A methodology to define the Cubic Equation of State of a simple fluid

Fluid Phase Equilibria 307 (2011) 190–196

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A methodology to define the Cubic Equation of State of a simple fluid F. de J. Guevara-Rodríguez ∗ Programa de Ingeniería Molecular, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, 07730, México, Distrito Federal, Mexico.

a r t i c l e

i n f o

Article history: Received 24 November 2010 Received in revised form 10 May 2011 Accepted 20 May 2011 Available online 30 May 2011 Keywords: Cubic equation Coexistence diagram Second virial coefficient Square-Well potential

a b s t r a c t A method to define the Cubic Equation of State (CES) of a simple substance is presented in this work. CES is constructed with only three parameters of the fluid, namely, the critical compressibility Zc ≡ Pc vc /RTc , the acentric factor ω ≡ − log (P(sat) /Pc ) − 1 (where P(sat) is the saturated vapor pressure), and the saturated vapor volume v(sat) at the temperature T(sat) /Tc = 0.7 (where Tc is the critical temperature, vc is the critical volume, and Pc is the critical pressure). The resulting CES is unique for each substance and, in general, it is different from other known CES in the literature. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Perhaps, Cubic Equations of State (CES) are the most useful tools to predict the thermodynamic equilibrium state of simple fluids and their mixtures. Their fundamental base, which has been accepted since the pioneer work of van der Waals [1,2], is the physical notion of an ensemble of spherical particles, which interact through an attractive potential and a hard core. The equation of state is not based on some particular molecular model, however, it allows us to improve the treatment of heavy hydrocarbons [3], and it describes the experimental measures of the pressure of the liquid phase of CO2 , H2 O, and their mixtures [4]. Those cases are mentioned as two examples among many other applications. The success of a CES comes from its simplicity and its capability to predict the thermodynamic equilibrium state of fluids with polar molecules, or nonpolar molecules, which can have, or not, spherical shape [5]. The CES is an empirical relation of the pressure P, the temperature T, the molar volume v, and it is formed by a repulsive and an attractive term, P(T, v) =

RT

v−b

−

a(T ) , (v − c)(v − d)

(1)

where R is the gas constant. Eq. (1) is valid in the interval b < v, where b corresponds to the occupied volume. b, c, and d are constant parameters, which define the kind of equation. For example, the attractive term of some known equations (van der Waals [1,2]

∗ Corresponding author. Tel.: +52 55 9175 8176. E-mail address: [email protected] 0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2011.05.012

(VDW), Soave–Redlich–Kwong [6,7] (SRK), Peng–Robinson [8] (PR), among others) are presented in Table 1. For a given substance, a unique CES predicts the thermodynamic equilibrium state of the fluid with certain accuracy [15]. A procedure to evaluate the goodness of a particular CES consists of analyzing the deviation of saturated liquid volume [13], as a function of the acentric factor of Pitzer ω. With the mentioned procedure, the results with the SRK equation are in agreement with the liquid phase of substances with a small acentric factor, i.e. ω ≈ 0. In a similar manner, the results with PR equation are in agreement with those substances with ω ≈ 0.35. With this fact in mind the question is: Which Cubic Equation of State is the best? [15] On the other hand, for a known CES, the critical compressibility Zc ≡ Pc vc /RTc is a well defined constant number. Thus, the value Zc ≈ 0.307 corresponds to PR equation. Other cases are in Table 1. In the previous example, the results with PR equation are in agreement with the experimental data of all those substances with (exp) ≈ 0.307. acentric factor ω ≈ 0.35 and critical compressibility Zc For other substances, PR equation gives inaccurate results. In this way, an important issue is to develop a methodology to construct a suitable CES for a fluid, which must be at least characterized with (exp) its acentric factor ω and its critical compressibility Zc . This is the main goal of this work. Another important element in Eq. (1) is the function a(T). There is not a formal manner to derive the function a(T) and therefore, it has many empirical definitions in the literature [16]. Recently, the Gibbs–Helmholtz equation has been used [4] as a restriction to define the function a(T). However, another restriction will be adopted in order to have an alternative definition of the function a(T), namely, the virial expansion. To clarify this point, let us write the virial expansion: P/RT =  + B(T)2 + · · ·, where  = 1/v is the

