Open access peer-reviewed chapter

Thermodynamic Properties and Applications of Modified van-der-Waals Equations of State

By Ronald J. Bakker

Submitted: November 14th 2011Reviewed: May 30th 2012Published: October 3rd 2012

DOI: 10.5772/50315

Downloaded: 3135

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).

p(T,Vm,x)E1

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].

dS=(pT)V,nTdVE2
dU=[T(pT)V,nTp]dVE3

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.

H=U+pVE4

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=UTSE5
dA=pdVE6

The Gibbs energy (G) is calculated in a similar procedure according to its definition in Eq. 7.

G=U+pVTSE7

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).

μi=(Ani)T,V,njE8

The fugacity (f) can be directly obtained from chemical potentials (Eq. 9) and from the definition of the fugacity coefficient (φi) with independent variables V and T (Eq. 10).

RTln(fifi0)=μiμi0E9

where µi0 and fi0 are the chemical potential and fugacity, respectively, of component i at standard conditions (0.1 MPa).

RTlnφi=V[(pni)T,V,njRTV]dVRTlnzE10
.

where φi and z (compressibility factor) are defined according to Eq. 11 and 12, respectively.

φi=fixipE11
z=pVnTRTE12

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.

p=RTVmζ1ζ2Vm(Vm+ζ3)+ζ4(Vmζ4)E13
p=nTRTVnTζ1nT2ζ2V2+nTζ3V+nTζ4VnT2ζ42E14

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 ζ1, ζ2, ζ3, and ζ4 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.

WRKSPR
ζ1bbbb
ζ2aa·T -0.5aa
ζ3-bbb
ζ4---b

Table 1.

Definitions of ζ1, ζ2, ζ3, and ζ4 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).

a0V3+a1V2+a2V+a3=0E15
b0z3+b1z2+b2z+b3=0E16

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.

a0=pE17
a1=nTp(ζ3+ζ4ζ1)nTRTE18
a2=nT2p(ζ42+ζ1ζ3+ζ1ζ4)nT2RT(ζ3+ζ4)+nT2ζ2E19
a3=nT3pζ1ζ42+nT3RTζ42nT3ζ1ζ2E20
b0=(RTp)3E21
b1=(RTp)2(ζ3+ζ4ζ1RTp)E22
b2=(RTp)(ζ42(ζ3+ζ4)(ζ1+RTp)+ζ2p)E23
b3=(ζ1+RTp)ζ42ζ1ζ2pE24

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].

p=pideal+presidualE25

where

pideal=nTRTVE26

The residual pressure (presidual) can be defined as the difference (Δp, Eq. 27) between ideal pressure and reel pressure as expressed in Eq. 14.

Δp=presidual=nTRTV+nTRTVnTζ1nT2ζ2V2+nTζ3V+nTζ4VnT2ζ42E27

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).

pT=pidealT+ΔpTE28

where

ΔpT=nTRV+nTRVnTζ1+nTRT(VnTζ1)2(nTζ1)T1V2+nTζ3V+nTζ4VnT2ζ42(nT2ζ2)T+nT2ζ2(V2+nTζ3V+nTζ4VnT2ζ42)2(nTζ3V+nTζ4VnT2ζ42)TE29

The parameters ζ1, ζ3, and ζ4 are usually independent of temperature, compare with the b parameter (Table 1). This reduces Eq. 29 to Eq. 30.

ΔpT=nTRV+nTRVnTζ11V2+nTζ3V+nTζ4VnT2ζ42(nT2ζ2)TE30

Other important equations to calculate thermodynamic properties of fluids are partial derivatives of pressure with respect to volume (Eq. 31 and 32).

pV=nTRT(VnTζ1)2+nT2ζ2(V2+nTζ3V+nTζ4VnT2ζ42)2(2V+nTζ3+nTζ4)E31
2pV2=2nTRT(VnTζ1)32nT2ζ2(V2+nTζ3V+nTζ4VnT2ζ42)3(2V+nTζ3+nTζ4)2+2nT2ζ2(V2+nTζ3V+nTζ4VnT2ζ42)2E32

Eqs. 31 and 32 already include the assumption that the parameters ζ1, ζ2, ζ3, and ζ4 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).

pni=RTVnTζ1+nTRT(VnTζ1)2(nTζ1)ni1V2+nTζ3V+nTζ4VnT2ζ42(nT2ζ2)ni+nT2ζ2(V2+nTζ3V+nTζ4VnT2ζ42)2[((nTζ3)ni+(nTζ4)ni)V2nTζ4(nTζ4)ni]E33

3. Thermodynamic parameters

The entropy (S) is obtained from the integration defined in Eq. 2 at constant temperature (Eqs. 34 and 35).

S0S1dS=V0V1(pT)V,nTdVE34
S1=S0+V0V1(pidealT+ΔpT)dVE35

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 ζ3 and ζ4: Eq. 36 for ζ3 = 0 and ζ4 = 0, and Eqs. 37 and 38 for ζ3 > 0.

