions are mutually independent and are sufficient to describe all thermodynamic real-gas corrections of humid air. It is the aim of this Appendix to provide rigorous relations for quantities of meteorological interest in terms of these seven VCs. Various empirical formulas for these VCs are available from the scientific literature; in Appendix 2 we shall focus on only those which are related the selected meteorological standard equations.
Expressing Eq. (A1.1) in terms of these seven VCs, the Helmholtz function of humid air reads:
$$ \begin{array}{@{}rcl@{}} f_{\text{AV}}(A,T,{\varrho}_{\text{AV}}) & = & f_{\text{AV,0}}(A,T) \\ & & + \frac{R T}{M_{\text{AV}}} \bigg[ x_{\mathrm{V}} \ln x_{\mathrm{V}} + (1-x_{\mathrm{V}}) \ln (1-x_{\mathrm{V}}) + \ln \frac{\widetilde{{\varrho}}_{\text{AV}}}{\widetilde{{\varrho}}_{0}} \\ & & + B(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}} + \frac{1}{2} C(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}}^{2} \bigg]. \end{array} $$
(A1.5)
In Eq. (A1.5) these seven VCs appear in two regular combinations of the so-called mixture virial coefficients B(xV,T) and C(xV,T) (Guggenheim 1950; Prausnitz et al. 1999, pp. 133–134, Eqs. (5.20), (5.22) therein):
$$ B(x_{\mathrm{V}},T) = x_{\mathrm{V}}^{2} B_{\text{WW}}(T) + 2 x_{\mathrm{V}} (1 - x_{\mathrm{V}}) B_{\text{AW}}(T) + (1 - x_{\mathrm{V}})^{2} B_{\text{AA}}(T) , $$
(A1.6)
$$ \begin{array}{@{}rcl@{}} C(x_{\mathrm{V}},T) & = & x_{\mathrm{V}}^{3} C_{\text{WWW}}(T) + 3 x_{\mathrm{V}}^{2} (1-x_{\mathrm{V}}) C_{\text{AWW}}(T)\\ & & + 3 x_{\mathrm{V}} (1-x_{\mathrm{V}})^{2} C_{\text{AAW}}(T) + (1-x_{\mathrm{V}})^{3} C_{\text{AAA}}(T) . \end{array} $$
(A1.7)
In Eqs. (A1.6) and (A1.7) the quantity xV is the mole fraction of water vapor in humid air,
$$ x_{\mathrm{V}} \equiv (1-A) \frac{M_{\text{AV}}(A)}{M_{\mathrm{W}}} , $$
(A1.8)
MAV is the mean molar mass of the binary gas mixture “humid air”,
$$ M_{\text{AV}}(A) \equiv \frac{{\varrho}_{\text{AV}}}{\widetilde{{\varrho}}_{\text{AV}}} = \left[ \frac{(1-A)}{M_{\mathrm{W}}} + \frac{A}{M_{\mathrm{A}}} \right]^{-1} , $$
(A1.9)
and
$$ f_{\text{AV,0}}(A,T) \equiv (1-A) f_{\text{V,0}}(T) + A f_{A,0}(T) $$
(A1.10)
is an abbreviation. Thermodynamic potentials may be mathematically transformed in various ways to be expressed in suitable independent variables (Alberty 2001). To replace the density argument of the Helmholtz function f(T,ϱ) by the more convenient pressure, by virtue of
$$ p = {\varrho}^{2} \left( \frac{\partial f}{\partial {\varrho}} \right)_{T} , $$
(A1.11)
a Gibbs function g(T,p) may be employed, which is obtained by the so-called Legendre transformation:
$$ g(T,p) = f + {\varrho} \left( \frac{\partial f}{\partial {\varrho}} \right)_{T} . $$
(A1.12)
The virial equation for the pressure of humid air is obtained from the density derivative, Eq. (A1.11), of the Helmholtz function, Eq. (A1.5):
$$ \begin{array}{@{}rcl@{}} p & = & {\varrho}^{2} \left( \frac{\partial f_{\text{AV}}}{\partial {\varrho}_{\text{AV}}} \right)_{T} \\ & = & \widetilde{{\varrho}}_{\text{AV}} R T \left[ 1 + B(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}} + C(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}}^{2} + D(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}}^{3} + {\ldots} \right] . \end{array} $$