F. de J. Guevara-Rodríguez / Fluid Phase Equilibria 307 (2011) 190–196

191

Table 1 Examples of the attractive term of some Cubic Equations of State (CES): Nasrifar–Moshfegian [9] (NM), Twu–Sim–Tassone [10] (TST), Kubic–Clausius [11,12] (KC), Schmidt–Wenzel [13] (SW), Harmes [14] (H), Peng–Robinson [8] (PR), Soave–Redlich–Kwong [6,7] (SRK), and van der Waals [1,2] (VDW). TST, H, PR and SRK are particular cases of the SW equation, and all of them fulfill the rule u + w = 1. The critical compressibility Zc , the reduced value of a(Tc ), and the reduced occupied volume b/vc are in the second, third, and fourth column, respectively. Attractive term

Zc ≡ Pc vc /RTc

˛3c ≡ a(Tc )Pc /R2 Tc2

b/vc

NM

a(T )/(v2 + 2bv − 2b2 )

0.302

0.498

0.313

TST

a(T )/(v + (5/2)bv − (3/2)b )

KC

a(T )/(v + u)

SW

a(T )/(v2 + ubv + wb2 ) with u + w = 1

2

2

3

8/27

2

(3/8) (1 + u/vc ) (3 − wb/vc )

−1

−1

(7/9)

1/4

(3/4)3

(1/3) (1 − 2u/vc )

(1 − Zc (1 − b/vc ))



3

1+

√ 3

2+u+r+

where r ≡ H

2

a(T )/(v + 3bv − 2b )

0.286

0.483

0.247

PR

a(T )/(v2 + 2bv − b2 )

0.307

0.457

0.253

SRK

a(T )/(v + bv)

1/3

0.427

0.260

VDW

a(T )/v2

3/8

(3/4)3

1/3

2

2

molar concentration, and B(T) is second virial coefficient (SVC). In this way, the SVC of Eq. (1) is



B(T ) = b 1 −



a(T ) . RTb

(2)

Eq. (2) is a direct relation between B(T) and the function a(T). Therefore, the function a(T) is defined through the SVC. Finally, B(T) can be obtained from its experimental measure, and/or it can be approached with a reasonable model for the interaction potential, for example, Square-Well, Lennard Jones, Yukawa, etc. 2. Equation of state 2.1. Critical point Eq. (1) is rewritten in the following manner



v3 − v2 b + c + d + −



bcd +

RT P





+ v bc + bd + cd +



RT a(T ) cd + b P P

= 0.

RT RT a(T ) c+ d+ P P P



(3)

In the critical point, Eq. (3) is the cube of a binomial, i.e. (v − vc )3 = 0, where vc is the critical molar volume. The coefficients of Eq. (3) are related to vc in the following manner 3vc = b + c + d +

RTc ; Pc

3v2c = bc + bd + cd +

v3c = bcd +

(4a)

RTc RTc a(Tc ) c+ d+ ; Pc Pc Pc

RTc a(Tc ) cd + b, Pc Pc

(4b)

(4c)

where Tc and Pc are the critical temperature and critical pressure, respectively. The formal expressions of the parameters b, c, and d are derived from Eqs. (4a)–(4c), and they are b

vc c

vc d

vc

=

=

=

Zc − 1 + ˛c ; Zc

(5a)

Zc − (1/2)˛c + ˛c Zc Zc − (1/2)˛c − ˛c Zc



˛c − (3/4)

;

(5b)

,

(5c)



˛c − (3/4)



−1 √ 3 2+u−r

u2 − 4w

where Zc = Pc vc /RTc is the critical compressibility and ˛c is a parameter defined by ˛3c ≡

a(Tc )Pc R2 Tc2

.

