A new equation of state for gaseous, liquid, and supercritical fluids

Accepted Manuscript A new equation of state for gaseous, liquid, and supercritical fluids N. Farzi, P. Hosseini PII:

S0378-3812(15)30097-2

DOI:

10.1016/j.fluid.2015.08.027

Reference:

FLUID 10734

To appear in:

Fluid Phase Equilibria

Received Date: 3 June 2015 Revised Date:

25 August 2015

Accepted Date: 25 August 2015

Please cite this article as: N. Farzi, P. Hosseini, A new equation of state for gaseous, liquid, and supercritical fluids, Fluid Phase Equilibria (2015), doi: 10.1016/j.fluid.2015.08.027. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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supercritical fluids

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A new equation of state for gaseous, liquid, and

a

P. Hosseini

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b

N. Farzi*

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[email protected]

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Fax: 0098-31-36689732

a,b

Department of Chemistry, University of Isfahan, Isfahan, Iran

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ABSTRACT In the present study, a new equation of state (EOS) was derived by using the thermodynamic

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equation of state and the intermolecular potential (3, 9, 12). It was shown that the EOS is applicable in low and high ranges of temperature, pressure and density for gaseous, liquid and supercritical fluids and even in liquid-gas phase transition region. The new EOS is applicable for

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a variety of fluids such as polar, nonpolar, rare gases, short-chain and long-chain hydrocarbon fluids. The absolute percent deviation of the calculated density for gaseous, liquid and

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supercritical fluids is very low. The common bulk modulus point and the common compression point regularities were predicted by the new EOS. The new EOS was compared with some equations of state which had been derived similarly. It is shown that the repulsive potential used

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in the EOS derivation is effective in predicting correct fluid properties.

Keywords: equation of state, intermolecular potential, dense fluid regularity, liquid-gas phase

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transition.

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1. Introduction The structure and properties of substances depend on intermolecular forces. Many studies

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for the careful formulation of the relationship between the bulk properties of substance and intermolecular forces have been done [1-5]. Such formulation is the ultimate goal of the molecular theory of substances. By developing such theories, all the bulk properties of substance

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can be evaluated just by knowing intermolecular forces [6-9].

Clearly, a close relationship between the macroscopic properties of fluids and the

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intermolecular forces can be established by the equation of state (EOS). Parsafar and Mason provided a general EOS for compressed solids using a universal binding energy [10]. The linear isotherm regularity, LIR, which is applicable for dense and supercritical fluids, was obtained based on a lattice model with Lennard-Jones potential (6, 12) [11] and then extended to mixtures

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[12]. The LIR predicts a linear behavior of ( Z − 1)v 2 isotherms versus ρ 2 for ρ 〉 ρ B and T 〈 2T B , where ρ = 1 v and T B are the molar density and Boyle temperature, respectively. Goharshadi et al. derived an EOS (GMA) by a similar method as the LIR, based on the average potential (9, 12)

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[13]. The GMA EOS predicts the linear behavior of ( 2 Z − 1)v 3 isotherms versus ρ in T 〈T C and ρ 〉 ρC where T c and ρC are the critical temperature and density, respectively. Ghatee and

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Bahadori [14] provided an EOS for liquid metals using the potential (3, 6) which predicts that the (Z − 1)v 2 isotherms versus 1 ρ are linear. Baniasadi and Ghader (BGH) [15] obtained an EOS

using the potential (3, 6, 9) that is applicable for the entire range of liquid and gas in high pressures and densities. Parsafar et al. [16] provided an EOS (PSP) using the potential (3, 6, 12). This equation holds for liquid metals, short-chain hydrocarbon, polar, nonpolar, and hydrogenbonding fluids. In addition, the behavior of metallic, covalent and ionic solids is justified by this

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equation. The application range variety related to the equations of state and obtained by the potentials (6, 12), (3, 6, 9) and (3, 6, 12) refers to the role of the potential type in the applicability of the EOS. The derivation method of the mentioned equations of state was almost equal;

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however, there are many equations of state that used the model potential functions for the prediction of the fluid properties in different temperatures and pressures like PC-SAFT and

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SAFT-VR descriptions [17-20].

In this work a new interaction potential, (3, 9, 12), was considered for fluids that had not

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been used before. The repulsion, the attraction and the long range interaction like polar-polar interactions are included in the potential function. The derived EOS parameters have the molecular basis and the fluid properties can be described in terms of the molecular interactions. This paper has been organized as follows: in part 2, an EOS with three temperature

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dependent parameters is derived by employing the thermodynamic equation of state and the interaction potential (3, 9, 12). Accordingly, the temperature dependencies of the new EOS parameters are obtained. In part 3, the applicability of the EOS for a variety of fluids in different

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temperatures and densities, and also in the liquid- gas transition region is investigated. In addition, the EOS ability predicting the density of fluid is determined and the common bulk

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modulus point and the common compression point regularities are investigated by the new EOS in this part. In part 4, the new EOS is compared with GMA, BGH and PSP equations of state, and the role of interaction potential is discussed. Finally, the conclusion is presented in part 5. 2. Equation of state derivation Equations of state are the key relations for the calculating the thermodynamic properties of substances. The properties of substances are related to their intermolecular potentials.

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Therefore, a close relationship between equations of state and the intermolecular forces is expected. As mentioned before, several equations of state were obtained, based on the

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intermolecular potential while each having its own advantages. Clearly, the attractive range increases with adding r −3 term in the functional form of the potential. It was shown that the application range of the extracted equations of state increased by

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using this form of potential in terms of fluid variety, temperature and density [15, 16]. So, in the present work, the potential (3, 9, 12) is used for deriving EOS and comparing it with any similar

is:

U =

N  C 12 (T ) C 9 (T ) C 3 (T )  z + + 2  r 12 r9 r 3 

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EOSes. The functional form of the potential (3, 9, 12) with temperature-dependent coefficients

(1)

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where, N is number of molecules, z is density independent coordination number, r is distance between two molecules, and C i is temperature-dependent coefficient of the potential function. The term r −3 in Eq. (1) relates to the long-range interactions and assigned to the polar-polar

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respectively.

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interactions. The terms r −9 and r −12 attribute to attraction and repulsion interactions,

Replacing r = k v 1/3 into Eq. (1) and rearranging the result, yield: U  C 12′ (T ) C 9′ (T ) C 3′ (T )  = + + N  v 4 v3 v 

(2)

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where, C 3′ (T ) , C 9′ (T ) and C 12′ (T ) are new temperature-dependent coefficients of the potential function. Here, we assumed that the temperature dependencies of C i′ coefficients are quadratic

2

C 9′ (T ) = B 1′′ + B 2′′T + B 3′′T

2

C 3′ (T ) = C 1′′+ C 2′′T + C 3′′T

2

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C 12′ (T ) = A1′′ + A 2′′T + A 3′′T

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as: (3)

(4)

(5)

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where A i′′ , B i′′ , and C i′′ are constant values that depend on the system. Replacing Eqs. (3) - (5) into Eq. (2), gives:

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U  A1′′+ A 2′′T + A3′′T 2 B1′′+ B 2′′T + B 3′′T 2 C 1′′+ C 2′′T + C 3′′T 2  = + +  N  v4 v3 v 

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 U   ∂ N   Pi =      ∂v     T

can be calculated by,

(7)

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Using Eq. (6), the internal pressure,

(6)

The following thermodynamic equation of state connects the internal pressure to the pressure:  ∂P  P =T  − Pi  ∂T  ρ

where T ( ∂P ∂T ) ρ is thermal pressure.

(8)

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Then, the EOS can be obtained by replacing Eq. (6) into (7) and gaining Eq. (8). Making some rearrangements gives: B (T ) C (T ) + 3

ρ

(9)

ρ

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( Z − 1)v 4 = A (T ) +

This is an EOS that is obtained by using the potential (3, 9, 12). This EOS exhibits the behavior

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of ( Z − 1)v 4 isotherms with density. The temperature dependency of the new EOS parameters,

A1 A2 ln T A3T − − RT R R

B (T ) = B0 +

B1 B2 ln T B3T − − RT R R

C ( T ) = C0 +

C1 C2 ln T C3T − − RT R R

(10)

(11)

(12)

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A(T ) = A0 +

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A(T), B(T) and C(T), are as follows:

where R is the universal constant of gases and A i , B i , and C i are constants of Eqs. (10)- (12)

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that i= 0, 1, 2, 3.