(A1.13)
In TEOS-10, the Gibbs function of liquid water, gW(T,p), is computed numerically from fW(T,ϱW), but for low-pressure conditions an analytical version of gW is available (Feistel 2003; IAPWS SR7-09 2009). The virial equation for the Gibbs function of humid air related to the Helmholtz function, Eq. (A1.1), takes the form
$$ \begin{array}{@{}rcl@{}} g_{\text{AV}}(A,T,p) = g_{0}(A,T) + a &\bigg[& \ln \frac{p}{RT \widetilde{{\varrho}}_{0}} + b \left( \frac{p}{RT} \right) + \frac{1}{2} c \left( \frac{p}{RT} \right)^{2}\\ &&+ \frac{1}{3} d \left( \frac{p}{RT} \right)^{3} \bigg] +\mathcal{O} \left\{ \left( \frac{p}{RT} \right)^{4} \right\} . \end{array} $$
(A1.14)
The virial equation for the mass density of humid air is obtained from gAV(A,T,p) via
$$ \frac{1}{{\varrho}_{\text{AV}}} = \left( \frac{\partial g_{\text{AV}}}{\partial p} \right)_{A,T} = \frac{a}{p} \left[1 + b \left( \frac{p}{RT} \right) + c \left( \frac{p}{RT} \right)^{2} + d \left( \frac{p}{RT} \right)^{3} + {\ldots} \right] . $$
(A1.15)
Introducing the variable y = p/(RT), Eq. (A1.13) can be brought into the following form:
$$ y = \widetilde{{\varrho}}_{\text{AV}} \chi , \quad \chi = 1 + B \widetilde{{\varrho}}_{\text{AV}} + C \widetilde{{\varrho}}_{\text{AV}}^{2} + D \widetilde{{\varrho}}_{\text{AV}}^{3} . $$
(A1.16)
Analogously, Eq. (A1.15) assumes the following form:
$$ p = a M_{\text{AV}} \widetilde{{\varrho}}_{\text{AV}} (1 + b y + c y^{2} + d y^{3}) . $$
(A1.17)
Equating the pressures defined by Eqs. (A1.16) and (A1.17) yields:
$$ R T \widetilde{{\varrho}}_{\text{AV}} \chi = a M_{\text{AV}} \widetilde{{\varrho}}_{\text{AV}} \left( 1 + b y + c y^{2} + d y^{3} \right) . $$
(A1.18)
Inserting y from Eq. (A1.16) into the right-hand side of Eq. (A1.18), one arrives at the following relation:
$$ \chi = \epsilon \left( 1 + b \widetilde{{\varrho}}_{\text{AV}} \chi + c \widetilde{{\varrho}}_{\text{AV}}^{2} \chi^{2} + d \widetilde{{\varrho}}_{\text{AV}}^{3} \chi^{3} \right) , \quad \epsilon = \frac{a M_{\text{AV}}}{RT} $$
(A1.19)
$$ \begin{array}{@{}rcl@{}} \leadsto \quad 1 &+& B \widetilde{{\varrho}}_{\text{AV}} + C \widetilde{{\varrho}}_{\text{AV}}^{2} + D \widetilde{{\varrho}}_{\text{AV}}^{3} \\ &=& \epsilon + \epsilon b \widetilde{{\varrho}}_{\text{AV}} \left( 1 + B \widetilde{{\varrho}}_{\text{AV}} + C \widetilde{{\varrho}}_{\text{AV}}^{2} + D \widetilde{{\varrho}}_{\text{AV}}^{3} \right)\\ &+& \epsilon c \widetilde{{\varrho}}_{\text{AV}}^{2} \left( 1 + B \widetilde{{\varrho}}_{\text{AV}} + C \widetilde{{\varrho}}_{\text{AV}}^{2} + D \widetilde{{\varrho}}_{\text{AV}}^{3} \right)^{2}\\ &+& \epsilon d \widetilde{{\varrho}}_{\text{AV}}^{3} \left( 1 + B \widetilde{{\varrho}}_{\text{AV}} + C \widetilde{{\varrho}}_{\text{AV}}^{2} + D \widetilde{{\varrho}}_{\text{AV}}^{3} \right)^{3}\\ &=& \epsilon + \epsilon b \widetilde{{\varrho}}_{\text{AV}} + (\epsilon bB + \epsilon