(6)

With Eqs. (5a)–(5c), we are able to obtain the parameters of the CES by using the critical compressibility Zc and the parameter ˛c , which are associated to the fluid. Thus, from experimental data, we can (exp) (exp) get the value Zc , and we will assume Zc = Zc from here on out. In this way, the CES takes into account the critical properties of the fluid, and it should give us a perfect agreement with these properties. The critical temperature, the critical pressure, the critical molar volume, and the critical compressibility of some substances are presented in Table 2. In general, the critical compressibility Zc , which is associated to other CES (see Table 1), is not in agreement with the experimental (exp) / Zc . This fact suggests that b, c, and d value of a substance, i.e. Zc = (exp) (which are calculated with Eqs.(5a)–(5c), and Zc ) may not correspond to a CES in Table 1. In other words, Eqs. (5a)–(5c) define the specific set of parameters of the CES of a given substance. Thus, the above methodology could be useful for mixtures since the mathematical formula of the CES does not change (only the parameters), and therefore, it allows using the well-known mixing rules. The second crucial element in Eqs. (5a)–(5c) is parameter ˛c . This parameter is related to the function a(T) at the critical temperature (see Eq. (6)). Later, in Section 3.1, ˛c will be exhibited as a function of the acentric factor ω. Finally, from the condition 0 < b < vc and Eq. (5a), ˛c is in the interval 1 − Zc < ˛c < 1. Table 2 The experimental data of the critical point of some substances are presented.[17].

Neon Argon Xenon Methane Nitrogen Ethane Propane Freon-12 Acetylene Benzene Carbon dioxide Ammonia Freon-113 n-Hexane Water Ethyl acetate

Tc (K)

Pc (bar)

vc (cm3 /mol)

Zc

44.4 150.9 289.7 190.6 126.3 305.5 370.0 385.0 308.7 562.6 304.2 405.6 487.3 507.9 647.3 523.3

26.5 50.0 58.3 46.4 34.0 49.1 42.6 41.3 62.4 49.2 73.8 112.9 34.1 30.3 221.2 38.5

41.8 74.5 118.3 99.0 92.2 141.8 196.0 215.2 113.2 256.9 94.0 72.5 325.1 368.3 56.7 286.3

0.300 0.297 0.286 0.290 0.298 0.274 0.272 0.278 0.275 0.270 0.275 0.243 0.274 0.264 0.233 0.253

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F. de J. Guevara-Rodríguez / Fluid Phase Equilibria 307 (2011) 190–196

2.2. Critical molar volume The critical molar volume vc is obtained from the first and the second volumetric derivative of the pressure at the critical point of the fluid



∂P ∂v



∂2 P ∂v2

= 0;

(7a)

Tc



3. Results = 0.

(7b) 3.1. The Reduced Equation of State

Tc

Cubic equation for the critical molar volume is derived from Eqs. (1), (7a), and (7b), and it is

v3c − 3bv2c + 3(bc + bd − cd)vc − (bc 2 + bcd + bd2 − c 2 d − cd2 ) = 0. (8) All the roots of this cubic equation are real, however, the unique physical root, which fulfills b < vc , is

vc = b + (b − c)2/3 (b − d)1/3 + (b − c)1/3 (b − d)2/3 .

(9)

The other two roots are discarded because vc < b. Finally, the set of Eqs. (5a)–(5c) is equivalent to Eq. (9).

In Section 1, the definition of function a(T), which is given in Eq. (2), was derived through the second virial coefficient [18,19]. Thus, a(T) can be obtained from experimental measures of B(T). However, many substances do not have available data for the function B(T), and in these cases, the second virial coefficient is determined, in an approximate way, with a proper model of the pair potential. Again, the goal of this work is to construct the simplest CES for a given substance, but it should be able to predict, with the most suitable accuracy, the features of the true equation of state of the fluid. With this purpose in mind, perhaps the Square-Well potential is the simplest model of a potential, which is not trivial. Its definition is:

⎧ ⎪ ⎨ +∞, 0 < r < ; ⎪ ⎩

−ε,

 < r < ;

0,

 < r,

(10)

where , , and ε are the diameter of the hard core, the attractive range, and the depth of the well in units of the energy, respectively. The Square-Well potential is a crude approximation of an intermolecular pair potential, where the liquid-vapor coexistence is due to the attractive well. Furthermore, the simplicity of the model enables us to have an analytical expression of the second virial coefficient [19], which is B(T ) =

  2  3 1 − (3 − 1)(eε/kB T − 1) . 3

(11)

The next two relations are derived from Eqs. (11) and (2), b=

2  3 ; 3

a(T ) = RTb(3 − 1)(eε/kB T − 1).

