Chapter 7 Thermodynamic Properties and Applications of Modified van-der-Waals Equations of State Ronald J. Bakker Additional information is available at the end of the chapter http://dx.doi.org/10.5772/50315 1. Introduction Physical and chemical properties of natural fluids are used to understand geological processes in crustal and mantel rock. The fluid phase plays an important role in processes in diagenesis, metamorphism, deformation, magmatism, and ore formation. The environment of these processes reaches depths of maximally 5 km in oceanic crusts, and 65 km in continental crusts, e.g. [1, 2], which corresponds to pressures and temperatures up to 2 GPa and 1000 ˚C, respectively. Although in deep environments the low porosity in solid rock does not allow the presence of large amounts of fluid phases, fluids may be entrapped in crystals as fluid inclusions, i.e. nm to µm sized cavities, e.g. [3], and fluid components may be present within the crystal lattice, e.g. [4]. The properties of the fluid phase can be approximated with equations of state (Eq. 1), which are mathematical formula that describe the relation between intensive properties of the fluid phase, such as pressure (p), temperature (T), composition (x), and molar volume (Vm). pTV , m , x (1) This pressure equation can be transformed according to thermodynamic principles [5], to calculate a variety of extensive properties, such as entropy, internal energy, enthalpy, Helmholtz energy, Gibbs energy, et al., as well as liquid-vapour equilibria and homogenization conditions of fluid inclusions, i.e. dew point curve, bubble point curve, and critical points, e.g. [6]. The partial derivative of Eq. 1 with respect to temperature is used to calculate total entropy change (dS in Eq. 2) and total internal energy change (dU in Eq. 3), according to the Maxwell's relations [5]. p dS dV (2) T Vn, T © 2012 Bakker, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 164 Thermodynamics – Fundamentals and Its Application in Science p dU T p dV (3) T Vn, T where nT is the total amount of substance in the system. The enthalpy (H) can be directly obtained from the internal energy and the product of pressure and volume according to Eq. 4. HUpV (4) The Helmholtz energy (A) can be calculated by combining the internal energy and entropy (Eq. 5), or by a direct integration of pressure (Eq. 1) in terms of total volume (Eq. 6). A UTS (5) dA pdV (6) The Gibbs energy (G) is calculated in a similar procedure according to its definition in Eq. 7. GUpVTS (7) The chemical potential (µi) of a specific fluid component (i) in a gas mixture or pure gas (Eq. 8) is obtained from the partial derivative of the Helmholtz energy (Eq. 5) with respect to the amount of substance of this component (ni). A i (8) ni TVn, , j The fugacity (f) can be directly obtained from chemical potentials (Eq. 9) and from the definition of the fugacity coefficient (i) with independent variables V and T (Eq. 10). f RT lni 0 (9) 0 ii fi where µi0 and fi0 are the chemical potential and fugacity, respectively, of component i at standard conditions (0.1 MPa). p RT RTln dV RT ln z . (10) i nV V i TVn,, j where i and z (compressibility factor) are defined according to Eq. 11 and 12, respectively. fi i (11) xpi Thermodynamic Properties and Applications of Modified van-der-Waals Equations of State 165 pV z (12) nRTT 2. Two-constant cubic equation of state The general formulation that summarizes two-constant cubic equations of state according to van der Waals [7], Redlich and Kwong [8], Soave [9], and Peng and Robinson [10] is illustrated in Eq. 13 and 14, see also [11]. In the following paragraphs, these equations are abbreviated with Weos, RKeos, Seos, and PReos. RT p 2 (13) Vm 1 VVmm 34 V m 4 nRT n2 p TT2 (14) Vn 222 T 1 VnVnVnTTT34 4 where p is pressure (in MPa), T is temperature (in Kelvin), R is the gas constant (8.3144621 J·mol-1K-1), V is volume (in cm3), Vm is molar volume (in cm3·mol-1), nT is the total amount of substance (in mol). The parameters 1, 2, 3, and 4 are defined according to the specific equations of state (Table 1), and are assigned specific values of the two constants a and b, as originally designed by Waals [7]. The a parameter reflects attractive forces between molecules, whereas the b parameter reflects the volume of molecules. W RK S PR 1 b b b b -0.5 2 a a·T a a 3 - b b b 4 - - - b Table 1. Definitions of 1, 2, 3, and 4 according to van der Waals (W), Redlich and Kwong (RK), Soave (S) and Peng and Robinson (PR). This type of equation of state can be transformed in the form of a cubic equation to define volume (Eq. 15) and compressibility factor (Eq. 16). 