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3. Evaluation of the equation of state In this part, the new EOS is investigated to specify its applications in terms of system

variety, temperature and density ranges. The fluid density and dense fluid regularities are also predicted by the new EOS. 3.1. Application range

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Perhaps the most reliable method to evaluate an EOS is to test it with experimental data. Although experimental data are not available in whole range of temperature, pressure and density, comparison with the experimental data is yet valuable for deciding the accuracy and

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precision related to the results of a theoretical method. Without major changes in the structure and properties of a substance, there is mainly a similar trend in different regions. This allows us to predict properties in inaccessible areas using the EOS. An accurate and reliable EOS will help

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us achieve this purpose.

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To find application range of the new EOS, the isotherms of ( Z − 1)v 4 in each density were calculated by using the P-v-T experimental data [21], for NH3, H2O, CH3OH, Ar, Ne, Kr, Xe, CH4, C2H6, C3H8, C4H10, Iso-C4H10, C5H12, C6H14, C7H16, C8H18, C9H20,CO, CO2, N2, H2 and O2 in

gaseous, liquid and supercritical states. The isotherms of ( Z − 1)v 4 versus ρ were plotted for

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these fluids in gaseous, liquid and supercritical states. Fitting the isotherms of ( Z − 1)v 4 with Eq. (1) gives the EOS parameters, A(T), B(T) and C(T). As an example, Figs. 1a-1c shows the isotherms of

(Z

− 1)v 4 versus ρ , for C9H20 in

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gaseous, liquid and supercritical states, respectively. In these plots, the symbols show the

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experimental data and the line determines the fitted experimental values with the new EOS. The fitted parameters, A(T), B(T) and C(T), and their coefficients of determination, R2 (sixth column), are given in Table 1 for Nonane, Krypton and Ammonia, as typical examples. The values of coefficients of determination show that the new EOS is in good accordance with the experimental data. The values of temperature-dependent parameters, A(T), B(T) and C(T), were plotted versus temperature in three states of substance and fitted with Eqs. (10)-(12). The parameters of fitting are the constants of Eqs. (10)-(12). Tables 2-4 contain a summary of the

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results related to fitting A(T), B(T) and C(T) parameters with temperature for gaseous, liquid and supercritical states of some substances. The coefficients of determination of fitting are given in the last column of Tables 2-4. The results in Tables 1-4 confirm that the new EOS can be applied

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for such substances in each three states of gaseous, liquid and supercritical.

In order to examine the applicability of the new EOS in the gas-liquid transition region,

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the isotherms of ( Z − 1)v 4 vs ρ were plotted for some fluids using their P-v-T experimental data in this region. Fig. 2 typically shows such plot for ethane in the gas-liquid phase transition region

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and has been fitted by the new EOS. Table 5 summarizes the results of such fitting for C2H6, CO2, NH3 and O2 in their gas-liquid phase transition region. The values of the coefficients of determination which are given in the last column of Table 5 show that the new EOS fairly holds for the gas-liquid phase transition region of these fluids.

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3.2. Density prediction

To evaluate the ability of the new EOS to predict the experimental density of fluids, the

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EOS was primarily rearranged as:

(13)

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RT  Aρ 5 + B ρ 4 + C ρ 2 + ρ  −1 = 0 P

By replacing A(T), B(T) and C(T) values in each temperature, density was calculated in the corresponding pressure.

The maximum, minimum and mean absolute percent deviation of the calculated density ( ( ρ exp − ρcal ρ exp ) × 100 ) in given ranges of pressure and temperature for some substances in gaseous, liquid and supercritical states, have been reported in Table 6. The values of absolute

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percent deviation show that the EOS can be used for predicting the density of substances in each three states. The results in Table 6 show that the mean deviation of the calculated density for

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liquids is lower than the gaseous and supercritical fluids. 3.3. Regularity prediction

Developing simple models to describe the thermodynamic behavior of dense fluids is a

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difficult task because of the diversity of molecular interactions that these fluids present. Nevertheless, there are many interesting regularities in physical properties of dense fluids and

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liquids [11, 13, 22-27]. To understand the role-played by the intermolecular forces, investigating the volumetric behavior of dense fluids can be useful. Since the volumetric properties of fluids are predictable by exact equations of state, it is expected that the dense fluid regularities are also predictable by these equations of state. In this work, the common bulk modulus point and the

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common compression point regularities have been investigated to confirm the new equation state in predicting such experimental regularities. 3.3.1. Common bulk modulus point

1  ∂P    RT  ∂ρ T

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Br =

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The reduced bulk modulus, B r , is defined as:

(14)

The experimental data show B r increases by increasing density at constant temperature. Huang and O’Connell examined the behavior of bulk modulus for more than 250 substances [22]. They concluded that for each substance, a density exists where bulk modulus is independent of temperature, i.e., the isotherms of bulk modulus also present a characteristic crossing point.

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Hence, it is expected that density is a dominant variable in determining the behavior of B and the temperature effects could be ignored. The crossing point of the isotherms of bulk modulus versus

placed in the minimum or maximum of B r isochors, i.e.:

(15)

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 ∂Br    =0  ∂T  ρ

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density was called common bulk modulus point. The common bulk modulus point density is

In other words, the reduced bulk modulus is independent of temperature at this density. The

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common intersection point of the isotherms of the experimentally reduced bulk modulus versus density for Ar is shown in Fig. 3. Since the new EOS fairly predicts the density of substances in different temperatures, it is expected that the reduced bulk modulus can be also calculated with the same accuracy as density by using the new EOS. To find out the ability of the new EOS

EOS as, Br = 5 Aρ 4 + 4 B ρ 3 + 2C ρ + 1

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showing the common intersection point, the reduced bulk modulus was calculated by the new

(16)

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Fig. 4 shows the isotherms of the calculated reduced bulk modulus versus density for Ar. As it

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can be seen, all isotherms intersect in a characteristic point. The common intersection point of the reduced bulk modulus which is calculated by the new EOS agrees with Huang and O’Connell’s crossing point. The calculated and the experimentally reduced bulk modulus and their corresponding common intersection point are compared in Fig. 5. This figure shows the good accordance between the calculated and the experimental values.

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Table 7 shows the calculated and experimental common bulk modulus point densities, ρOB , for Ar, N2, and CH4. The good agreement between the calculated and experimental values of

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ρOB implies that the new EOS can successfully predict the common intersection point regularity.

Since the common intersection point in the linear form of plots can be seen better, we rearranged Eq. (16) as:

ρ3

= 5 Aρ + 4 B

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Br − 1 − 2C ρ

(17)

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The isotherms of ( B r − 1 − 2C ρ ) ρ 3 versus ρ for Ar were plotted and placed at the left corner of Fig. 4. However, the intersection point densities of Br and ( B r − 1 − 2C ρ ) ρ 3 isotherms are not equal; the linear form of Eq. (16) is very useful in the qualitative evaluation of the intersection

(18)

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  Br − 1 − 2C ρ    ∂  ρ3  =0     ∂T     ρOB

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point. So, the common intersection point density of the linear plots, ρOB , can be calculated by,

ρOB

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Thus, ρOB is obtained as:

 ∂B    4  ∂T  ρOB 4 B1 + B2T + B3T 2 =− =− 5  ∂A  5 A1 + A2T + A3T 2    ∂T  ρOB

(19)

Where B i and A i are the constants of Eqs. (10) and (11). The obtained relation for ρOB , has two implicit outcomes. First, ρOB is not exactly independent of temperature. However, in short range of temperature the ratio of nominator to dominator in Eq. (19) is nearly constant but the common

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intersection point varies with variation of the temperature range. This result also had been pointed out by Boushehri et al. previously [28]. They performed a theoretical analysis using a statistical- mechanical-based EOS and showed that the isotherms of B do not intersect at exactly

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one point, but rather over a small range of density. Second, if B2 and B3 and also A2 and A3 are small in comparison to B1 and A1, respectively, they can be neglected in the low temperatures

4 B1 5 A1

(20)

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ρOB = −

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and we will have,

Therefore, the common intersection point density has been related to ratios of B1 and A1 which are constants and are related to the attraction and the repulsion interactions, respectively. 3.3.2. Common compression point

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It was experimentally seen that the compressibility factor isotherms of liquids and dense fluids intersect in a characteristic point which is called the common compression point. The density of this point, ρOZ , is temperature independent. Fig. 6 shows the experimental isotherms

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of the compressibility factor of Ar in the liquid state. In this figure the circles show the values of the experimental compressibility factor and the line is given to follow the compressibility factor

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in high densities while the arrow shows the increasing direction of temperature. The common compression point can be seen clearly from Fig. 6. To find out the ability of the new EOS in predicting common compression point, the

compressibility factor was calculated from the new EOS as, Z = A ρ4 + B ρ3 +C ρ +1

(21)

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The calculated compressibility factor isotherms are plotted in Fig. 7 for Ar. The common intersection point is also seen in Fig. 7. The calculated and the experimental common

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intersection point densities, ρOZ , are given in Table 7 for Ar, N2 and CH4. Since the common intersection point can be seen better in the linear form of plots, Eq. (21) was rearranged as,

( Z − 1 − C ρ ) = B + Aρ

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ρ3

(22)

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The isotherms of ( Z − 1 − C ρ ) ρ 3 versus ρ for Ar are given in the left corner of Fig. 7 for comparison.