For a given substance, the CES is defined if the parameters Zc , ˛c , ε, and  are determined. In this way, the experimental value (exp) , and the of critical compressibility is assigned to Zc , i.e. Zc = Zc other parameters ˛c , ε, and , will be determined with the help of the Reduced Equation of State (RES). With this purpose, let us define the reduced variables vr ≡ v/vc , Tr ≡ T/Tc , and Pr ≡ P/Pc , where the molar volume v, the temperature T, and the pressure P are rescaled with its corresponding critical value. Eq. (1) is rewritten as Pr =

Tr ˛3 (Tr ) , − 2 Zc (vr − B) Zc (vr − C)(vr − D)

(13)

where B = b/vc , C = c/vc , D = d/vc , and ˛3 (Tr ) ≡ a(T )Pc /R2 Tc2 . Eq. (12b) is substituted into the definition of the function ˛3 (Tr ), and therefore, the result is

2.3. Second virial coefficient

u(r) =

volume, the critical temperature, and the acentric factor. Furthermore, in that work, , , and ε were only correlated with the experimental data of the second virial coefficient of some chemicals. In this work, the parameters ˛c , ε, and  are also correlated with experimental data of coexistence diagram. Finally, the procedure to calculate the parameters will be described in the next section.

(12a) (12b)

Clearly, the diameter of the hard core  is related to the parameter b, which is calculated with Eq. (5a). The resulting value of the hard core diameter for some substances is presented in Table 5. On the other hand, McFall et al. [20] derived a mean-field second virial coefficient. In their work, they take into account the anisotropic effects of the potential, and they found that the parameters , , and ε can be expressed in terms of the critical

˛3 (Tr ) = ˛3c Tr

 (1 + f )1/Tr − 1  c fc

,

(14)

where fc ≡ eε/kB Tc − 1. The most important consequence of Eqs. (13) and (14) is that the RES is independent of the parameter. In other words, the RES fulfills the corresponding-states principle. At this point, the next comment is relevant: The CES is not the equation of state of the Square-Well fluid. In fact, the virial expansion of the CES is identical to the virial expansion of the Square-Well model but only up the second coefficient. Moreover, the Square-Well model does not fulfill the corresponding-states principle because any change of the parameter  corresponds to a non-conformal change of the potential. The last statement has been demonstrated in the work of Fernando del Rio et al. [21] Therefore, the CES does not describe correctly the Square-Well model, however, the purpose in this work is to construct (in a systematic manner) the proper CES of a fluid. Despite the RES is independent from , the parameter  does not have an arbitrary value. In fact,  is determined through the experimental data of the second virial coefficient B(T). The procedure to calculate the parameter  with the experimental data of the SVC will be addressed in another future work. (sat) (sat) On the other hand, the pressure Pr , and the molar volume vr (sat) of the saturated vapor phase at the reduced temperature Tr = 0.7 are used to calculate the parameters ˛c , and ε. The acentric factor (sat) of Pitzer [22], which is defined as ω ≡ − log(Pr ) − 1, will be used (sat) along the rest of the work instead of the reduced pressure Pr . The (sat) molar volume vr , and the acentric factor ω of some substances (sat) (sat) are in Table 3. With these data (Pr and vr ), and the critical (exp) compressibility Zc of the substance, the parameters ˛c and ε are calculated to obtain a minimal deviation of the molar volume ( v) and the pressure ( P). The previous two quantities are defined by



P = 100 ×



,

(sat)

v = 100 ×

(sat)

Pr − Pr Pr

vr − v(sat) r v(sat) r

(15a)

.

(15b)

F. de J. Guevara-Rodríguez / Fluid Phase Equilibria 307 (2011) 190–196 Table 3 (sat) The experimental data of the acentric factor ω, and the reduced molar volume vr (sat) of the saturated vapor phase at the reduced temperature Tr = 0.7 of some substances are in the first two columns. The parameters ˛c and ε, in the last two columns, were calculated to achieve the minimal deviation of the molar volume ( v) and the pressure ( P).