c) \widetilde{{\varrho}}_{\text{AV}}^{2} + (\epsilon bC + 2 \epsilon c B + \epsilon d) \widetilde{{\varrho}}_{\text{AV}}^{3}\\ &+& \mathcal{O} \left\{ \widetilde{{\varrho}}_{\text{AV}}^{4} \right\}\\ \leadsto \quad 0 &=& (1 - \epsilon) + (B -\epsilon b) \widetilde{{\varrho}}_{\text{AV}} + (C - \epsilon b B - \epsilon c) \widetilde{{\varrho}}_{\text{AV}}^{2}\\ &+& (D - \epsilon b C - 2 \epsilon c B - \epsilon d) \widetilde{{\varrho}}_{\text{AV}}^{3} + \mathcal{O} \left\{ \widetilde{{\varrho}}_{\text{AV}}^{4} \right\} . \end{array} $$
The satisfaction of the last relation in Eq. (A1.19) requires the fulfillment of the following relations between the coefficients of the pressure expansion, Eq. (A1.15), and the original VCs introduced by Eq. (A1.13):
$$ \begin{array}{@{}rcl@{}} 1 - \epsilon = 0 & \quad \leadsto & \quad \epsilon = 1 \quad \leadsto \quad a = \frac{RT}{M_{\text{AV}}} , \\ B - \varepsilon b = 0 & \quad \leadsto & \quad b = B , \\ C - \varepsilon bB - \varepsilon c = 0 & \quad \leadsto & \quad c = C - B^{2} , \\ D - \epsilon b C - 2 \epsilon c B - \epsilon d = 0 & \quad \leadsto & \quad d = D - 3BC + 2B^{3} . \end{array} $$
(A1.20)
The inverse relations read:
$$ B = b , \quad C= c+b^{2} , \quad D=d+3bc+b^{3} . $$
(A1.21)
Also, from Eq. (A1.12), g = f + p/ϱ, it follows that
$$ g_{0}(A,T) \equiv f_{\text{AV,0}}(T) + \frac{R T}{M_{\text{AV}}} \left[ 1 + x_{\mathrm{V}} \ln x_{\mathrm{V}} + (1- x_{\mathrm{V}}) \ln (1-x_{\mathrm{V}})\right] . $$
(A1.22)
By virtue of Eq. (A1.20) the virial equation for the Gibbs function of humid air, Eq. (A1.14), can be expressed in terms of the original VCs:
$$ \begin{array}{@{}rcl@{}} g_{\text{AV}}(A,T,p) & = & g_{0}(A,T) + \frac{RT}{M_{\text{AV}}} \bigg[ \ln \frac{p}{RT \widetilde{{\varrho}}_{0}} + B \left( \frac{p}{RT} \right) + \frac{1}{2} (C-B^{2}) \left( \frac{p}{RT} \right)^{2} \\ & & + \frac{1}{3} (D- 3 BC + 2 B^{3}) \left( \frac{p}{RT} \right)^{3} \bigg] + \mathcal{O} \left\{ \left( \frac{p}{RT} \right)^{4} \right\} . \end{array} $$
(A1.23)
Analogously, the virial equation for the mass density of humid air, Eq. (A1.15), can be rewritten as follows:
$$ \begin{array}{@{}rcl@{}} \frac{1}{{\varrho}_{\text{AV}}} & = & \frac{RT}{p M_{\text{AV}}} \bigg[1 + B \left( \frac{p}{RT} \right) + (C-B^{2}) \left( \frac{p}{RT} \right)^{2} \\ & & + (D - 3BC + 2B^{3}) \left( \frac{p}{RT} \right)^{3} + {\ldots} \bigg] . \end{array} $$
(A1.24)
A1.3 Virial form of thermodynamic properties
A1.3.1 Basic remarks
Virial coefficients are basic thermodynamic quantities which contain the full information about the real-gas behavior of a vapor or a vapor mixture. These coefficients represent a complete, self-consistent, and independent description of real gases, while thermodynamic derivables given in form of empirical or semiempirical functions do neither fulfill the criterion of independency from each other, nor the criterion of completeness. In other words, it is impossible to provide separate empirical formulae for such derivables, which do not lead to discrepancies in the corresponding VCs retrieved from these formulae.