The intersection point density of ( Z − 1 − C ρ ) ρ 3 isotherms can be calculated by using

(23)

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  (Z −1− Cρ )    ∂  ρ3  =0     ∂T       ρOZ

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Eq. (22) as,

ρOZ

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This condition gives the common compression point density, ρOZ , as,  ∂B     ∂T  ρOB B + B2T + B3T 2 =− =− 1 A1 + A2T + A3T 2  ∂A     ∂T  ρOB

Comparison of Eq. (19) with (24) reveals that,

(24)

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4 5

ρOB = ρOZ

(25)

Eq. (25) shows the relation between ρOB and ρOZ which were obtained by incorporating the linear

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form of the reduced bulk modulus and the compressibility factor isotherms, namely as Eqs. (17) and (22).

It must be noticed that by the rearranging Eqs. (16) and (21) to the linear forms, namely

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Eqs. (17) and (22), we could obtain the analytical relations for ρOB and ρOZ and Eq. (25) gives

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the connection between them. Comparison of the experimental and the calculated values of ρOB with the corresponding values of ρOZ in Table 7 shows that relation (25) is almost general and even

holds for the common intersection point densities that are obtained from the nonlinear form of

4. Discussion

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the reduced bulk modulus and the compressibility factor isotherms Eqs. (16) and (21)).

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To evaluate the new EOS, the equation was compared with similar equations of state such as BGH and PSP in derivation of which the potential term r −3 has been used in while it has

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not been incorporated in the GMA EOS. The coefficients of determination of the fitted P-v-T experimental data of NH3, Kr, and C9H20 in gaseous, liquid and supercritical states with GMA, BGH, and PSP equations of state have been compared with the new EOS in Table 1. Results show that the GMA EOS which has not employed the potential term r −3 in its derivation is not applicable for gaseous and supercritical fluids. However, BGH, PSP and the new equations of state which have used the potential term r −3 in their extractions are applicable in a wide range of temperature, density and pressure for gaseous, liquid and supercritical states. This subject reflects

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the important role of the long-range interaction in some states of substances. It is noticeable that PSP and the new equations of state are preferred to BGH EOS in all three states. This can be

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observed from Table 8 by comparison of the values of absolute percent deviations. The potentials used in derivation of several recent equations of state have been compared in Fig. 8 to find the reason behind the behavioral differences of the equations of state. In Fig. 8,

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the intermolecular potential (3, 6, 12) in PSP, (3, 9, 12) in this work, and (3, 6, 9) in BGH have been compared with each other and with the potential (6, 9, 12) in GMA which does not use the

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potential term r −3 . It can be seen that the term r −3 in the mentioned potentials brings about increases in the attractive interaction range. Therefore, it is expected that the fluids with strong long range interactions can better predict by relying on the derived equations of state which used the potential term r −3 . It is observable from Fig. 8 that the intermolecular potentials which contain polar-polar interactions, the term r −3 , are almost equal in attraction range. Thus, the

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differences in prediction of the fluid properties by the equations of state are probably related to the repulsive part of the potentials. In Fig. 8, the repulsion part of (3, 9, 12) intermolecular

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potential is located between (3, 6, 12) and (3, 6, 9) potentials. Therefore, the measure and the slope of the repulsive part of potential is the main reason for the prediction of properties by the

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equation of state. It is expected that the intermolecular interactions and the distance of molecules in the supercritical region are so that the intermolecular potentials (3, 9, 12) and (3, 6, 12) can better describe the bulk properties of fluids.

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5. Conclusion An EOS was obtained by using the intermolecular potential (3, 9, 12) which is applicable

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for gaseous, liquid and supercritical fluids in low and high ranges of temperature, pressure and density. The new EOS is also applicable in the gas-liquid phase transition region. The EOS can be used for a variety of fluids like polar, nonpolar, rare gases, short-chain and long-chain

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hydrocarbon fluids. It is shown that the experimental density of fluids can be fairly predicted by using the new EOS. The highest value of mean deviation of the predicted density is 5.7% for Ar

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in supercritical state and the minimum is 0.02% for NH3 in liquid state. The results show that the absolute percent deviation of the calculated density of substances in the liquid state is lower than the gaseous and supercritical states.

The new EOS is able to predict the dense fluids regularities such as the common bulk

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modulus and common compression points. A relation between the common bulk modulus point and common compression point was provided by applying the new EOS that had been confirmed by their experimental values.

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In addition to the type of the intermolecular potential which is used in this study, an important feature of the EOS is that the A, B and C parameters are explicitly related to the

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repulsion, the attraction and the long range interactions, respectively. This is the characterization of the equations of state which are obtained using the specific potential function with respect to other equations of state. This allows us to describe the volumetric and thermodynamic properties of the fluids by the intermolecular interactions. For example, in this work, it is shown that how the common bulk modulus and the common compression points are related to the ratio of the

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attraction and the repulsion interactions. However, the classical equations of state are not able to describe the fluid’s properties from the viewpoint of molecular.

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Adding the term r −3 in the potential function led to an increase in the application range of the new EOS which is derived by this potential. As an example, the LIR EOS that was obtained by (12, 6) potentials is applicable in ρ 〉 ρ B and T 〈 2T B and the GMA EOS that was obtained by (9,

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6) potential is applicable in liquid state of substances. This is also confirmed by comparing the application range of GMA EOS with BGH, PSP, and new equations of state in this work.

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Furthermore, the coefficients of determination values of the fitted P-v-T experimental data with GMA, BGH, PSP, and the new equations of state show that they can be fitted better with PSP and new equations of state in gaseous, liquid and supercritical states that are probably due to other contributions in the potential function.

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It was shown that for potentials with similar attraction range, the type of employed repulsive potential in derivation of EOS is effective in accurately predicting the fluid properties. The results show that the new EOS can better describe the properties of the hydrocarbons

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compared to the PSP EOS in the liquid and supercritical regions. Although the calculations are time consuming with increasing the parameters of EOS, the model

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can predict the fluid properties more accurately. The calculation using the new EOS is very simple.

The new EOS parameters in each temperature are determined just by fitting the available experimental data in the EOS. Then, the constants of the temperature dependent relation of the parameters are determined by a simple fitting. The resulting constants can be used in the temperatures and pressures within which the experimental data are not reported. Because of the

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complete description of the volumetric properties of the fluids by the new EOS, this equation has practical importance in science and industry.

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The derivation method of the new EOS is different from the equations of state such as SoaveBenedict-Webb-Rubin (SBWR) EOS [29]. The most important character of the equations of state which are derived by intermolecular potential function is that the intermolecular contributions are known in the EOS formulation. Each parameter in the new EOS is related to the specific interaction which exists in the

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fluid. As previously mentioned, the A, B and C parameters are explicitly related to the repulsion, the attraction and the long range interactions, respectively. Some equations of state have been generalized

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with a general parameter like acentric factor. The generalization made the molecular validity of the EOS diminished; in addition, the EOS is limited to a specific state and a group of substances in each general parameter. It should be noted that using the acentric factor parameter also has its own problems. It is possible that the acentric factor is justified by fitting the same equations with the experimental data that we are again engaged in fitting which must be determined for each fluid. Otherwise, it must be taken from

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other studies in which there are different values by different researches. We believe that working with the new equation of state is simpler than the SBWR EOS in spite of its multi-parameter essence. The new

simultaneously.