Neon Argon Xenon Methane Nitrogen Ethane Propane Freon-12 Acetylene Benzene Carbon dioxide Ammonia Freon-113 n-Hexane Water Ethyl acetate

v(sat) r

ω

˛c

ε/kB Tc

19.01 21.40 22.10 22.70 23.05 28.96 34.17 34.97 36.11 39.23 38.70 46.89 43.14 49.25 61.67 60.24

−0.037 −0.003 0.002 0.016 0.036 0.100 0.160 0.175 0.188 0.207 0.224 0.252 0.255 0.299 0.344 0.367

0.8070 0.8079 0.8140 0.8114 0.8009 0.8247 0.8258 0.8181 0.8188 0.8219 0.8191 0.8382 0.8208 0.8269 0.8411 0.8317

0.1828 0.2383 0.2037 0.2469 0.3570 0.2892 0.3742 0.4535 0.4672 0.4729 0.5171 0.4146 0.5494 0.5640 0.5170 0.6171

193

0.7 0.6 0.5 0.4 0.3 0.2 0.1 -0.1

0.0

0.1

0.2

0.3

0.4

Fig. 2. The parameter ε as a function of the acentric factor ω. The solid-circles correspond to a minimal deviation of v and P. The open-circles is the Taylor expansion (Eq. (17)).

(sat)

The results for the parameters ˛c , and ε are given in Table 3. The corresponding graphs of these parameters are in Figs. 1 and 2, and their values are plotted with solid-circles. The parameters exhibit an important dispersion, and this feature shows us that ˛c and ε (sat) depend on the molar volume vr , and on the acentric factor ω of the substance. Despite the fact that the functions are unknown, the values of ˛c and ε can be approached with their Taylor expansion in two variables, (sat) 2

˛c ≈ a1 [vr

(sat)

] + a2 vr

a1 = −0.000086 a2 = 0.008928 a3 = −0.669840

b1 = 0.000602 b2 = −0.057395 b3 = 3.780987

+ a5 ω + a6 ,

(16)

(sat)

(17)

a4 = 0.009406 a5 = −0.488859 a6 = 0.647036

(sat) 2

ε/kB Tc ≈ b1 [vr

(sat)

ω + a3 ω2 + a4 vr

(sat)

] + b2 vr

ω + b3 ω2 + b4 vr

+ b5 ω + b6 ,

b4 = −0.068888 b5 = 5.241789 b6 = 1.434793

where the coefficients ai , and bi (with i = 1, 2, . . ., 6) are calculated with the method of least-squares. With Eqs. (16) and (17), the 0.84

Table 4 The deviation of the pressure P˜ = 100 × (P˜ − P (sat) )/P (sat) , the molar volume of the (sat) (sat) saturated liquid phase ˜vl = 100 × (˜vl − vl )/vl , and the molar volume of the (sat)

(sat)

saturated vapor phase ˜vv = 100 × (˜vv − vv )/vv at the reduced temperature: (sat) = 0.7 are presented. The pressure P˜ and the volume v˜ have been calculated Tr with the approximate values of the parameters ˛c and ε, which are given in Eqs. (16) and (17), respectively.

0.82

0.80 -0.1

parameters (˛c and ε) can be estimated with the molar volume vr and the acentric factor ω of the saturated vapor phase at the tem(sat) perature Tr = 0.7 of a fluid. The approximate values of ˛c , and ε are also plotted in the same Figs. 1 and 2 as open-circles. Clearly, the main behavior of the previous values (solid-circles) is captured with the corresponding Taylor expansion. Thus, Eqs. (16) and (17) give a good estimation of the parameters ˛c and ε, respectively. For a given substance, Eqs. (16) and (17) are used to calculate the approximate value of ˛c , and ε, respectively. After that, the parameters are used to construct the RES with Eqs. (13) and (14). ˜ the molar volume of the In this way, the coexistence pressure P, saturated liquid v˜ l , and the molar volume of the saturated vapor v˜ v are calculated through the Maxwell construction. The deviation ˜ the liquid volume v˜ l , and the vapor volume v˜ v of the pressure P, with respect to their experimental values are in Table 4. Clearly, the results differs from its experimental value in less than 2% in almost all the cases, and principally with the coexistence pressure P˜ and the molar volume of the saturated vapor phase v˜ v . Due to the fact that the experimental data of the molar volume of the saturated liquid phase v˜ l were not used to determine the parameters ˛c and ε, they must be used to evaluate the goodness of the RES. In this way, the deviation of the volume of the saturated

0.0

0.1

0.2

0.3

0.4

Fig. 1. The parameter ˛c as a function of the acentric factor ω. The solid-circles correspond to a minimal deviation of v and P. The open-circles is the Taylor expansion (Eq. (16)).