S1=S0+nTRln(V1V0)+nTRln(V1nTζ1V0nTζ1V0V1)+(1V11V0)(nT2ζ2)TE36
S1=S0+nTRln(V1V0)+nTRln(V1nTζ1V0nTζ1V0V1)1q(nT2ζ2)Tln(2V1+nT(ζ3+ζ4)q2V1+nT(ζ3+ζ4)+q)+1q(nT2ζ2)Tln(2V0+nT(ζ3+ζ4)q2V0+nT(ζ3+ζ4)+q)E37

where

q=nT4ζ42+(ζ3+ζ4)2E38

The RKeos and Seos define q asnTb, whereas in the PReos q is equal tonTb8, according to the values for ζ3 and ζ4 listed in Table 1. Eqs. 36 and 37 can be simplified by assuming that the lower limit of the integration corresponds to a large number of V0. As a consequence, part of the natural logarithms in Eqs. 36 and 37 can be replaced by the unit value 1 or 0 (Eqs. 39, 40, and 41).

limV0(V0V0nTζ1)=1E39
limV0(1V0)=0E40
limV0(2V0+nT(ζ3+ζ4)q2V0+nT(ζ3+ζ4)+q)=1E41

The entropy change that is caused by a volume change of ideal gases corresponds to the second term on the right-hand side of Eqs. 36 and 37. This term can be used to express the behaviour of an ideal mixture of perfected gases. Each individual gas in a mixture expands from their partial volume (vi) to the total volume at a pressure of 0.1 MPa, which results in a new expression for this term (Eq. 42)

nTRln(V1V0)ideal.mix=i[niRln(V1vi)]E42

where ni is the amount of substance of component i in the fluid mixture. In addition, the partial volume of an ideal gas is related to the standard pressure p0 (0.1 MPa) according to the ideal gas law (Eq. 43, compare with Eq. 26).

vi=niRTp0E43

Finally, the entropy of fluid phases containing gas mixtures at any temperature and total volume according to the two-constant cubic equation of state is given by Eq. 44 for ζ3 = 0 and ζ4 = 0, and Eq. 45 for ζ3 > 0.

S=S0+i[niRln(p0VniRT)]+nTRln(VnTζ1V)+1V(nT2ζ2)TE44
S=S0+i[niRln(p0VniRT)]+nTRln(VnTζ1V)1q(nT2ζ2)Tln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)E45

The subscripts "1" for the upper limit of integration is eliminated to present a pronounced equation. The standard state entropy (S0) of a mixture of ideal gases is defined according to the arithmetic average principle (Eq. 46).

S0=inisi0E46

where si0 is the molar entropy of a pure component i in an ideal gas mixture at temperature T.

The internal energy (U, see Eq. 3) is obtained from the pressure equation (Eq. 14) and its partial derivative with respect to temperature (Eqs. 28 and 30):

U0U1dU=V0V1(TpTp)dVE47
U1=U0+V0V1(1V2+nTζ3V+nTζ4VnT2ζ42(nT2ζ2T(nT2ζ2)T))dVE48

Similar to the integral in the entropy definition (see Eqs. 44 and 45), Eq. 48 has different solutions dependent on the values of ζ3 and ζ4: Eq. 49 for ζ3 = 0 and ζ4 = 0, and Eq. 50 for ζ3 > 0.

U=U01V(nT2ζ2T(nT2ζ2)T)E49
U=U0+1q(nT2ζ2T(nT2ζ2)T)ln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)E50

The definition of q is given in Eq. 38. The standard state internal energy (U0) of a mixture of ideal gases is defined according to the arithmetic average principle (Eq. 51).

U0=iniui0E51

where ui0 is the molar internal energy of a pure component i in an ideal gas mixture at temperature T.

Enthalpy (Eq. 52 for ζ3 = 0 and ζ4 = 0, and Eq. 53 for ζ3 > 0), Helmholtz energy (Eq. 55 for ζ3 = 0 and ζ4 = 0, and Eq. 56 for ζ3 > 0), and Gibbs energy (Eq. 58 for ζ3 = 0 and ζ4 = 0, and Eq. 59 for ζ3 > 0) can be obtained from the definitions of pressure, entropy and internal energy according to standard thermodynamic relations, as illustrated in Eq. 4, 5, and 7. Standard state enthalpy (H0), standard state Helmholtz energy (A0), and standard state Gibbs energy (G0) of an ideal gas mixture at 0.1 MPa and temperature T are defined in Eqs. 54, 57, and 60, respectively.

H=U0+nTRTVVnTζ11V(2nT2ζ2T(nT2ζ2)T)E52
H=U0+nTRTVVnTζ1nT2ζ2VV2+nTζ3V+nTζ4VnT2ζ42+1q(nT2ζ2T(nT2ζ2)T)ln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)E53
H0=U0+nTRTE54
A=U0TS0i[niRTln(p0VniRT)]nTRTln(VnTζ1V)nT2ζ2VE55
A=U0TS0i[niRTln(p0VniRT)]nTRTln(VnTζ1V)+nT2ζ2qln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)E56
A0=U0TS0E57
G=U0TS0RTi[niln(p0VniRT)]nTRTln(VnTζ1V)+nTRTVVnTζ12nT2ζ2VE58
G=U0TS0RTi[niln(p0VniRT)]nTRTln(VnTζ1V)+nTRTVVnTζ1+nT2ζ2qln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)nT2ζ2VV2+nTζ3V+nTζ4VnT2ζ42E59
G0=U0TS0+nTRTE60

The Helmholtz energy equation (Eqs. 55, 56, and 57) is used for the definition of chemical potential (μi) of a component in either vapour or liquid phase gas mixtures (compare with Eq. 8), Eq. 61 for ζ3 = 0 and ζ4 = 0, and Eq. 62 for ζ3 > 0, calculated with two-constant cubic equations of state.