The thermodynamic equilibrium conditions given by Eqs. (9) and (10) allow the determination of the mole fraction of water vapor in saturated humid air, \(x_{\text {V,sat}}^{\text {(c)}}\), and saturation pressure of water vapor, \(e_{\text {sat}}^{\text {(c)}}\), respectively. In TEOS-10, the chemical potentials of liquid water and ice, μW(T,p) and μIh(T,p), are available as functions of temperature and pressure. Furthermore, in Feistel et al. (2015a, 2015a) also virial approximations of the TEOS-10 equations for the fugacity and the chemical potential of water vapor in humid, \(\mu _{\mathrm {W}}^{\text {AV}}(x_{\mathrm {V}},T,p)\), are derived. This preparatory effort suggests the possibility of expressing \(x_{\text {V,sat}}^{\text {(c)}}\) and \(e_{\text {sat}}^{\text {(c)}}\) in terms of VCs. The derivation of virial representations for these two quantities, however, is a nontrivial and pending task, which is beyond the scope of the present study. The same holds true for the enhancement factor, f(c)(T,p), defined by Eq. (11). Therefore, below we restrict our consideration to the derivation of the virial forms of the compressibility coefficient and the virtual temperature, which can be obtained in a more straightforward manner.
A1.3.2 Virial form of the compressibility coefficient
The compressibility coefficient of humid air, ZAV, is defined by Eq. (14) as
$$ Z_{\text{AV}} \equiv \frac{p}{\widetilde{{\varrho}}_{\text{AV}} R T} . $$
(A1.25)
By virtue of Eq. (A1.13), ZAV can be expressed as a series expansion with respect to the molar density:
$$ Z_{\text{AV}}(\widetilde{{\varrho}}_{\text{AV}}) \equiv \frac{p(\widetilde{{\varrho}}_{\text{AV}})}{\widetilde{{\varrho}}_{\text{AV}} R T} \approx 1 + B(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}} + C(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}}^{2} + D(x_{\mathrm{V}},T) \widetilde{{\varrho}}_{\text{AV}}^{3} . $$
(A1.26)
Analogously, by virtue of Eqs. (A1.15) and (A1.24), ZAV can be expressed as a series expansion with respect to pressure:
$$ \begin{array}{@{}rcl@{}} Z_{\text{AV}}(p) & \equiv & \frac{p}{\widetilde{{\varrho}}_{\text{AV}}(p) R T} \approx 1 + b(x_{\mathrm{V}},T) \left( \frac{p}{RT} \right) + c(x_{\mathrm{V}},T) \left( \frac{p}{RT} \right)^{2} \\ & & + d(x_{\mathrm{V}},T) \left( \frac{p}{RT} \right)^{3} \\ & = & 1 + B(x_{\mathrm{V}},T) \left( \frac{p}{RT} \right) \\ & & + \left( C(x_{\mathrm{V}},T)-B^{2}(x_{\mathrm{V}},T) \right) \left( \frac{p}{RT} \right)^{2} \\ & & + \left( D(x_{\mathrm{V}},T) - 3 B(x_{\mathrm{V}},T) C(x_{\mathrm{V}},T) + 2 B^{3}(x_{\mathrm{V}},T) \right) \left( \frac{p}{RT} \right)^{3}. \end{array} $$