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EOS relations are much simpler and can be used in gaseous, liquid and supercritical states

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The new EOS has a zero value of the co-volume like SBWR EOS. The SBWR EOS was used for investigation of the phase equilibria of some mixtures [30]. It was shown that in spite of the successful predicting of the VLE critical loci, the SBWR EOS is not able to correctly describe the LLE critical loci of the asymmetric mixtures. This behavior has been related to the zero value of the co-volume in the SBWR EOS. However, it must be mentioned that this behavior which has been seen for asymmetric mixtures, cannot diminish the EOS validity in predicting thermodynamic properties. In our work, we have just shown that the new equation of state can be used for pure substances and also in the vapor-liquid region and we did not intend to investigate the phase equilibria in pure and mixtures of fluids. When the

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new EOS is extended to the mixtures, the LLE and VLE critical loci of the mixtures can be investigated. Such investigations are not within the scope of this work and require more work in a separate paper like

Nomenclature

SC

RI PT

Ref. 30.

coordination number

Z

compressibility factor

V

molar volume

ρ

molar density

U

potential energy

C 3 ,C 9 ,C 12

temperature-dependent coefficients of Eq. (1)

C 3′ ,C 9′ ,C 12′

temperature-dependent coefficients of Eq. (2)

TE D

EP

constants of Eq. (3)

AC C

A1 , A 2 , A 3

M AN U

z

B1, B 2 , B 3

constants of Eq. (4)

C 1 ,C 2 ,C 3

constants of Eq. (5)

Pi

internal pressure

A(T), B(T), C(T)

the new equation of state parameters

ρexp

experimental density

ρcal

calculated density

Br

reduced bulk modulus

ρOB

common bulk modulus point density

ρOZ

common compression point density

SC

constants of Eqs. (10)- (12)

M AN U

Ai , Bi , Ci

RI PT

ACCEPTED MANUSCRIPT

TE D

References

AC C

EP

[1] W.L. Jorgensen, C.J. Swenson, Optimized intermolecular potential functions for amides and peptides. Structure and properties of liquid amides, J. Am. Chem. Soc., 107 (1985) 569-578. [2] J. Reimers, R. Watts, M. Klein, Intermolecular potential functions and the properties of water, Chem. Phys., 64 (1982) 95-114. [3] J. Fischer, R. Lustig, H. Breitenfelder-Manske, W. Lemming, Influence of intermolecular potential parameters on orthobaric properties of fluids consisting of spherical and linear molecules, Mol. Phys., 52 (1984) 485-497. [4] X.-P. Li, J.P. Lu, R.M. Martin, Ground-state structural and dynamical properties of solid C 60 from an empirical intermolecular potential, Physical Review B, 46 (1992) 4301. [5] E. Bourasseau, P. Ungerer, A. Boutin, A.H. Fuchs, Monte Carlo simulation of branched alkanes and long chain n-alkanes with anisotropic united atoms intermolecular potential, Molecular Simulation, 28 (2002) 317-336. [6] C.F. Roche, A.S. Dickinson, A. Ernesti, J.M. Hutson, Line shape, transport and relaxation properties from intermolecular potential energy surfaces: The test case of CO2–Ar, The Journal of chemical physics, 107 (1997) 1824-1834. [7] M. Klein, H. Hanley, Selection of the intermolecular potential. Part 2.—From data of state and transport properties taken in pairs, Transactions of the Faraday Society, 64 (1968) 2927-2938. [8] W. Allen, R.L. Rowley, Predicting the viscosity of alkanes using nonequilibrium molecular dynamics: Evaluation of intermolecular potential models, The Journal of chemical physics, 106 (1997) 1027310281. [9] E.A. Mason, Transport Properties of Gases Obeying a Modified Buckingham (Exp-Six) Potential, The Journal of Chemical Physics, 22 (1954) 169-186.

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

[10] G. Parsafar, E. Mason, Universal equation of state for compressed solids, Physical Review B, 49 (1994) 3049-3060. [11] G. Parsafar, E. Mason, Linear isotherms for dense fluids: a new regularity, The Journal of Physical Chemistry, 97 (1993) 9048-9053. [12] G. Parsafar, E. Mason, Linear Isotherms for Dense Fluids: Extension to Mixtures, The Journal of Physical Chemistry, 98 (1994) 1962-1967. [13] E.K. Goharshadi, A. Morsali, M. Abbaspour, New regularities and an equation of state for liquids, Fluid Phase Equilib., 230 (2005) 170-175. [14] M. Ghatee, M. Bahadori, New thermodynamic regularity for cesium over the whole liquid range, The Journal of Physical Chemistry B, 105 (2001) 11256-11263. [15] M. Baniasadi, M. Baniasadi, S. Ghader, New isotherm regularity and an equation of state for gases and liquids, Journal of Industrial and Engineering Chemistry, 18 (2012) 474-482. [16] G. Parsafar, H. Spohr, G. Patey, An accurate equation of state for fluids and solids, The Journal of Physical Chemistry B, 113 (2009) 11977-11987. [17] W.A. Burgess, D. Tapriyal, B.D. Morreale, Y. Wu, M.A. McHugh, H. Baled, R.M. Enick, Prediction of fluid density at extreme conditions using the perturbed-chain SAFT equation correlated to high temperature, high pressure density data, Fluid Phase Equilib., 319 (2012) 55-66. [18] C. McCabe, S.B. Kiselev, A crossover SAFT-VR equation of state for pure fluids: preliminary results for light hydrocarbons, Fluid Phase Equilib., 219 (2004) 3-9. [19] F. Tumakaka, J. Gross, G. Sadowski, Thermodynamic modeling of complex systems using PC-SAFT, Fluid Phase Equilib., 228 (2005) 89-98. [20] G. Watson, T. Lafitte, C.K. Zéberg-Mikkelsen, A. Baylaucq, D. Bessieres, C. Boned, Volumetric and derivative properties under pressure for the system 1-propanol+ toluene: A discussion of PC-SAFT and SAFT-VR, Fluid Phase Equilib., 247 (2006) 121-134. [21] NIST:, http://www.nist.gov/. [22] Y.H. Huang, J.P. O'Connell, Corresponding states correlation for the volumetric properties of compressed liquids and liquid mixtures, Fluid Phase Equilib., 37 (1987) 75-84. [23] A.T.J. Hayward, Compressibility equations for liquids: a comparative study, British Journal of Applied Physics, 18 (1967) 965. [24] D. Ben-Amotz, D.R. Herschbach, Correlation of Zeno (Z= 1) Line for Supercritical Fluids with Vapor-Liquid Rectilinear Diameters, Isr. J. Chem., 30 (1990) 59-68. [25] J. Xu, D.R. Herschbach, Correlation of Zeno line with acentric factor and other properties of normal fluids, The Journal of Physical Chemistry, 96 (1992) 2307-2312. [26] E. Goharshadi, A. Boushehri, Common intersection point independent of mole fraction: A new regularity, Int. J. Thermophys., 18 (1997) 1517-1526. [27] E. Goharshadi, A. Naseri Mood, Common intersection point independent of pressure, a new regularity, J. Mol. Liq., 113 (2004) 133-141. [28] A. Boushehri, F.M. Tao, E. Mason, Common bulk modulus point for compressed liquids, The Journal of Physical Chemistry, 97 (1993) 2711-2714. [29] G.S. Soave, An effective modification of the Benedict–Webb–Rubin equation of state, Fluid Phase Equilib., 164 (1999) 157-172. [30] I. Polishuk, Implementation of SAFT+ Cubic, PC-SAFT, and Soave–Benedict–Webb–Rubin Equations of State for Comprehensive Description of Thermodynamic Properties in Binary and Ternary Mixtures of CH4, CO2, and n-C16H34, Industrial & Engineering Chemistry Research, 50 (2011) 14175-14185.

ACCEPTED MANUSCRIPT

Figure Caption Fig. 1. The isotherms of ( Z − 1)v 4 vs ρ for C9H20 in a) gaseous, b) liquid, and c) supercritical states.

RI PT

The symbols show the experimental values and the lines show the fitted values with the new EOS.

Fig. 2. The plot of ( Z − 1)v 4 vs ρ for C2H8 in the gas- liquid phase transition region at T=300 K. The symbol (●) shows the experimental values and the line shows the fitted values with the new EOS.

SC

Fig. 3. The isotherms of the experimental reduced bulk modulus, B r vs ρ for Ar. The plots are

M AN U

shown just by line in the left corner to better see the common intersection point. Fig. 4. The isotherms of the calculated reduced bulk modulus, B r vs ρ for Ar. The isotherms of

( B r − 1 − 2C ρ ) ρ 3 vs

ρ for Ar are given in the left corner for comparison.

Fig. 5. Comparison of the experimental (black line) and the calculated (red line) reduced bulk

TE D

modulus isotherms of Ar.

Fig. 6. The isotherms of the experimental compressibility factor, Z vs ρ for Ar. The symbol (●) shows the experimental values and the arrow indicates the increases in temperature.

EP

Fig. 7. The isotherms of the calculated compressibility factor, Z vs ρ for Ar. The symbol (●) shows the experimental values and the arrow indicates the temperature increases. The isotherms

AC C

of ( Z − 1 − C ρ ) ρ 3 vs ρ for Ar are given in the left corner, for comparison. Fig. 8. Comparison of the intermolecular potentials (3, 6, 12), (3, 9, 12), (3, 6, 9), and (6, 9, 12).