Substance

P˜ (%)

˜vl (%)

˜vv (%)

Neon Argon Xenon Methane Nitrogen Ethane Propane Freon-12 Acetylene Benzene Carbon dioxide Ammonia Freon-113 n-Hexane Water Ethyl acetate

−0.10 0.07 0.15 0.11 −0.07 0.16 0.13 0.09 0.08 0.08 0.03 −0.11 0.07 0.09 0.13 0.34

−0.72 1.15 −1.16 0.00 3.85 −5.08 −3.10 1.03 0.72 3.38 −1.79 0.61 1.55 0.17 0.90 −1.98

−0.24 −2.46 −0.89 −1.27 −0.16 0.80 0.03 −0.78 0.13 −1.64 2.40 2.22 −0.66 0.24 1.06 0.43

194

F. de J. Guevara-Rodríguez / Fluid Phase Equilibria 307 (2011) 190–196

1.0

H2 O P / Pc

Δvliq [%]

10

Soave-Redlich-Kwong Peng-Robinson Schmidt-Wenzel This CES

0

0.5

-10 0.0

0.1

0.2

0.3

0.0 0.1

0.4

1

ω

liquid phase vl is plotted as a function of the acentric factor ω in Fig. 3. In the same figure, the deviation of the volume with other theoretical predictions is also plotted. Those cases correspond to the equations of Soave–Redlich–Kwong (SRK), Peng–Robinson (PR) and Schmidt–Wenzel (SW). In the case of the SRK equation, the minimal deviation of the volume occurs for the fluids with a small value of the acentric factor, i.e. ω ≈ 0. In the second case, the PR equation gives the best results for the fluids with ω ≈ 0.35. In the third case (SW equation), and our RES (Eq. (13)), the results for vl are similar, and they have a completely different behavior with respect to the previous two cases. The results in the last two cases have an almost null mean and a similar dispersion. Therefore, the SW equation and our RES are equivalent if the reduced variables of the fluid are taken into account. However, they are different because the RES is constructed with the experimental value of the critical compressibility (exp) Zc . 3.2. Coexistence diagram In the previous section, the methodology to construct the RES has been described. In the methodology, the inputs are the vol(sat) ume of the saturated vapor phase vr at the reduced temperature (sat) Tr = 0.7, the acentric factor ω, and the critical compressibility Zc of the fluid. The procedure to construct the RES consists of two (sat) steps: In the first one, the volume vr , and the acentric factor ω are used to determine the parameters ˛c and ε with Eqs. (16) and (17), respectively. In the second step, ˛c , ε, and Zc are used to construct the RES with Eqs. (13) and (14). In this way, the resulting RES is constructed with two states, namely, the critical point, and the sat(sat) urated vapor at the temperature Tr = 0.7. Other thermodynamic properties of the fluid should be used to check the goodness of the RES. Thus, the theoretical coexistence diagram and the experimental data for water are presented in Fig. 4. The experimental data and the theoretical results are given with solid-circles, and a solid-line, respectively. Other results, with the Peng–Robinson (PR) equation (open-circles), are also plotted in the same Fig. 4. Clearly, the theoretical results with Eq. (13) are in much better agreement with the experimental data than the results with the PR equation. This scenario is very similar for other substances. For example, the coexistence diagram of the methane is illustrated in Fig. 5. Again, the results with the RES are in better agreement with the experimental data than the other theoretical results with Soave–Redlich–Kwong [6,7] equation. In both cases, the volume translated cubic equation of state [15,23] is not necessary in order to correct the volume of the saturated vapor phase and the saturated liquid phase. In this point,

100

Fig. 4. Coexistence diagram of the water. The solid- and the open-circles are the experimental data [25,17] and the results with the Peng–Robinson [8] equation, respectively. Our results, which are obtained with Eq. (13), are plotted with the curve.