(A1.27)
These equations permit the a posteriori derivation of associated VCs from several published empirical functions ZAV. Such VCs, in turn, permit comparison with VCs derived from other empirical formulas, such as for the density. The special case of the compressibility factor for dry air can be recovered from Eqs. (A1.26) and (A1.27) for xV = 0:
$$ Z_{\mathrm{A}}(\widetilde{{\varrho}}_{\text{AV}}) \approx 1 + B(0,T) \widetilde{{\varrho}}_{\mathrm{A}} + C(0,T) \widetilde{{\varrho}}_{\mathrm{A}}^{2} + D(0,T) \widetilde{{\varrho}}_{\mathrm{A}}^{3} , $$
(A1.28)
$$ \begin{array}{@{}rcl@{}} Z_{\mathrm{A}}(p) & \approx & 1 + b(0,T) \left( \frac{p}{RT} \right) + c(0,T) \left( \frac{p}{RT} \right)^{2} + d(0,T) \left( \frac{p}{RT} \right)^{3} \\ & = & 1 + B(0,T) \left( \frac{p}{RT} \right) + \left( C(0,T)-B^{2}(0,T) \right) \left( \frac{p}{RT} \right)^{2} \\ & & + \left( D(0,T) - 3 B(0,T) C(0,T) + 2 B^{3}(0,T) \right) \left( \frac{p}{RT} \right)^{3} . \end{array} $$
(A1.29)
A1.3.3 Virial form of the virtual temperature
The general expression of the virtual temperature, Tv, is defined by Eq. (17):
$$ T_{\mathrm{v}} = T \left( \frac{Z_{\text{AV}}}{Z_{\mathrm{A}}} \right) \left( \frac{R_{\text{AV}}}{R_{\mathrm{A}}} \right) = T \left( \frac{Z_{\text{AV}}}{Z_{\mathrm{A}}} \right) \left( \frac{M_{\mathrm{A}}}{M_{\text{AV}}} \right) . $$
(A1.30)
For meteorological applications this temperature is typically evaluated as function of the pressure. By virtue of ZAV(p) from Eq. (A1.27) and ZA(p) from Eq. (A1.29) the virial form of the virtual temperature reads:
$$ \begin{array}{@{}rcl@{}} T_{\mathrm{v}} & = & T \left( \frac{M_{\mathrm{A}}}{M_{\text{AV}}} \right) \times \bigg\{ 1 + B(x_{\mathrm{V}},T) \left( \frac{p}{RT} \right) \\ & & + \left( C(x_{\mathrm{V}},T)-B^{2}(x_{\mathrm{V}},T) \right) \left( \frac{p}{RT} \right)^{2} \\ & & + \left( D(x_{\mathrm{V}},T) - 3 B(x_{\mathrm{V}},T) C(x_{\mathrm{V}},T) + 2 B^{3}(x_{\mathrm{V}},T) \right) \left( \frac{p}{RT} \right)^{3} \bigg\} \\ & & \times \bigg\{ 1 + B(0,T) \left( \frac{p}{RT} \right) + \left( C(0,T)-B^{2}(0,T) \right) \left( \frac{p}{RT} \right)^{2} \\ & & + \left( D(0,T) - 3 B(0,T) C(0,T) + 2 B^{3}(0,T) \right) \left( \frac{p}{RT} \right)^{3} \bigg\}^{-1} . \end{array} $$
(A1.31)
Empirical equations available for Tv permit the estimation of associated VCs by means of Eqs. (A1.27)–(A1.28) or (A1.31), respectively, or by suitable low-pressure series expansions of those equations.