μi=U0niTS0niRTln(p0VniRT)+RTRTln(VnTζ1V)+nTRTVnTζ1(nTζ1)ni1VnT2ζ2niE61
μi=U0niTS0niRTln(p0VniRT)+RTRTln(VnTζ1V)+nTRTVnTζ1(nTζ1)ni+(nT2ζ2ni1qnT2ζ2q2qni)ln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)+nT2ζ2q12V+nT(ζ3+ζ4)q(nTζ3ni+nTζ4niqni)nT2ζ2q12V+nT(ζ3+ζ4)+q(nTζ3ni+nTζ4ni+qni)E62

where

qni=1q[4nTζ4nTζ4ni+nT(ζ3+ζ4)(nTζ3ni+nTζ4ni)]E63

The definitions of the partial derivative of q in respect to amount of substance (Eq. 63) according to ζ3 = b and ζ4 = 0 [8, 9] is illustrated in Eq. 64, and ζ3 = b and ζ4 = b [10] in Eq. 65.

qni=(nTb)niE64
qni=8(nTb)niE65

The fugacity coefficient (φi) is defined according to Eqs. 9 and 10 from the difference between the chemical potential of a real gas mixture and an ideal gas mixture at standard conditions (0.1 MPa), see Eq. 66 for ζ3 = 0 and ζ4 = 0, and Eq. 67 for ζ3 > 0. Fugacity coefficient defined in Eq. 66 is applied to Weos and Eq. 67 is applied to RKeos, Seos, and PReos.

RTln(φi)=RTln(pVnTRT)RTln(VnTζ1V)+nTRTVnTζ1(nTζ1)ni1VnT2ζ2niE66
RTln(φi)=RTln(pVnTRT)RTln(VnTζ1V)+nTRTVnTζ1(nTζ1)ni+(nT2ζ2ni1qnT2ζ2q2qni)ln(2V+nT(ζ3+ζ4)q2V+nT(ζ3+ζ4)+q)+nT2ζ2q12V+nT(ζ3+ζ4)q(nTζ3ni+nTζ4niqni)nT2ζ2q12V+nT(ζ3+ζ4)+q(nTζ3ni+nTζ4ni+qni)E67

4. Spinodal

The stability limit of a fluid mixture can be calculated with two-constant cubic equations of state, e.g. see [6]. This limit is defined by the spinodal line, i.e. the locus of points on the surface of the Helmholtz energy or Gibbs energy functions that are inflection points, e.g. see [12] and references therein. The stability limit occurs at conditions where phase separation into a liquid and vapour phase should take place, which is defined by the binodal. Metastability is directly related to spinodal conditions, for example, nucleation of a vapour bubble in a cooling liquid phase within small constant volume cavities, such as fluid inclusions in minerals (< 100 µm diameter) occurs at conditions well below homogenization conditions of these phases in a heating experiment. The maximum temperature difference of nucleation and homogenization is defined by the spinodal. In multi-component fluid systems, the partial derivatives of the Helmholtz energy with respect to volume and amount of substance of each component can be arranged in a matrix that has a determinant (Dspin) equal to zero (Eq. 68) at spinodal conditions.

Dspin=|AVVAn1VAn2VAVn1An1n1An2n1AVn2An1n2An2n2|=0E68

This matrix is square and contains a specific number of columns that is defined by the number of differentiation variables, i.e. volume and number of components in the fluid mixture minus 1. The individual components of this matrix are defined according to Eqs. 69, 70, 71, 72, 73, and 74. The exact definition of these components according to two-constant cubic equations of state can be obtained from the web site http://fluids.unileoben.ac.at (see also [6]).

AVV=(2AV2)n1,n2,E69
An1n1=(2An12)n2,V,E70
An2n2=(2An22)n1,V,E71
An1V=(2An1V)n2,=AVn1E72
An2V=(2An2V)n1,=AVn2E73
An1n2=(2An1n2)V,=An2n1E74

The determinant in Eq. 68 is calculated with the Laplacian expansion that contains "minors" and "cofactors", e.g. see [13]. The mathematical computation time increases exponential with increasing number of components. Therefore, the LU decomposition [14] can be applied in computer programming to reduce this time.

The spinodal curve, binodal curve and critical point of a binary CO2-CH4 mixture with x(CO2) = 0.9 are illustrated in Figure 1, which are calculated with the PReos [10]. The spinodal has a small loop near the critical point, and may reach negative pressures at lower temperatures. The binodal remains within the positive pressure part at all temperatures. The binodal is obtained from equality of fugacity (Eq. 66 and 67) of each component in both liquid and vapour phase, and marks the boundary between a homogeneous fluid mixture and fluid immiscibility [6, 15].