in the work of Peneloux et al. [23], the corrected volume vˆ = v − t vc is proposed and Eq. (1) is rewritten as ˆ vˆ ) = P(T,

RT

vˆ + t vc − b

−

a(T ) , (ˆv + t vc − c)(ˆv + t vc − d)

(18)

where t vc is a small volume correction factor. Thus, the parameters ˆ vc ≡ b/vc − t; of the cubic equation (18) are redefined, namely, b/ ˆ cˆ /vc ≡ c/vc − t; and d/vc ≡ d/vc − t. Moreover, equations (5a)–(5c) are valid for the new parameters ˛ ˆ c , and Zˆ c at the critical point. ˛ ˆc and Zˆ c are related to ˛c and Zc through the equations ˛ ˆc =

˛c + 2tZc ; 1 + 3tZc

(19a)

Zˆ c =

Zc . 1 + 3tZc

(19b)

Clearly, if t = / 0 then the critical compressibility of the volume translated equation differs from the experimental value Zc . On the other hand, the parameter ˛c corresponds to the minimum of Eqs. (15a) and (15b), and therefore, ˛ ˆ c does not fulfill with the above condition and the volume translated equation does not improve the cubic equation with parameters ˛c , and Zc . Recently, in the work of Kurt Frey et al. [24] the volume translations, which depend

1.0

CH4

P / Pc

Fig. 3. Deviation of the molar volume of the saturated liquid phase at the reduced temperature T(sat) /Tc = 0.7.

10

v / vc

0.5

0.0 0.1

1

10

100

v / vc Fig. 5. Coexistence diagram of the methane. The solid- and the open-circles are the experimental data [25,17] and the results with the Soave–Redlich–Kwong [6,7] equation, respectively. Our results, which are obtained with Eq. (13), are plotted with the curve.

F. de J. Guevara-Rodríguez / Fluid Phase Equilibria 307 (2011) 190–196 Table 5 The hard sphere diameter  is calculated with Eq. (12a). The sum of the parameters u and w of Eq. (20) is in the second column.

Neon Argon Xenon Methane Nitrogen Ethane Propane Freon-12 Acetylene Benzene Carbon dioxide Ammonia Freon-113 n-Hexane Water Ethyl acetate

 (Å)

u+w

2.272 2.771 3.184 3.018 2.955 3.351 3.764 3.911 3.141 4.176 2.916 2.683 4.499 4.659 2.437 4.204

−0.3522 −0.4001 −0.3189 −0.3504 −0.3520 −0.2790 −0.3460 −0.3810 −0.3526 −0.4395 −0.2619 −0.3722 −0.4062 −0.3852 −0.3009 −0.2804

on the temperature, are insufficient to correct CES, and therefore, the authors suggest a temperature- and density-dependent volume correction. 3.3. Schmit–Wenzel equation The Cubic Equation of State (CES), which is given in Eq. (1), is also viewed in the works of Esmaeilzadeh–Roshanfekr [26] and Schmit–Wenzel [13]. In the first case, the experimental value of the critical compressibility Zc is always overestimated. In the second case, Eq. (1) is rewritten as P(T, v) =

RT

v−b

−

a(T )

v2 + ubv + wb2

,

(20)

where the parameters u and w are related to c and d through the relations 1 c = − b(u − r); 2

(21a)

1 d = − b(u + r), 2

(21b)



u2 − 4w. In that work, Schmit and Wenzel [13] detected and r = the relation u + w = 1 in some known equations, namely, Harmes [14], Peng–Robinson [8], and Soave–Redlich–Kwong [6,7] (see Table 1). Thus, the authors analyzed the Eq. (20) with the restriction u + w = 1. On the other hand, the resulting CES, which is determined with (exp) the experimental value of the critical compressibility Zc , does not fulfill the relation u + w = 1 for any of the substances given in Table 5. This fact demonstrates that the CES is unique for a substance, and the resulting CES is none of the usual equations found in the literature. Finally, Eqs. (21a) and (21b), and the restriction u + w = 1 are substituted in Eq. (9). Therefore, the analytical expression of the rate ˇc ≡ b/vc is obtained, and it corresponds to ˇc =



1+

 3

2+u+r+

 3

2+u−r

−1

.