Appendix 2. Compressibility factor employed in the LB-1987 approach
A2.1 Compressibility factor of nitrogen
As a first guess, Herbert (1987, p. 74 therein) approximated the compressibility factor of humid air, ZAV, by the one of nitrogen, serving in turn as proxy of dry air. The corresponding virial form of ZAV is given by the following relation:
$$ Z_{\text{AV}} \approx Z_{\text{N}_{2}}(T,p) = 1 + \sum\limits_{\text{k=1}}^{3} B^{\prime}_{\mathrm{k}}(T) p^{\mathrm{k}} . $$
(A2.1)
The coefficients \(B^{\prime }_{\mathrm {k}}(T)\) are presented in Table 8. Equation (A2.1) is equivalent to the textbook-like form, Eq. (A1.27), with the pressure-expanded VCs b, c, and d determined by the following transformations:
$$ b = RT B_{1}^{\prime}(T) , \quad c = (RT)^{2} B_{2}^{\prime}(T) , \quad d = (RT)^{3} B_{3}^{\prime}(T) . $$
(A2.2)
By virtue of Eq. (A1.21), the density-expanded VCs read
$$ \begin{array}{@{}rcl@{}} B & = & RT B_{1}^{\prime}(T) , \\ C & = & (RT)^{2} \bigg[ (B_{1}^{\prime}(T))^{2} + B_{2}^{\prime}(T) \bigg] , \\ D & = & (RT)^{3} \bigg[ (B_{1}^{\prime}(T))^{3} + 3 B_{1}^{\prime}(T) B_{2}^{\prime}(T) + B_{3}^{\prime}(T) \bigg] . \end{array} $$
(A2.3)
Table 8 Coefficients \(B^{\prime }_{1}\), \(B^{\prime }_{2}\), and \(B^{\prime }_{3}\) of the compressibility factor of nitrogen, Eq. (A2.1) For temperatures in the interval 200K<T< 300K the VCs \(B^{\prime }_{1}\), \(B^{\prime }_{2}\), and \(B^{\prime }_{3}\) in Table 8 must be interpolated. As the VCs vary by more than one order of magnitude over this range and as the corresponding temperature coefficients differ by their signs (\(\mathrm {d}B^{\prime }_{1}/\mathrm {d}T>0\), \(\mathrm {d}B^{\prime }_{2}/\mathrm {d}T>0\), and \(\mathrm {d}B^{\prime }_{3}/\mathrm {d}T<0\)) it would be desirable to have physical a priori information for the specification of the appropriate interpolation. However, due to lack of such guiding information in Herbert (1987, p. 74 therein) as a first guess we applied linear interpolation appearing the simplest one. One can at least expect that the uncertainty of such practice is not larger than those originating from linear interpolation of look-up table values. However, the analysis of the sensitivity of \(Z_{\text {N}_{2}}(T,p)\) against the interpolation method for the coefficients \(B^{\prime }_{1}\), \(B^{\prime }_{2}\), and \(B^{\prime }_{3}\) as well as against the application of alternative expressions for the temperature dependence of the VCs is beyond the scope of the present study, but is subject of an ongoing work.
A2.2 Compressibility factor of humid air
For a refined guess of the compressibility factor, Herbert (1987, Section 2.3.3, Table 15 therein) presented table values of ZAV with reference to WMO (1968). In the source (WMO 1968, Table 4.12.2 therein) in turn, reference is given to SMT (1951, pp. 331–333 and Table 84 therein)Footnote 3, which again relegates to the Goff–Gratch formulation of the thermodynamic properties of air and water vapor as published in Goff and Gratch (1945, 1946), and Goff(1949a)Footnote 4.