Figure 1.

a) Temperature-pressure diagram of a binary CO2-CH4 fluid mixture, with x(CO2) = 0.9. The shaded area illustrates T-p condition of immiscibility of a CO2-rich liquid phase and a CH4-rich vapour phase (the binodal). The red dashed line is the spinodal. All lines are calculated with the equation of state according to PReos [10]. The calculated critical point is indicated with cPR. cDK is the interpolated critical point from experimental data [16]. (b) enlargement of (a) indicated with the square in thin lines.

5. Pseudo critical point

The pseudo critical point is defined according to the first and second partial derivatives of pressure with respect to volume (Eqs. 31 and 32). This point is defined in a p-V diagram where the inflection point and extremum coincide at a specific temperature, i.e. Eqs. 31 and 32 are equal to 0. The pseudo critical point is equal to the critical point for pure gas fluids, however, the critical point in mixtures cannot be obtained from Eqs. 31 and 32. The pseudo critical point estimation is used to define the two-constants (a and b) for pure gas fluids in cubic equations of state according to the following procedure. The molar volume of the pseudo critical point that is derived from Eqs 31 and 32 is presented in the form of a cubic equation (Eq. 75).

0=Vm33ζ1Vm2+[3ζ423ζ1(ζ3+ζ4)]Vmζ1[ζ42+(ζ3+ζ4)2]+ζ42(ζ3+ζ4)E75

The solution of this cubic equation can be obtained from its reduced form, see page 9 in [15]:

x3+fx+g=0E76

where

f=3[ζ42ζ1(ζ1+ζ3+ζ4)]E77
g=2ζ13+ζ1[2ζ42(ζ3+ζ4)2]+(ζ3+ζ4)[ζ423ζ12]E78
Vm=x+ζ1E79

The values of f and g in terms of the b parameters for the individual two-constant cubic equations of state are given in Table 2. The molar volume at pseudo critical conditions is directly related to the b parameter in each equation of state: Weos Eq. 80; RKeos Eq. 81; Seos also Eq. 81; and PReos Eq. 82.

Weos:Vmpc=3bE80
RKeos:Vmpc=b2313.847322bE81
PReos:Vmpc=[1+Q]b3.951373bE82

where Q is defined according to Eq. 83, the superscript "pc" is the abbreviation for "pseudo critical".

Q=(4+8)13+(48)13E83
Equation of statefgb in Eqs. 80-82b in Eqs. 94-96difference
van der Waals [7]-3b2-2b331.372742.845337 %
Redlich and Kwong [8]-6b2-6b324.463329.697121 %
Soave [9]-6b2-6b324.463329.697121 %
Peng and Robinson [10]-6b2-8b323.819126.665612 %

Table 2.

Definitions of f and g according to Eq. 77 and 78, respectively. The values of b are calculated for the critical conditions of pure CO2: Vm,C = 94.118 cm3 mol-1, TC = 304.128 K and pC = 7.3773 MPa [18]. The last column gives the percentage of difference between the values of b (Eqs. 80-82 and 94-96).

The temperature at pseudo critical conditions is obtained from the combination of Eqs. 80-82 and the first partial derivative of pressure with respect to volume (Eq. 31).

Weos:Tpc=827ζ2bR0.29629630ζ2bRE84
RKeos:Tpc=(231)2(3ζ2bR)0.20267686ζ2bRE85
PReos:Tpc=2Q+4(Q+4+2Q)2(3ζ2bR)0.17014442ζ2bRE86

where Q is defined according to Eq. 83. The order of equations (84, 85, 86) is according to the order of equations of state in Eq. 80, 81,and 82. The parameter ζ2 is used in Eqs. 84, 85 and, 86 instead of the constant a (see Table 1). Eq. 87 illustrates the transformation of Eq. 85 for the RKeos [8] by substitution of ζ2 according to its value given in Table 1.

Tpc=(231)43(3abR)23E87

Any temperature dependency of the a constant has an effect on the definition of the pseudo critical temperature. The pressure at pseudo critical condition (Eqs. 88-90) is obtained from a combination of the pressure equation (Eq.14), pseudo critical temperature (Eqs. 84-87) and pseudo critical molar volume (Eqs. 80-82).

Weos:ppc=127ζ2b20.03703704ζ2b2E88
RKeos:ppc=(231)3ζ2b20.01755999ζ2b2E89
PReos:ppc=Q22(Q2+4Q+2)2ζ2b20.01227198ζ2b2E90

where Q is defined according to Eq. 83. The order of equations (88, 89, and 90) is according to the order of equations of state in Eqs. 80, 81, and 82. These equations define the relation between the a and b constant in two-constant cubic equations of state and critical conditions, i.e. temperature, pressure, and molar volume of pure gas fluids. Therefore, knowledge of these conditions from experimental data can be used to determine the values of a (or ζ2) and b, which can be defined as a function of only temperature and pressure (Eqs. 91-93, and 94-96, respectively).