ACCEPTED MANUSCRIPT

Table Caption Table 1

RI PT

2 The A, B, and C parameters of the new EOS in given temperatures and pressure range. R 2 , R SPS ,

2 2 R BGH and RGMA are the fitting coefficients of determination of the experimental data with the

states are denoted with G, L, and SC, respectively.

M AN U

Table 2

SC

new, PSP, BGH and GMA equations of state, respectively. Gaseous, liquid and supercritical

The constants of Eqs. (10)- (11) for some fluids in gaseous state and the given range of temperature. The fitting coefficients of determination, R 2 , were given in the last column. Table 3

TE D

The constants of Eqs. (10)- (11) for some fluids in liquid state and the given range of temperature. The fitting coefficients of determination, R 2 , were given in the last column.

EP

Table 4

AC C

The constants of Eqs. (10)- (11) for some fluids in supercritical state and the given range of temperature. The fitting coefficients of determination, R 2 , were given in the last column. Table 5

The A, B, and C parameters of the new EOS for C2H8, CO2, NH3, and O2, in the gas-liquid phase transition region. The fitting coefficients of determination of the experimental data, R 2 , with the new EOS were given in the last column.

ACCEPTED MANUSCRIPT

Table 6 The maximum (Max), minimum (Min), and mean absolute percent deviation of the calculated

( ∆ρ ρ )

in the given ranges of pressure and temperature for gaseous, liquid and

RI PT

density,

supercritical states of some substances. Table 7

SC

The experimental and the calculated common bulk modulus point and common compression

M AN U

point densities ( ρOZ and ρOB ). Table 8

Comparison of the absolute percent deviations of densities, ( ∆ρ ρ ) which were calculated by

AC C

EP

TE D

using the new, BGH, and PSP equations of state.

ACCEPTED MANUSCRIPT

Table 1 2 The A, B, and C parameters of the new EOS in given temperatures and pressure range. R 2 , R PSP ,

RI PT

2 2 and RGMA are the fitting coefficients of determination of the experimental data with the new, PSP, R BGH

BGH and GMA equations of state, respectively. Gaseous, liquid and supercritical states denote with G, L, and SC, respectively.

(L)

NH3 (SC)

Kr (G)

Kr (L)

Kr (SC)

2 RGMA

∆P (MPa)

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 0.9990 0.9975 0.9984 0.9970 0.9974

0.9815 0.6213 0.3477 0.3264 0.1625 0.1413

0.13-0.16 0.16-0.55 0.20-1.42 0.40-3.08 0.50-7.14 0.80-10.3

1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 0.9998

0.9999 0.9979 0.9951 0.9858 0.8982

1-56 1-201 1-201 1-201 1-201

-0.056052 -0.051867 -0.042119 -0.033364 -0.026116 -0.020184

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9999 0.9998 0.9995 0.9994 0.9993 0.9993

0.8825 0.7557 0.6818 0.6454 0.6275 0.6182

20.1-100.1 20.1-100.1 20.1-100.1 20.1-100.1 20.1-100.1 20.1-100.1

-0.083706 -0.013696 -0.001731 -0.001731 0.002541 0.004149

-0.260841 -0.210251 -0.173939 -0.173939 -0.146753 -0.132173

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9999 0.9990 0.9973 0.9965 0.9987 0.9981

0.8674 0.5922 0.3741 0.2629 0.3539 0.2447

0.11-0.21 0.14-0.50 0.16-1.03 0.20-1.87 0.40-2.65 0.50-4.27

3.0486e-5 2.6116e-5 2.2586e-5 1.9217e-5 1.6940e-5 1.3494e-5

-9.2088e-4 -7.5207e-4 -6.1701e-4 -4.8848e-4 -4.0248e-4 -2.7482e-4

0.019500 0.006661 -0.003104 -0.014822 -0.023704 -0.037197

1.0000 1.0000 1.0000 1.0000 0.9999 0.9999

1.0000 1.0000 1.0000 0.9999 0.9999 0.9999

1.0000 1.0000 1.0000 1.0000 0.9999 0.9998

0.9996 0.9984 0.9950 0.9891 0.9813 0.9463

1-49 1-101 1-161 1-191 1-191 1-191

4.6922e-6 3.0067e-7 -1.7693e-6 -3.1567e-6 -4.3677e-6

2.0726e-5 1.3182e-4 1.7000e-4 1.8951e-4 2.0377e-4

-0.054501 -0.041802 -0.028526 -0.018337 -0.010541

0.9999 1.0000 1.0000 1.0000 0.9999

1.0000 1.0000 1.0000 1.0000 1.0000

0.9997 0.9995 0.9991 0.9986 0.9979

0.6744 0.5220 0.4392 0.3852 0.3384

10-190 10-190 10-190 10-190 10-190

(L .mol )

(L .mol )

7.978937 0.663605 0.065898 -0.004499 -0.002324 -0.001159

-1.539257 -0.221678 -0.030629 0.009433 0.007153 0.005069

-0.444163 -0.307289 -0.226281 -0.174397 -0.128893 -0.112415

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

200 250 300 350 400

4.4858e-6 4.2470e-6 3.5299e-6 2.9963e-6 1.7714e-6

-8.4913e-5 -1.1881e-4 -9.3671e-5 -7.2205e-5 -1.5275e-5

-0.218748 -0.099624 -0.066174 -0.048070 -0.050123

450 500 550 600 650 700

-4.8530e-7 -3.0900e-6 -4.4603e-6 -5.2565e-6 -5.8672e-6 -6.4538e-6

7.1398e-5 1.4365e-4 1.6474e-4 1.6857e-4 1.6820e-4 1.6784e-4

130 145 160 175 185 200

0.148313 0.020057 0.005216 0.005216 -0.000072 -0.001406

130 145 160 175 185 200

250 300 350 400 450

-5

7

-7

2 R PSP

SC

(L .mol )

250 280 310 340 380 400

5

M AN U

NH3

C

2 RBGH

-4

TE D

(G)

B

R2

4

EP

NH3

َ◌ A

T (K)

AC C

Fluid

ACCEPTED MANUSCRIPT

(SC)

250 300 350 400 450 500 550

0.129605 0.102112 0.082834 0.067802 0.055487 0.045477 0.036242

-0.898913 -0.675153 -0.521755 -0.404408 -0.310433 -0.236141 -0.169663

4.608832 3.149184 2.213434 1.517885 0.982881 0.592699 0.257888

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

50 100 150 200 250 300 350

3.7194e-7 -1.1683e-7 -2.8915e-7 -4.4649e-7 -6.2254e-7 -8.2330e-7 -1.0485e-6

-9.1757e-6 1.7899e-5 2.2509e-5 2.5723e-5 2.8973e-5 3.2386e-5 3.5902e-5

-0.021065 -0.003308 0.005121 0.008934 0.010939 0.012084 0.012766

0.9999 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9999 0.9992 0.9987 0.9973 0.9972 0.9961 0.9970

0.8585 0.6822 0.5023 0.3632 0.2685 0.1925 0.1482

0.10-0.19 0.10-0.29 0.10-0.44 0.10-0.63 0.10-0.88 0.10-1.20 0.10-1.61

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

1.0000 1.0000 0.9998 0.9996 0.9991 0.9977 0.9910

1-46 1-111 1-171 1-201 1-201 1-201 1-201

0.9997 0.9998 0.9971 0.9992 0.9988 0.9986 0.9985

0.8878 0.3434 0.3350 0.2872 0.2624 0.2464 0.2352

5- 75 5-200 5-200 5-200 5-200 5-200 5-200

RI PT

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

SC

C9H20

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

M AN U

(L)

-1.611865 -1.429244 -1.274428 -1.142118 -1.028079 -0.929010 -0.842370

TE D

C9H20

-0.457948 0.338431 0.655181 0.773797 0.850444 0.824103 0.868532

EP

(G)

18.642723 7.094531 2.373522 0.637589 -0.441755 -0.529086 -0.915259

AC C

C9H20

450 470 490 510 530 550 570

ACCEPTED MANUSCRIPT

Table 2 The constants of Eqs. (10)- (11) for some fluids in gaseous state and the given range of