The form of ZAV used in SMT (1951, Table 84 therein) can be derived from Goff and Gratch (1945, Eq. (13.8) therein):
$$ \begin{array}{@{}rcl@{}} \frac{p}{\widetilde{{\varrho}}_{\text{AV}}} & = & RT - \bigg[x_{\mathrm{A}}^{2} A_{\text{AA}}^{\prime} + 2 x_{\mathrm{A}} (1-x_{\mathrm{A}}) A_{\text{AW}}^{\prime} + (1-x_{\mathrm{A}})^{2} A_{\text{WW}}^{\prime} \bigg] p \\ & & - \bigg[(1-x_{\mathrm{A}})^{3} A_{\text{WWW}}^{\prime} \bigg] p^{2} . \end{array} $$
(A2.4)
Here, xA = 1 − xV denotes the mole fraction of dry air in humid air. By virtue of the definition \(Z_{\text {AV}}=p/(RT \widetilde {{\varrho }}_{\text {AV}})\), Eq. (A2.4) can be rewritten as follows:
$$ \begin{array}{@{}rcl@{}} Z_{\text{AV}}(x_{\mathrm{V}},T,p) & \approx & 1 - \bigg[ (1-x_{\mathrm{V}} )^{2} A_{\text{AA}}^{\prime} + 2 (1 - x_{\mathrm{V}}) x_{\mathrm{V}} A_{\text{AW}}^{\prime} \\ & & + x_{\mathrm{V}}^{2} A_{\text{WW}}^{\prime} \bigg] \frac{p}{R T} - \bigg[ x_{\mathrm{V}}^{3} A_{\text{WWW}}^{\prime} \bigg] \frac{p^{2}}{R T} . \end{array} $$
(A2.5)
Here, \(A_{\text {AA}}^{\prime }\), \(A_{\text {WW}}^{\prime }\), and \(A_{\text {AW}}^{\prime }\) (in units of a molar volume, m3mol− 1) are coefficients, which are related to the second molar virial and cross-virial coefficients of molecular air–air, water–water, and air–water interactions, BAA, BWW, and BAW, respectively, and \(A_{\text {WWW}}^{\prime }\) (in units of m3mol− 1Pa− 1) is related to the third molar VC of interactions between three water molecules, CWWW. As Goff and Gratch (1945, p. 139 therein) did neither had experimental nor theoretical information regarding the triple-interactions coefficients \(A_{\text {AAA}}^{\prime }\), \(A_{\text {AAW}}^{\prime }\), and \(A_{\text {AWW}}^{\prime }\), they were entirely neglected in Eq. (A2.4).
The LB-1987 form of ZAV, Eq. (A2.5), is equivalent to the textbook-like form, Eq. (A1.27),
$$ \begin{array}{@{}rcl@{}} Z_{\text{AV}} & \approx & 1 + b \left( \frac{p}{RT} \right) + c \left( \frac{p}{RT} \right)^{2} \\ & = & 1 + B \left( \frac{p}{RT} \right) + (C-B^{2}) \left( \frac{p}{RT} \right)^{2} , \end{array} $$
(A2.6)
with the second and third VCs defined by Eqs. (A1.6) and (A1.7),
$$ \begin{array}{@{}rcl@{}} B & = & x_{\mathrm{V}}^{2} B_{\text{WW}} + 2 x_{\mathrm{V}} (1 - x_{\mathrm{V}}) B_{\text{AW}} + (1 - x_{\mathrm{V}})^{2} B_{\text{AA}} , \\ C & = & x_{\mathrm{V}}^{3} C_{\text{WWW}} + 3 x_{\mathrm{V}}^{2} (1-x_{\mathrm{V}}) C_{\text{AWW}} \\ & & + 3 x_{\mathrm{V}} (1-x_{\mathrm{V}})^{2} C_{\text{AAW}} + (1-x_{\mathrm{V}})^{3} C_{\text{AAA}} \end{array} $$
(A2.7)
and the replacements
$$ \begin{array}{@{}rcl@{}} b & = & B = - \bigg[ x_{\mathrm{V}}^{2} A_{\text{WW}}^{\prime} + 2 x_{\mathrm{V}} (1 - x_{\mathrm{V}}) A_{\text{AW}}^{\prime} + (1 - x_{\mathrm{V}})^{2} A_{\text{AA}}^{\prime} \bigg] \\ \leadsto \quad B_{\text{WW}} & = & - A_{\text{WW}}^{\prime} , \quad B_{\text{AW}} = - A_{\text{AW}}^{\prime} , \quad B_{\text{AA}} = - A_{\text{AA}}^{\prime} , \end{array} $$
(A2.8)
and
$$ \begin{array}{@{}rcl@{}} c \left( \frac{p}{RT} \right)^{2} & = & C \bigg(1 - \underbrace{\frac{B^{2}}{C}}_{\ll 1} \bigg) \left( \frac{p}{RT} \right)^{2} \approx C \left( \frac{p}{RT} \right)^{2} \\ & \approx & x_{\mathrm{V}}^{3} C_{\text{WWW}} \left( \frac{p}{R T} \right)^{2} = - \bigg[ x_{\mathrm{V}}^{3} A_{\text{WWW}}^{\prime} \bigg] \frac{p^{2}}{R T} \\ \leadsto \quad C_{\text{WWW}} & = & - R T A_{\text{WWW}}^{\prime} . \end{array} $$
(A2.9)
While \(A_{\text {WWW}}^{\prime }\) is given in units of m3mol− 1Pa− 1, the commonly used third VC, CWWW, is given in units of m6mol− 2.