32 (15) aV0123 aV aV a 0 32 (16) bz0123 bz bz b 0 where a0, a1, a2, and a3 are defined in Eq. 17, 18, 19, and 20, respectively; b0, b1, b2, and b3 are defined in Eq. 21, 22, 23, and 24, respectively. (17) ap0 anp1341TT nRT (18) 166 Thermodynamics – Fundamentals and Its Application in Science 22 2 2 anp2TTT 4 13 14 nRTn 3 4 2 (19) 32323 anpnRTn314412TTT (20) 3 RT b0 (21) p 2 RT RT b1341 (22) pp RT RT b 2 2 24341 (23) ppp RT 2 12 b31 4 (24) pp The advantage of a cubic equation is the possibility to have multiple solutions (maximally three) for volume at specific temperature and pressure conditions, which may reflect coexisting liquid and vapour phases. Liquid-vapour equilibria can only be calculated from the same equation of state if multiple solution of volume can be calculated at the same temperature and pressure. The calculation of thermodynamic properties with this type of equation of state is based on splitting Eq. 14 in two parts (Eq. 25), i.e. an ideal pressure (from the ideal gas law) and a departure (or residual) pressure, see also [6]. ppideal p residual (25) where nRT p T (26) ideal V The residual pressure (presidual) can be defined as the difference (p, Eq. 27) between ideal pressure and reel pressure as expressed in Eq. 14 . nRT nRT n2 pp TT T2 (27) residual VVn 222 T 1 VnVnVnTTT34 4 The partial derivative of pressure with respect to temperature (Eq. 28) is the main equation to estimate the thermodynamic properties of fluids (see Eqs. 2 and 3). pp p ideal (28) TTT Thermodynamic Properties and Applications of Modified van-der-Waals Equations of State 167 where p nR nR nRT () n TT T T1 TVVn 2 T T 1 Vn T 1 1 ()n 2 T 2 222 VnVnVnTTT34 4T (29) 2 22 n nVnVnTTT34 4 T 2 2 222 T VnVnVnTTT34 4 The parameters 1, 3, and 4 are usually independent of temperature, compare with the b parameter (Table 1). This reduces Eq. 29 to Eq. 30. p nR nR1 () n2 TT T2 (30) 222 TVVnT 1 VnVnVnTTT34 4 T Other important equations to calculate thermodynamic properties of fluids are partial derivatives of pressure with respect to volume (Eq. 31 and 32). p nRT n2 TT 2 2Vn n (31) 22TT34 V Vn 222 T 1 VnVnVnTTT34 4 2 2 p 22nRT n 2 TT2 2Vn n 23 3TT34 V Vn 222 T 1 VnVnVnTTT34 4 (32) 2n 2 T 2 2 222 VnVnVnTTT34 4 Eqs. 31 and 32 already include the assumption that the parameters 1, 2, 3, and 4 are independent of volume. Finally, the partial derivative of pressure in respect to the amount of substance of a specific component in the fluid mixture (ni) is also used to characterize thermodynamic properties of fluid mixtures (Eq. 33). p RT nRT () n TT1 nVn 2 n iT1 Vn T 1 i 1 ()n 2 T 2 (33) 222 VnVnVnTTT34 4ni nnn2()n () () TTT244T 3 Vn2 2 T 4 222 nnii n i VnVnVnTTT34 4 168 Thermodynamics – Fundamentals and Its Application in Science 3. Thermodynamic parameters The entropy (S) is obtained from the integration defined in Eq. 2 at constant temperature (Eqs. 34 and 35). SV 11 p dS dV (34) T SV00Vn, T V 1 p p ideal SS10 dV (35) TT V0 The limits of integration are defined as a reference ideal gas at S0 and V0, and a real gas at S1 and V1. This integration can be split into two parts, according to the ideal pressure and residual pressure definition (Eqs. 25, 26, and 27). The integral has different solutions dependent on the values of 3 and 4: Eq. 36 for 3 = 0 and 4 = 0, and Eqs. 37 and 38 for 3 > 0. VVn V 11 () n2 111 TT 0 2 S10 S nRTTln nR ln (36) VVnVVVT001110 T VVn V 111 T 0 S10 S nRTTln nR ln VVnV0011 T (37) 22 11()nnTT222()Vn134TT q () 2() Vn 034 q ln ln qT2() Vn134TT qqT 2() Vn 034 q where 22 (38) qnT 4(434 ) The RKeos and Seos define q as nbT , whereas in the PReos q is equal to nbT 8 , according to the values for 3 and 4 listed in Table 1. Eqs.