∆T (K)

i

constants 0

Ar

100-150

C4H10

300-420

CO

90-130

CO2

260-300

C2H6

220-300

C7H16

CH4

Ai Bi Ci

-47.8230 20.6517 4.0919

53.3680 -23.7410 -7.1946

Ai Bi Ci

-92810.1595 169.9264 38.5762

2403.1319 -443.0214 -143.4196

Ai Bi Ci

-46.7112 46.1391 5.8525

Ai Bi Ci

370-490

300-405

130-200

140-190

-172.2365 28.1612 0.7923

3

0.2523 -0.0409 -1.0256e-3

R2

0.9876 0.9929 1.0000

-0.8019 0.3436 0.0650

2.9948e-3 -1.2390e-3 -2.0513e-4

0.9966 0.9994 1.0000

-128.4939 23.3691 0.5104

0.1710 -0.0307 -5.5794e-4

0.9825 0.9920 1.0000

46.3324 -46.6331 -8.8342

-0.8075 0.7938 0.0959

3.4953e-3 -3.3641e-3 -3.5282e-4

0.9800 0.9921 1.0000

-14190.5561 329.2595 10.2633

2965.4328 -690.9518 -31.7794

-20.5470 4.7577 0.1410

0.0353 -8.0409e-3 -1.9692e-4

0.9989 0.9998 1.0000

Ai Bi Ci

-30700.2214 214.0188 10.5623

628.3680 -407.3034 -34.9402

-4.4882 3.1691 0.1446

8.2871e-3 -6.2440e-3 -1.9692e-4

0.9839 0.9623 1.0000

Ai Bi Ci

-47595.9959 4304.9223 111.6742

1557.3358 -137.7678 -489.1440

-629.5496 57.1793 1.4214

0.6371 -0.0594 -1.1815e-3

0.9975 0.9989 1.0000

Ai Bi Ci

-25418.6584 43.9223 -12.5990

7893.9158 -137.7678 35.7602

-339.9267 57.1793 -0.1706

3.6886 -0.0594 1.9692e-4

0.9987 0.9996 0.9998

Ai Bi Ci

-88990.1898 1231.8209 30.5528

2318.4625 -323.0187 -115.4000

-123.1381 17.0309 0.4042

0.1639 -0.0226 -4.4307e-4

0.9940 0.9915 1.0000

Ai Bi Ci

-221.6971 116.4562 66.0045

305.4027 -160.8962 -91.3415

-3.5277 18.0980 1.0566

0.0101 -5.3168e-3 -3.1425e-3

0.9914 0.9945 0.9823

Ai Bi Ci

7.2254 -53.2578 50.6872

4.6868 641.6821 -12.3918

0.1529 -8.2558 0.0857

-9.7640e-4 2.8471e-3 -2.0513e-4

0.9996 0.9989 1.0000

EP

Iso-C4H10

Kr

2918.0968 -482.3771 -163.5773

AC C

C6H14

400-540

-12236.4096 20.2841 57.7142

2

M AN U

250-400

Ai Bi Ci

TE D

NH3

1

SC

Gas

RI PT

temperature. The fitting coefficients of determination, R 2 , were given in the last column.

ACCEPTED MANUSCRIPT

C5H12

C3H8

H2O

Xe

H2

350-465

250-360

450-640

180-280

23-33

0.9988 0.9973 1.0000

Ai Bi Ci

-136165.2595 963.1332 170.3538

4757.5258 -336.1194 -795.7269

-177.8108 12.5666 2.1410

1.6618 -0.1177 -1.6410e-3

0.9977 0.9976 1.0000

Ai Bi Ci

-703070.1210 486.1712 137.1853

2425.7150 -165.6816 -627.5759

-92.1044 63.9359 1.7311

0.8764 -0.0621 -1.3620e-3

0.9962 0.9917 1.0000

Ai Bi Ci

-18.8420 210.7042 0.9679

22.1777 -239.5968 -3.9652

-0.3126 32.8573 0.0120

1.1241e-3 -1.4031e-3 8.2050e-7

0.9973 0.9891 1.0000

Ai Bi Ci

-9172.6435 148.2460 55.3831

2668.9746 -425.1452 -221.2149

Ai Bi Ci

-11289.5920 162.6443 19.9409

2613.2621 -379.1135 -70.5194

Ai Bi Ci

-3612.2049 970.4056 37.3616

Ai Bi Ci

0.1451 -0.0241 -6.9743e-4

0.9942 0.9737 1.0000

-16.0131 22.8928 0.2685

0.2448 -0.0346 -3.2820e-4

0.9971 0.9964 1.0000

1354.3016 -371.8308 -166.4316

-46.5468 12.4568 0.4710

0.0399 -0.0104 -3.6102e-4

0.9992 0.9999 1.0000

-332.9065 115.5207 12.2031

654.4020 -224.9888 -31.9016

-4.8900 16.4692 0.1745

9.1978e-3 -3.2246e-3 -3.0359e-4

0.9955 0.9876 1.0000

Ai Bi Ci

0.2606 -0.4686 0.7455

-0.0891 0.1498 -0.5818

6.4480e-3 -0.0119 0.0163

-1.1679e-4 2.3179e-4 -2.4048e-4

1.0000 1.0000 1.0000

Ai Bi Ci

-2.8909 4.7470 1.0822

1.2306 -2.0260 -0.8452

-0.0654 0.1073 0.0219

8.6941e-4 -1.4256e-3 -2.1546e-4

0.9999 1.0000 1.0000

30-41

N2

90-125

Ai Bi Ci

-182.0443 177.4790 -1.2937

25518.3778 -24673.7428 -270.6861

-473.8976 462.9445 -0.4294

-2.1889 -577.3561 0.03290

0.9972 0.9873 1.0000

AC C

Ne

RI PT

100-150

0.2761 -0.1929 -1.3702e-3

SC

O2

430-565

-279.0176 18.4756 1.5323

-124.1748 20.2184 0.7205

M AN U

C8H18

450-570

70792.7519 -443.5983 -449.1887

EP

C9H20

400-510

-21176.3669 138.4081 118.5035

TE D

CH3OH

Ai Bi Ci

ACCEPTED MANUSCRIPT

Table 3 The constants of Eqs. (10)- (11) for some fluids in liquid state and the given range of

i

constants 0

90-140

C4H10

150-400

CO

80-130

CO2

250-300

C2H6

100-300

C7H16

Iso-C4H10

Kr

CH4

200-500

-4.0000e-4 0.0184 -5.4030

4.7589e-4 -0.0226 5.1405

Ai Bi Ci

-8.0000e-4 -0.2424 33.5826

0.0635 -0.5339 -49.1527

Ai Bi Ci

-3.0000e-4 8.0000e-4 6.6489

Ai Bi Ci

200-400

130-200

110-170

3

7.3803e-9 -9.0559e-8 -3.3641e-4

0.9917 0.9793 0.9984

-6.6187e-6 3.2000e-4 -0.0949

3.2415e-8 -1.5343e-6 4.5948e-4

1.0000 1.0000 0.9998

-3.2820e-5 -3.4789e-3 0.4904

5.1375e-7 9.2454e-7 -7.5486e-4

1.0000 1.0000 0.9999

5.0051e-4 -8.9681e-3 -5.4022

-6.6421e-6 2.4615e-5 0.1174

5.2364e-8 -7.3566e-7 -4.9230e-4

0.9997 0.9997

-0.0312 1.0289 -147.9697

0.0608 -2.0092 287.3875

-4.5948e-4 0.0152 -2.1810

8.8704e-7 -3.2820e-5 4.2010e-3

0.9997 0.9997 0.9995

-7.4000e-3 0.2080 -25.0912

0.0167 -0.4687 47.9243

-1.1487e-4 3.1261e-3 -0.3751

2.8963e-7 -7.4581e-6 8.2050e-4

1.0000 1.0000 1.0000

-0.1395 1.4487 -22.8365

1.3924 -14.6761 183.3589

-1.8232e-4 0.0170 -0.2487

3.3220e-6 -2.4615e-5 5.7435e-5

0.9999 0.9999 0.9998

Ai Bi Ci

-0.1923 1.8650 -29.7302

0.9204 -10.2367 160.3035

-2.6912e-4 0.0247 -0.3654

4.8088e-6 -4.1025e-5 3.2000e-4

0.9997 0.9998 0.9997

Ai Bi Ci

-0.1170 0.8803 21.5048

0.2722 -2.4222 -38.3223

-0.0176 0.0133 0.2991

3.9329e-6 -3.2820e-5 -2.8718e-4

0.9998 0.9999 0.9999

Ai Bi Ci

-1.8000e-3 0.0617 -9.2642

2.4861e-3 -0.0887 11.8094

-3.2820e-5 1.0092e-3 -0.1525

1.0205e-7 -3.5657e-6 5.4974e-4

1.0000 1.0000 0.9996

Ai Bi Ci

-2.2000e-3 0.0899 -20.4348

2.9702e-3 -0.1162 22.4572

-3.2820e-5 1.4851e-3 -0.3416

1.3301e-7 -5.3545e-6 1.2472e-3

0.9999 0.9999 0.9978

Ai Bi Ci Ai Bi Ci

AC C

C6H14

200-500

Ai Bi Ci

2

-2.9474e-6 -3.2820e-5 0.2609

M AN U

Ar

3.1179e-4 0.0149 -50.9636

TE D

200-400

-2.0000e-4 -3.5000e-3 18.9133

EP

NH3

1

Ai Bi Ci

R2

SC

∆T (K)

Liquid

RI PT

temperature. The fitting coefficients of determination, R 2 , were given in the last column.