Weos:ζ2=2764R2TC2pC=0.421875R2TC2pCE91
RKeos:ζ2=19(231)R2TC2pC0.42748024R2TC2pCE92
PReos:ζ2=(Q2+4Q+2)2(Q22)4Q2(Q+2)2R2TC2pC0.45723553R2TC2pCE93
Weos:b=18RTCpC=0.125RTCpCE94
RKeos:b=(231)3RTCpC0.08664035RTCpCE95
PReos:b=(Q22)2Q2(Q+2)RTCpC0.07779607RTCpCE96

where TC and pC are the critical temperature and critical pressure, and Q is defined according to Eq. 83. The order of equations (91-93, and 94-96) is according to the order of equations of state in Eqs. 80-82. Comparison of the value of b calculated with experimental critical volume (Eqs. 80, 81 and 82) and critical temperature and pressure (Eqs. 94, 95, and 96) is illustrated in Table 2. The difference indicates the ability of a specific equation of state to reproduce fluid properties of pure gases. A large difference indicates that the geometry or morphology of the selected equation of state in the p-V-T-x parameter space is not exactly reproducing fluid properties of pure gases. The empirical modifications of the van-der-Waals equation of state according to Peng and Robinson [10] result in the most accurate equation in Table 2 (11% for pure CO2).

6. Critical point and curve

The critical point is the highest temperature and pressure in a pure gas system where boiling may occur, i.e. where a distinction can be made between a liquid and vapour phase at constant temperature and pressure. At temperatures and pressures higher than the critical point the pure fluid is in a homogeneous supercritical state. The critical point of pure gases and multi-component fluid mixtures can be calculated exactly with the Helmholtz energy equation (Eqs. 55-57) that is obtained from two-constant cubic equations of state, e.g. see [17, 18], and it marks that part of the surface described with a Helmholtz energy function where two inflection points of the spinodal coincide. Therefore, the conditions of the spinodal are also applied to the critical point. In addition, the critical curve is defined by the determinant (Dcrit) of the matrix illustrated in Eq. 97, see also [6].

Dcrit=|AVVAn1VAn2VAVn1An1n1An2n1DVDn1Dn2|=0E97

The number of rows in Eq.97 is defined by the differentiation variables volume and number of components minus 2. The last row is reserved for the partial derivatives of the determinant Dspin from Eq. 68:

DV=DspinVE98
Dn1=Dspinn1E99
Dn2=Dspinn2E100

The derivatives of the spinodal determinant (Eqs. 98-100) are calculated from the sum of the element-by-element products of the matrix of "cofactors" (or adjoint matrix) of the spinodal (Eq. 101) and the matrix of the third derivatives of the Helmholtz energy function (Eq. 102).

|CVVCn1VCn2VCVn1Cn1n1Cn2n1CVn2Cn1n2Cn2n2|E101
|AVVKAn1VKAn2VKAVn1KAn1n1KAn2n1KAVn2KAn1n2KAn2n2K|E102

Figure 2.

Calculated critical points of binary CO2-CH4 fluid mixtures in terms of temperature (red line) and pressure (green line), obtained from the PReos [10]. Solid circles are experimental data [16, 19]. The open squares are the critical point of pure CO2 [20].

where Cxy are the individual elements in the matrix of "cofactors", as obtained from the Laplacian expansion. The subscript K refers to the variable that is used in the third differentiation (volume, amount of substance of the components 1 and 2. To reduce computation time in software that uses this calculation method, the LU decomposition has been used to calculate the determinant in Eq. 97. The determinants in Eqs. 68 and 97 are both used to calculate exactly the critical point of any fluid mixture and pure gases, based on two-constant cubic equations of state that define the Helmholtz energy function.

An example of a calculated critical curve, i.e. critical points for a variety of compositions in a binary fluid system, is illustrated in Figure 2. The prediction of critical temperatures of fluid mixtures corresponds to experimental data [16, 19], whereas calculated critical pressures are slightly overestimated at higher fraction of CH4. This example illustrates that the PReos [10] is a favourable modification that can be used to calculate sub-critical conditions of CO2-CH4 fluid mixtures.

7. Mixing rules and definitions of ζ1 and ζ2

All modifications of the van-der-Waals two-constant cubic equation of state [7] have an empirical character. The main modifications are defined by Redlich and Kwong, Soave and Peng and Robinson (see Table 1), and all modification can by summarized by specific adaptations of the values of ζ1, ζ2, ζ3, and ζ4 to fit experimental data. The original definition [7] of ζ1 (b) and ζ2 (a) for pure gases is obtained from the pseudo critical conditions (Eqs. 91-93, and 94-96). This principle is adapted in most modifications of the van-der-Waals equation of state, e.g. RKeos [8]. Soave [9] and Peng and Robinson [10] adjusted the definition of ζ2 with a temperature dependent correction parameter α (Eqs. 103-105).

ζ2=aCαE103
α=[1+m(1TTC)]2E104
m=i=0,1,2miωiE105

where ac is defined by the pseudo critical conditions (Eqs. 91-93), and ω is the acentric factor. The summation in Eq. 105 does not exceed i = 2 for Soave [9] and Peng and Robinson [10]. The definition of the acentric factor is arbitrary and chosen for convenience [5] and is a purely empirical modification. These two equations of state have different definitions of pseudo critical conditions (see Eqs. 91-93 and 94-96), therefore, the values of mi must be different for each equation (Table 3).

Soave [9]Peng and Robinson [10]
m00.4800.37464
m11.5741.54266
m2-0.176-0.26992

Table 3.

Values of the constant mi in Eq. 105.