Parameterizations of the virial-like coefficients \(A_{\text {AA}}^{\prime }\), \(A_{\text {WW}}^{\prime }\), \(A_{\text {WWW}}^{\prime }\), and \(A_{\text {AW}}^{\prime }\) used in the LB-1987 formulation are given in Goff and Gratch (1945, Eqs. (2.4), (8.1), (8.2), and (16.2) therein). However, as SMT (1951, pp. 331–333 and Table 84 therein) cited the later publication of Goff (1949a), here the virial-like coefficients were taken from this source. Correspondingly, the second virial-like coefficient of dry air reads (Goff 1949a, Table 3 therein):
$$ \begin{array}{@{}rcl@{}} \frac{A_{\text{AA}}^{\prime}}{\text{m}^{3} \text{mol}^{-1}} & = & -4.07 {\cdot} 10^{-5} + \frac{1.3116{\cdot}10^{-2}}{T/\mathrm{K}} + \frac{120}{(T/\mathrm{K})^{3}} ,\\ & & 183 \mathrm{K} \le T \le 363 \mathrm{K} . \end{array} $$
(A2.10)
For the second and third virial-like coefficients of water vapor, Goff (1949a, Tables 4 and 5 therein) referred to the formulation provided by Goff and Gratch (1946, Eq. (20) therein)Footnote 5:
$$ \begin{array}{@{}rcl@{}} \frac{A_{\text{WW}}^{\prime}}{\text{m}^{3} \text{mol}^{-1}} & = & -3.397 {\cdot} 10^{-5} + \frac{5.5306 {\cdot} 10^{-2}}{T/\mathrm{K}} \times 10^{72000/(T/\mathrm{K})^{2}} ,\\ & & 263 \mathrm{K} \le T \le 363 \mathrm{K} , \end{array} $$
(A2.11)
$$ \begin{array}{@{}rcl@{}} \frac{A_{\text{WWW}}^{\prime}}{\text{m}^{3} \text{mol}^{-1}\text{Pa}^{-1}} & = & \frac{3.43449{\cdot}10^{5}}{(T/\mathrm{K})^{2}} \left( \frac{A_{\text{WW}}}{\text{m}^{3} \text{mol}^{-1}} \right)^{3} ,\\ & & 293 \mathrm{K} \le T \le 363 \mathrm{K} . \end{array} $$
(A2.12)
For the second cross virial-like coefficient of air–water interactions, Goff (1949a, Table 6 therein) referred to the formulation provided by Goff and Gratch (1945, Eq. (16.2) therein):
$$ \begin{array}{@{}rcl@{}} \frac{A_{\text{AW}}^{\prime}}{\text{m}^{3} \text{mol}^{-1}} & = & -2.953 {\cdot} 10^{-5} + 6.69 {\cdot} 10^{-9} \left( \frac{T}{\mathrm{K}} \right) \left[ 1 - \exp \left( -\frac{4416.5}{T/\mathrm{K}} \right) \right]\\ & & + \frac{1.7546 {\cdot} 10^{-2}}{T/\mathrm{K}} + \frac{9.53 {\cdot} 10^{-2}}{(T/\mathrm{K})^{2}} + \frac{85.15}{(T/\mathrm{K})^{3}} ,\\ & & 183 \mathrm{K} \le T \le 363 \mathrm{K} . \end{array} $$
(A2.13)
To avoid spurious misfits outside the declared definition ranges of \(A_{\text {AA}}^{\prime }\), \(A_{\text {AW}}^{\prime }\), \(A_{\text {WW}}^{\prime }\), and \(A_{\text {WWW}}^{\prime }\), the VCs were set to zero beyond the declared temperature intervals (instead of extrapolating the corresponding formulas beyond their definition ranges). Zeroing simultaneously all of the VCs corresponds to idealization of the gas-phase mixture.