0.9984

ACCEPTED MANUSCRIPT

C5H12

C3H8

H2O

Xe

H2

180-450

100-350

290-600

180-280

17-31

30-44

N2

70-120

Ai Bi Ci

-0.6832 5.5230 -47.2074

4.7657 -39.3482 292.5557

-9.3209e-3 0.0715 -0.5852

1.6410e-6 -9.8460e-5 5.9897e-4

1.0000 1.0000 1.0000

Ai Bi Ci

0.0567 -0.4776 3.7136

1.8117 -17.1487 160.0101

-0.0104 -0.0118 0.1386

Ai Bi Ci

-1.0000e-4 1.3000e-3 4.7784

1.3948e-4 -1.5097e-3 -7.0055

-0.0237 4.1025e-5 0.0740

Ai Bi Ci

1.4190 -36.2613 -36.2613

-6.1987 138.5383 138.5383

Ai Bi Ci

-0.0262 0.5737 -38.8049

0.0641 -1.3559 72.9268

Ai Bi Ci

-3.0000e-4 0.0333 -47.2168

Ai Bi Ci Ai Bi Ci Ai Bi Ci Ai Bi Ci

AC C

Ne

0.9997 0.9996 0.9950

RI PT

80-145

3.1788e-8 -1.2017e-6 1.8872e-4

-2.7837e-7 2.4615e-5 -5.2512e-4

1.0000 1.0000 0.9999

1.7889e-8 -5.4287e-7 -1.4769e-4

0.9999 0.9999 0.9999

SC

O2

250-500

-2.4615e-4 8.1230e-4 -0.1375

-0.0195 -0.4839 -0.4839

-3.2820e-6 6.4820e-4 6.4820e-4

1.0000 1.0000 1.0000

M AN U

C8H18

250-550

4.5948e-3 -0.1739 26.1292

-4.0204e-4 8.6399e-3 -0.5862

9.8230e-7 -1.6410e-5 1.3210e-3

1.0000 0.9999 0.9987

8.9435e-4 -0.0868 123.3573

-4.7178e-6 4.5948e-4 -0.6461

5.8902e-9 -5.6659e-7 7.8768e-4

0.9993 0.9994 0.9997

-6.9000e-3 0.1913 -19.1511

0.0124 -0.3486 32.0807

-1.0667e-4 2.9374e-3 -0.2943

2.6402e-7 -7.2897e-6 7.3845e-4

1.0000 1.0000 0.9998

1.0000e-4 -8.4000e-3 4.0716

-3.2820e-5 2.3138e-3 -1.4227

3.8188e-6 -2.2154e-4 0.1055

-8.2419e-8 4.7623e-6 -2.2728e-3

0.9998 0.9992 0.9998

9.0000e-4 -0.0100 14.5211

-4.1025e-5 3.9794e-3 -5.9798

0.0215 -2.2154e-4 0.3288

-2.4520e-8 2.7228e-6 -4.2502e-3

0.9992 0.9991 0.9967

-1.2000e-3 0.0549 -15.5894

1.3046e-3 -0.0577 14.0528

-2.4615e-4 9.6819e-4 -0.2764

1.0530e-7 -4.7472e-6 1.3702e-3

0.9999 0.9999 0.9983

TE D

C9H20

300-500

-1.5000e-3 0.0579 -9.9658

EP

CH3OH

Ai Bi Ci

ACCEPTED MANUSCRIPT

Table 4 The constants of Eqs. (10)- (11) for some fluids in supercritical state and the given range of

i

∆T (K)

constants

450-700

Ai Bi Ci

-2.7000e-3 0.1182 -19.7511

Ai Bi Ci

-2.0000e-4 2.9000e-3 0.1829

3.6102e-4 -6.8102e-3 -1.8616

Ai Bi Ci

-0.1353 5.7520 -201.5772

Ai Bi Ci

0

Ar

200-500

C4H10

460-570

CO

CO2

400-800

C2H6

C7H16

AC C

C6H14

550-600

Iso-C4H10

530-600

440-560

Kr

250-500

CH4

300-500

2.7234e-8 -1.2024e-6

1.0000 0.9990

71.0667

-0.2543

2.1333e-4

0.9992

5.7012e-9 -7.4908e-8 -9.2766e-7

1.0000 0.9995 1.0000

0.7931 -22.8876 695.2300

-9.7730e-3 0.0732 -2.6304

4.1036e-7 -5.7435e-5 2.4287e-3

1.0000 1.0000 0.9999

-8.7000e-3 0.0970 1.1916

0.0192 -0.3033 -8.0638

-1.3128e-4 1.3866e-3 0.0122

2.9969e-7 -3.2169e-6 8.2050e-6

0.9997 0.9990 1.0000

Ai Bi Ci

-3.0000e-3 0.0822 0.0822

0.0104 -0.2786 -0.2786

-4.1025e-5 1.0749e-3 1.0749e-3

4.0492e-8 -1.0570e-6 -1.0570e-6

1.0000 1.0000 0.9999

Ai Bi Ci

-0.0213 0.4389 -12.2856

0.0724 -1.4427 31.4154

-2.7897e-4 5.7763e-3 -0.1658

2.9051e-7 -5.7776e-6 1.8051e-4

0.9998 0.9984 0.9988

Ai Bi Ci

63.2370 -500.5857 62.0814

-233.7242 1839.5316 68.2534

0.0816 -6.4693 0.9639

-7.0563e-4 5.6286e-3 -1.5672e-3

1.0000 1.0000 0.9992

Ai Bi Ci

-4.5423 76.3105 -1841.7211

21.4849 -325.2833 7267.1447

-0.0559 0.9600 -23.5007

3.2820e-5 -7.0563e-4 0.0189

0.9999 0.9999 0.9998

Ai Bi Ci

-0.4688 8.2980 -200.1750

1.8060 -30.1593 671.4529

-6.0307e-3 0.1076 -2.6262

5.2756e-6 -9.8460e-5 2.5107e-3

1.0000 1.0000 0.9997

Ai Bi Ci

-1.3000e-3 0.0415 -4.5279

1.6410e-5 -5.6615e-4 0.0635

2.3196e-8 -7.2721e-7 8.2050e-5

-3.5381e-3 1.1402e-3 -9.4183

1.0000 0.9999 0.9999

Ai

-7.8080e-5

1.9364e-3

1.4773e-10

-1.8975e-13

1.0000

EP

310-600

3

-3.2820e-5 1.5015e-3

-2.3758e-6 4.1025e-5 1.7887e-3

TE D

200-450

2

0.0108 -0.4687

M AN U

NH3

1

R2

SC

Supercritical

RI PT

temperature. The fitting coefficients of determination, R 2 , were given in the last column.