The two-constant cubic equation of state can be applied to determine the properties of fluid mixtures by using "mixing rules" for the parameters ζ1 and ζ2 which are defined for individual pure gases according to pseudo critical conditions. These mixing rules are based on simplified molecular behaviour of each component (i and j) in mixtures [21, 22] that describe the interaction between two molecules:

ζ1mix=ijxiζ1(i)E106
ζ2mix=ijxixjζ2(i,j)E107

where

ζ2(i,j)=ζ2(i)ζ2(j)E108

These mixing rules have been subject to a variety of modifications, in order to predict fluid properties of newly available experimental data of mixtures. Soave [9] and Peng and Robinson [10] modified Eq. 108 by adding an extra correction factor (Eq. 109).

ζ2(i,j)=(1δij)ζ2(i)ζ2(j)E109

where δij has a constant value dependent on the nature of component i and j.

8. Experimental data

As mentioned before, modifications of two-constant cubic equation of state was mainly performed to obtain a better fit with experimental data for a multitude of possible gas mixtures and pure gases. Two types of experimental data of fluid properties were used: 1. homogeneous fluid mixtures at supercritical conditions; and 2. immiscible two-fluid systems at subcritical conditions (mainly in petroleum fluid research). The experimental data consist mainly of pressure, temperature, density (or molar volume) and compositional data, but can also include less parameters. Figure 3 gives an example of the misfit between the first type of experimental data for binary CO2-CH4 mixtures [19] and calculated fluid properties with RKeos [8] at a constant temperature (15 ˚C). The RKeos uses the pseudo critical defined parameters ζ1 and ζ2 (Eqs. 92 and 95) and mixing rules according to Eqs. 106-108 and is only approximately reproducing the fluid properties of CO2-CH4 mixtures at subcritical conditions

Experimental data of homogeneous supercritical gas mixtures in the ternary CO2-CH4-N2 system [23] are compared with the two-constant cubic equations of state in Table 4. The Weos [7] clearly overestimates (up to 14.1 %) experimentally determined molar volumes at 100 MPa and 200 ˚C. The Seos [9] is the most accurate model in Table 4, but still reach deviations of up to 2.3 % for CO2-rich gas mixtures. The PReos [10] gives highly underestimated molar volumes at these conditions.

Figure 3.

Modelled immiscibility of binary CO2-CH4 gas mixtures (shaded areas) in a pressure - amount CH4 fraction diagram (a) and amount CH4 fraction - molar volume diagram (b) at 15 ˚C. The solid and open circles are experimental data [16]. The red squares are the properties of pure CO2 [20]. The yellow triangle (Cexp) is the interpolated critical point for experimental data, and the green triangle (CRK) is the calculated critical point [8]. tie1 and tie 2 in (b) are calculated tie-lines between two phases at constant pressures 6.891 and 6.036 MPa, respectively.

compositionVm(exp)
cm3·mol-1
WRKSPR
CO2CH4N2
0.80.10.156.6464.61 (14.1%)54.90 (-3.1%)57.94 (2.3%)53.59 (-5.4%)
0.80.20.258.9265.81 (11.7%)56.61 (-3.9%)59.61 (1.2%)56.93 (-6.1%)
0.40.30.361.0867.08 (9.6%)58.27 (-4.6%)61.12 (0.1%)56.93 (-6.8%)
0.20.40.462.9068.28 (8.6%)59.83 (-4.9%)62.42 (-0.8%)58.28 (-7.3%)

Table 4.

Comparison of supercritical experimental molar volumes [23] at 100 MPa and 200 ˚C with two-constant cubic equations of state (abbreviations see Table 1). The percentage of deviation from experimentally obtained molar volumes is indicated in brackets.

Figure 3 and Table 4 illustrate that these modified two-constant cubic equations of state still need to be modified again to obtain a better model to reproduce fluid properties at sub- and supercritical conditions.

9. Modifications of modified equations of state

The number of publications that have modified the previously mentioned two-constant cubic equations of state are numerous, see also [11], and they developed highly complex, but purely empirical equations to define the parameters ζ1 and ζ2. A few examples are illustrated in the following paragraphs.

9.1. Chueh and Prausnitz [24]

The constant values in the definition of ζ1 and ζ2 (Eqs. 92 and 95) are modified for individual gases by Chueh and Prausnitz [24]. This equation is an arbitrary modification of the RKeos [8]. Consequently, the calculation of the value of ζ1 and ζ2 is not any more defined by pseudo critical conditions, which give exact mathematical definition of these constants. Although the prediction of fluid properties of a variety of gas mixtures was improved by these modifications, the morphology of the Helmholtz energy equation in the p-V-T-x parameter space is not any more related to observed fluid properties. The theory of pseudo critical conditions is violated according to these modifications.

The mixing rules in Eqs. 106-108 were further refined by arbitrary definitions of critical temperature, pressure, volume and compressibility for fluid mixtures.

ζ2(i,j)=Ωi+Ωj2R2TCij2pCijE110
aij=Ωi+Ωj2R2TCij2.5pCijE111

where Ωi and Ωj are the newly defined constant values of component i and j, and TCij and pCij are defined according to complex mixing rules [see 24]. The values of TCij and pCij are not related to true critical temperatures and pressures of specific binary gas mixtures.