(22)

This is the formal solution of the equation (6ω + 1)ˇc3 + 3ˇc2 + 3ˇc − 1 = 0, which corresponds to Eq. (7) in the work of Schmit–Wenzel [13]. 4. Conclusions A new Cubic Equation of State for a simple substance is presented in this work. The CES of a fluid is determined with the experimental values of the acentric factor ω, the reduced volume

195 (sat)

at the reduced temperature of the saturated vapor phase vr (sat) Tr = 0.7, and the critical compressibility Zc . with those inputs, the parameters ˛c and ε are determined with Eqs. (16) and (17), respectively. In the final step, the previous inputs, and the parameters are used to construct the Reduced Equation of State (RES) with Eqs. (5a)–(5c), (13), and (14). In the RES, the function ˛3 (Tr ) ≡ a(T )Pc /R2 Tc2 , which is given in Eq. (14), takes into account the relation between the function a(T) and the second virial coefficient B(T) of the Square-Well potential (Eq. (12b)). However, the RES does not correspond to a fluid with the Square-Well potential. In fact, the RES fulfills with the corresponding-states principle, and the fluid with the Square-Well potential does not. With the resulting RES of a fluid, the deviation of the coexistence pressure, the volume of the saturated vapor, and the volume (sat) of the saturated liquid at the reduced temperature Tr = 0.7 were recalculated. For several substances, the theoretical predictions are in agreement with their corresponding experimental values, and the deviation is less than 2% in almost all cases. Furthermore, the coexistence diagram for a polar substance as water was calculated with its corresponding RES, and the Peng–Robinson equation. The theoretical results with Eq. (13) are in agreement with the experimental data, and they offer us small deviations when those results are compared to the results of the Peng–Robinson equation. This scenario is fulfilled for other substances as methane, which is a nonpolar chemical. In this case, the results with Eq. (13) are in much better agreement with the experimental data than the results with Soave–Redlich–Kwong equation. Clearly, the RES well works to predict the coexistence diagram, and it should work to evaluate the residual properties of the fluid. In general, the RES of a given fluid is unique, and it is notably different when it is compared to other Cubic Equations of State found in the literature. On the other hand, Eq. (1) can be extended to mixtures by using an appropriate mixing rule. Perhaps, the most commonly mixing rules are mentioned in the work of Twu et. al. [27], but the extended RES will be addressed in a future work. Finally, the fundamental base of the CES is an ensemble of spherical particles, which interact with a hard core and an attractive pair potential. Under this physical notion, a CES describes all the thermodynamic states, where the molecular details are not relevant. This condition can be false if the system concentration is high and/or if the fluid is constituted by complex molecules, and therefore, in those cases the CES must not be accurate. Acknowledgment The author thanks to Instituto Mexicano del Petróleo for the support to this work through the grants: D.00406 and D.00264. Dedicated to Debi. References [1] J.S. Rowlinson, J.D. van der Waals, On the Continuity of the Gaseous and Liquid States, Elsevier, Amsterdam, 1988. [2] J.D. van der Waals, Doctoral Dissertation, Leiden, Holland, 1873. [3] G. Soave, Fluid Phase Equilib. 84 (1993) 339. [4] A. Lucia, J. Thermodynam. (2010) 238365 (2010). [5] K.C. Chao, R.L. Robinson, Jr., Equations of State American Chemical Society, Washington, DC, 1986. [6] G. Soave, Chem. Eng. Sci. 27 (1972) 1197. [7] O. Redlich, J.N.S. Kwong, Chem. Rev. 44 (1949) 233. [8] D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59. [9] K. Nasrifar, M. Moshfeghian, Fluid Phase Equilib. 190 (2001) 73. [10] C.H. Twu, V. Tassone, W.D. Sim, AIChE J. 49 (2003) 2957. [11] W.L. Kubic, Fluid Phase Equilib. 9 (1982) 79. [12] R. Clausius, Ann. Phys. 9 (1880) 337. [13] G. Schmidt, H. Wenzel, Chem. Eng. Sci. 35 (1979) 1503. [14] A. Harmes, Cryogenics 17 (1977) 519. [15] J.J. Martin, Ind. Eng. Chem. Fundam. 18 (1979) 81.

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