The prediction of the properties of homogeneous fluids at supercritical conditions (Table 5) is only slightly improved compared to RKeos [10], but it is not exceeding the accuracy of the Seos [11]. At sub-critical condition (Figure 4), the Chueh-Prausnitz equation is less accurate than the Redlich-Kwong equation (compare Figure 3) in the binary CO2-CH4 fluid mixture at 15 ˚C.

compositionVm(exp)
cm3·mol-1
CPHB1B2
CO2CH4N2
0.80.10.156.6456.42 (-0.4%)55.96 (-0.6%)56.84 (0.4%)56.53 (-0.2%)
0.80.20.258.9257.85 (-1.8%)57.68 (-2.1%)59.43 (0.9%)58.81 (-0.2%)
0.40.30.361.0859.21 (-3.1%)59.17 (-3.1%)61.67 (1.0%)60.79 (-0.5%)
0.20.40.462.9060.44 (-3.9%)60.38 (-4.0%)63.45 (0.9%)62.40 (-0.8%)

Table 5.

The same experimental molar volumes as in Table 4 compared with two-constant equations of state according to Chueh and Prausnitz [24] (CP), Holloway [25, 26] (H), Bakker [27] [B1], and Bakker [28] (B2). The percentage of deviation from experimentally obtained molar volumes is indicated in brackets.

9.2. Holloway [25, 26] and Bakker [27]

The equation of Holloway [25, 26] is another modification of the RKeos [8]. The modification is mainly based on the improvement of predictions of homogenous fluid properties of H2O and CO2 mixtures, using calculated experimental data [29]. The value for ζ1 and ζ3 (both b) of H2O is arbitrarily selected at 14.6 cm3 mol-1, whereas other pure gases are defined according to pseudo critical conditions. The definition of ζ2 (i.e. a) for H2O as a function of temperature was subjected to a variety of best-fit procedures [25, 26]. The fitting was improved from four experimental data points [25] to six [26] (Figure 5), but was restricted to temperatures above 350 ˚C. Bakker [27] improved the best-fit equation by including the entire data set [29], down to 50 ˚C (Eq. 112).

Figure 4.

See Figure 3 for details. The RKeos is indicated by dashed lines in (a) and (b). The shaded areas are immiscibility conditions calculated with the Chueh-Prausnitz equation. tie1 and tie 2 in (b) are calculated tie-lines between two phases at constant pressures 6.944 and 5.984 MPa, respectively.

Figure 5.

Temperature dependence of the a constant for pure H2O in the modified cubic equation of state [25, 26]. The open circles are calculated experimental data [29]. fit [25] is the range of fitting in the definition of Holloway [25], and fit [26] of Holloway [26]. RK illustrates the constant value calculated from pseudo critical condition [8].

aH2O=(9.46542.0246103T+1.4928106T2+7.57108T3)106E112

where T is temperature in Kelvin, and the dimension of a is cm6 MPa K0.5 mol-2. The properties of homogeneous pure CO2, CH4 and N2 fluids [27] were also used to obtain a temperature dependent a constant (Eqs. 113, 114, and 115, respectively).

aCO2=(1.2887+5.9363103T1.4124106T2+1.1767108T3)106E113
aCH4=(1.1764+3.5216103T1.155106T2+1.1767108T3)106E114
aN2=(0.0601910.20059103T+0.15386106T2)106E115

The aij value of fluid mixtures with a H2O and CO2 component (as in Eqs. 106-108 and 110-111) is not defined by the value of pure H2O and CO2 (Eqs. 112 and 113), but from a temperature independent constant value (Eqs. 116 and 117, respectively). In addition, a correction factor is used only for binary H2O-CO2 mixtures, see [25, 29].

a0(H2O)=3.5464106MPacm6K0.5mol2E116
a0(CO2)=4.661106MPacm6K0.5mol2E117

Table 5 illustrates that the equation of Holloway [25] is not improving the accuracy of predicted properties of supercritical CO2-CH4-N2 fluids, compared to Chueh-Prausnitz [24] or Seos [9], and it is only a small improvement compared to the RKeos [8]. The accuracy of this equation is highly improved by using the definitions of a constants according to Bakker [27] (see Eqs. 112-115), and result in a maximum deviation of only 1% from experimental data in Table 5.

Experimental data, including molar volumes of binary H2O-CO2 fluid mixtures at supercritical conditions [30, 31, 32] are used to estimate fugacities of H2O and CO2 according to Eq. 118 (compare Eq. 10).

RTlnφi=0p[Vm,iVmideal]dpE118

where Vm,i - Vmideal is the difference between the partial molar volume of component i and the molar volume of an ideal gas (see also Eq. 43). The difference between Eqs. 118 and 10 is the mathematical formulation and the use of different independent variables, which are temperature and pressure in Eq. 118. The integration to calculate the fugacity coefficient can be graphically obtained by measuring the surface of a diagram of the difference between the ideal molar volume and the partial molar volume (i.e. Vm,i - Vmideal) as a function of pressure (Figure 6). The surface obtained from experimental data can be directly compared to calculated curves from equations of state, according to Eq. 10 (Table 6).

The dashed line in Figure 6 is calculated with another type of equation of state: a modification of the Lee-Kesler equation of state [33] that is not treated in this manuscript because it is not a two-constant cubic equation of state. Fugacity estimations of H2O are similar according to both equations, and reveal only a minor improvement for the two-constant cubic equation of state [27]. The experimental data to determine fugacity of CO2 in this fluid mixture is inconsistent at relative low pressures (< 100 MPa). The calculated fugacity [27] is approximately compatible with the experimental data from [31, 32].

Figure 6.

Fugacity estimation in a pressure - dv diagram at 873 K and a composition of x(CO2) = 0.3 in the binary H2O-CO2 system, where dv is the molar volume difference of an ideal gas and the partial molar volume of either H2O or CO2 in binary mixtures. Experimental data are illustrated with circles, triangles and squares (solid for CO2 and open for H2O. The red lines are calculated with Bakker [27], and the shaded area is a measure for the fugacity coefficient of H2O (Eq. 118).

Pressure (MPa)Exp. fugacity (MPa)B1 fugacity (MPa)
106.6926.659 (-0.5%)
5027.96227.3061 (-2.3%)
10045.34144.6971 (-1.4%)
20077.27875.0515 (-2.9%)
300114.221111.072 (-2.8%)
400160.105157.145 (-1.8%)
500219.252216.817 (-1.1%)
600295.350294.216 (-04%)

Table 6.

Fugacities of H2O in H2O-CO2 fluid mixtures, x(CO2) = 0.3, at 873.15 K and variable pressures. B1 fugacity is calculated with Bakker [27]. The deviation (in %) is illustrated in brackets.

9.3. Bowers and Helgeson [34] and Bakker [28]

Most natural occurring fluid phases in rock contain variable amounts of NaCl, which have an important influence on the fluid properties. Bowers and Helgeson [34] modified the RKeos [8] to be able to reproduce the properties of homogeneous supercritical fluids in the H2O-CO2-NaCl system, but only up to 35 mass% NaCl. The model is originally restricted between 350 and 600 ˚C and pressures above 50 MPa, according to the experimental data [35] that was used to design this equation. This model was modified by Bakker [28] including CH4, N2, and additionally any gas with a2) and b (ζ1) constants defined by the pseudo critical conditions (Eqs. 91-93 and 94-96). Experimental data in this multi-component fluid system with NaCl can be accurately reproduced up to 1000 MPa and 1300 K. Table 5 illustrates that this modification results in the best estimated molar volumes in the ternary CO2-CH4-N2 fluid system at 100 MPa and 673 K. Similar to all modifications of the RKeos [8], this model cannot be used in and near the immiscibility conditions and critical points (i.e. sub-critical conditions).

10. Application to fluid inclusion research

Knowledge of the properties of fluid phases is of major importance in geological sciences. The interaction between rock and a fluid phase plays a role in many geological processes, such as development of magma [36], metamorphic reactions [37] and ore formation processes [38]. The fluid that is involved in these processes can be entrapped within single crystal of many minerals (e.g. quartz), which may be preserved over millions of years. The information obtained from fluid inclusions includes 1. fluid composition; 2. fluid density; 3. temperature and pressure condition of entrapment; and 4. a temporal evolution of the rock can be reconstructed from presence of various generation of fluid inclusions. An equation of state of fluid phases is the major tool to obtain this information. Microthermometry [39] is an analytical technique that directly uses equations of state to obtain fluid composition and density of fluid inclusions. For example, cooling and heating experiment may reveal fluid phase changes at specific temperatures, such as dissolution and homogenization, which can be transformed in composition and density by using the proper equations of state.

The calculation method of fluid properties is extensive and is susceptible to errors, which is obvious from the mathematics presented in the previous paragraphs. The computer package FLUIDS [6, 40, 41] was developed to facilitate calculations of fluid properties in fluid inclusions, and fluids in general. This package includes the group "Loners" that handles a large variety of equations of state according to individual publications. This group allows researchers to perform mathematical experiments with equations of state and to test the accuracy by comparison with experimental data.

The equations of state handled in this study can be downloaded from the web site http://fluids.unileoben.ac.at and include 1. "LonerW" [7]; 2. "LonerRK" [8]; 3. "LonerS" [9]; 4. "LonerPR" [10]; 5. "LonerCP" [24]; 6. "LonerH" [25, 26, 27]; and 7. "LonerB" [28, 34]. Each program has to possibility to calculate a variety of fluid properties, including pressure, temperature, molar volume, fugacity, activity, liquid-vapour equilibria, homogenization conditions, spinodal, critical point, entropy, internal energy, enthalpy, Helmholtz energy, Gibbs energy, chemical potentials of pure gases and fluid mixtures. In addition, isochores can be calculated and exported in a text file. The diagrams and tables presented in this study are all calculated with these programs.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Ronald J. Bakker (October 3rd 2012). Thermodynamic Properties and Applications of Modified van-der-Waals Equations of State, Thermodynamics - Fundamentals and Its Application in Science, Ricardo Morales-Rodriguez, IntechOpen, DOI: 10.5772/50315. Available from:

chapter statistics

3135total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Next chapter

Thermodynamics Simulations Applied to Gas-Solid Materials Fabrication Processes

By Elisabeth Blanquet and Ioana Nuta

Related Book

First chapter

Recent Studies on Fundamentals and Application of Microwave Processing of Materials

By Noboru Yoshikawa

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us