ACCEPTED MANUSCRIPT

C5H12

C3H8

H2O

Xe

H2

Ne

N2

520-600

400-500

750-950

300-700

1.0000 1.0000

Ai Bi Ci

-5.6747 115.2811 -264.3117

21.6562 -438.6283 753.9443

-0.0729 1.4816 -3.5330

5.7435e-5 -1.2472e-3 3.6430e-3

1.0000 1.0000 1.0000

Ai Bi Ci

-7.2581e-6 3.0000e-4 -0.1028

8.2050e-6 -4.7589e-4 -0.0397

-1.2141e-7 4.9976e-6 -1.9938e-3

6.5095e-10 -1.7471e-8 5.9857e-6

1.0000 0.9996 0.9997

Ai Bi Ci

4.3978 -29.6145 70.2601

-19.1981 119.0935 -199.7358

0.0536 -0.3677 0.9069

-2.4615e-5 2.0513e-4 -5.9897e-4

1.0000 1.0000 1.0000

Ai Bi Ci

-1.3349e-4 5.5339e-4 0.1464

2.9060e-4 -1.2244e-3 -1.7518

Ai Bi Ci

1.0865 -6.1525 -96.6840

-3.1843 13.6062 478.7742

Ai Bi Ci

0.0559 0.6720 -71.5316

Ai Bi Ci

100-600

150-550

300-1500

RI PT

3.9164e-12 -7.0884e-11

SC

300-700

-3.0491e-9 5.5186e-8

4.8698e-9 -2.6118e-8 -4.8653e-7

0.9998 0.9996 1.0000

0.0145 -0.0846 -1.1701

-1.4691e-5 9.8153e-5 6.0202e-4

1.0000 1.0000 1.0000

-0.0788 -3.2011 215.7853

8.1230e-4 8.1065e-3 -0.9542

-1.2504e-6 -3.8081e-6 1.0010e-3

1.0000 1.0000 1.0000

0.0334 0.2128 0.2128

-0.1716 -10.6998 -10.6998

4.0204e-4 6.3999e-4 6.3999e-4

-2.2493e-7 4.0564e-6 4.0564e-6

1.0000 1.0000 1.0000

Ai Bi Ci

-6.1327e-4 0.0358 -9.2604

4.0959e-3 -0.1728 31.4161

-7.2208e-6 4.4035e-4 -0.1200

3.6601e-9 -3.0263e-7 1.0204e-4

1.0000 1.0000 0.9993

Ai Bi Ci

4.8850e-7 -3.7898e-5 0.1182

1.2089e-6 1.3419e-4 -0.3548

8.6997e-9 -7.4704e-7 1.3058e-3

-3.3879e-12 9.8502e-10 -1.1742e-6

1.0000 0.9985 1.0000

Ai Bi Ci

-3.9239e-6 5.5455e-5 0.0549

1.0562e-4 -9.7501e-4 -3.8354

-8.8054e-7 1.069e-5 0.0068

2.6177e-9 -5.3015e-8 -3.3930e-7

1.0000 1.0000 1.0000

Ai Bi Ci

-3.0000e-4 0.0121 -2.0158

8.2050e-4 -0.0291 3.1859

-0.4699 1.7230e-4 -0.0296

8.7941e-9 -2.6992e-7 4.9230e-5

0.9998 0.9992 0.9985

TE D

O2

579-600

-0.0489 5.3291

EP

C8H18

50-350

AC C

C9H20

300-500

2.1000e-3 -0.2508

-1.9642e-6 6.8411e-6 1.2951e-3

M AN U

CH3OH

Bi Ci

ACCEPTED MANUSCRIPT

Table 5 The A, B, and C parameters of the new EOS for C2H8, CO2, NH3, and O2, in the gas-liquid

the new equation of state, R 2 , are given in the last column. Fluid

T (K)

∆P (MPa)

A (L4 .mol-4 )

B (L5 .mol -5 )

C2 H6

250 260 270 290 300

0.31-5.01 0.31-6.01 0.41-7.01 0.41-8.01 0.41-10.01

-1.633000e-3 -1.962521e-3 -1.633111e-3 -2.499999e-3 -2.206000e-3

0.016623 0.016914 0.014569 0.015819 0.014528

CO2

275 280 285 290 295

0.30-8.10 0.30-8.10 0.30-8.10 0.30-8.10 0.30-8.10

3.422714e-4 -5.444111e-4 -1.818888e-3 -1.918000e-3 -2.092762e-3

NH3

320 340 360 380

0.40-4.00 1.00-5.00 3.00-7.00 3.50-11.00

O2

120 130 140 150

0.30-3.50 0.40-4.00 0.40-4.50 0.40-6.50

R2

-0.265216 -0.244444 -0.225839 -0.194837 -0.181315

1.0000 1.0000 1.0000 1.0000 1.0000

2.188752e-3 5.131247e-3 8.525000e-3 8.212524e-3 8.212000e-3

-0.146442 -0.140846 -0.135435 -0.130000 -0.125000

1.0000 1.0000 1.0000 1.0000 1.0000

1.719717e-4 -2.576427e-4 -6.826237e-5 -6.173111e-5

3.578888e-3 4.772618e-3 2.070636e-3 1.578717e-3

-0.206728 -0.173938 -0.146121 -0.125664

1.0000 1.0000 1.0000 1.0000

-6.5890e-5 -8.7682e-5 -1.7464e-4 -3.8961e-5

1.4995e-3 1.5510e-3 1.7402e-3 7.4222e-4

-0.1363 -0.1172 -0.1018 -0.0882

1.0000 1.0000 1.0000 1.0000

M AN U

SC

C (L7 .mol-7 )

TE D

EP AC C

RI PT

phase transition region. The fitting coefficients of determination of the experimental data with

ACCEPTED MANUSCRIPT

Table 6 The maximum (Max), minimum (Min), and mean absolute percent deviation of the

(G), liquid (L) and supercritical (SC) states of some substances.

RI PT

calculated density, ( ∆ρ ρ ) in the given ranges of pressure and temperature for gaseous

ρ cal − ρ exp

Ar

C4H10

C2H6

C7H16

C6H14

Iso-C4H10

× 100

0.33 0.38) 0.002 (0.09) 0.77 (7.11)

0.35 0.02 1.78

0.12-4.73 1.0-201.0 5.1-200.1

0.34 (0.61) 0.01(0.06) 4.21 (6.88)

0.43 0.03 5.71

0.1-2.3 1.0-96.0 5.1-65.0

0.33 (2.22) 0.01 (0.06) 0.92 (2.49)

0.92 0.02 1.86

90-130 80-130 200-450

0.1-2.2 1.0-96.0 10.1-90.1

0.36 (2.03) 0.02 (0.15) 0.26 (0.63)

0.43 0.05 0.56

260-300 250-300 400-800

0.1-3.0 1.7-201.0 20.0-800.0

2.85 (7.08) 0.03 (0.23) 0.42 (1.14)

4.43 0.20 0.79

G L SC

220-300 100-300 310-600

0.1-3.7 1.0-66.0 10.1-65.0

0.32 (3.43) 0.01 (0.16) 0.04 (1.06)

1.67 0.04 0.57

G L SC

400-540 200-500 550-600

0.1-2.7 1.0-96.0 5.1-95.1

0.35 (5.71) 0.003 (0.07) 0.08 (0.96)

1.78 0.02 0.38

G L SC

370-490 200-500 530-600

0.1-2.3 1.0-96.0 5.1-95.1

0.38 (5.722) 0.004 (0.21) 0.19 (0.96)

1.82 0.07 0.54

G L

300-405 200-400

0.1-2.6 1.0-31.0

0.33 (1.81) 0.005 (0.161)

0.83 0.04

G L SC

250-350 200-400 450-700

G L SC

100-150 90-140 200-500

G L SC

300-420 150-400 460-570

EP

G L SC

0.06-7.69 1.0-201.0 20.1-100.1

SC

Mean

AC C

CO2

ρ exp

Min (max)

G L SC

CO

∆ P ( M Pa )

M AN U

NH3

∆T ( K )

State

TE D

Fluid

ACCEPTED MANUSCRIPT

C9H20

C8H18

N2

C3H8

C5H12 Xe

0.1-2.6 1.0-191.0 10.1-190.1

0.33 (2.89) 0.01 (0.18) 0.51 (2.04)

1.37 0.05 1.43

G L SC

140-190 110-170 300-450

0.1-3.2 1.0-201.0 30.1-990.0

0.02 (0.10) 0.01 (0.04) 0.099 (0.27)

0.03 0.34 0.15

G L SC

400-510 300-500 560-615

0.10-7.70 1.00-781.0 30.0-780.0

0.45 (8.72) 0.002 (0.26) 0.08 (0.20)

5.72 0.08 0.08

G L SC

450-570 250-550 50-350

0.10-1.40 1.0-201.0 5.1-155.1

G L SC

430-565 250-500 579-600

G L SC

90-125 70-120 300-1500

G L SC

250-360 100-350 400-500

RI PT

130-200 130-200 250-500

0.32 2(0.69) 0.01 (0.07) 0.32 (2.67)

0.39 0.03 1.86

0.1-1.9 1.0-96.0 5.1-95.1

0.33 (1.90) 0.01 (0.05) 0.08 (0.27)

0.86 0.04 0.14

0.1-3.1 1.0-201.0 30.0-2190.0

0.34 (2.86) 0.01 (0.05) 0.17 (3.13)

1.22 0.03 1.67

0.17-3.48 1.0-91.0 5.10-95.10

0.38 (3.34) 0.006 (0.076) 1.02 (3.74)

1.03 0.03 2.84

0.2-20.1 1.0-961.0 40.0-965.0

0.33 (6.33) 0.03 (0.10) 1.33 (2.97)

2.83 0.05 2.10

23-33 17-31 100-600

0.10-1.22 1.0-77.0 40.0-1990.0

0.37 (1.21) 0.01 (0.14) 0.09 (2.23)

0.66 0.06 5.09

G L SC

30-41 30-44 150-550

0.10-1.36 1.0-61.0

0.33 (0.52) 0.02 (0.13)

0.48 0.05

70.0-640.0

0.37 (2.63)

1.02

G L SC

100-150 80-145 300-700

0.10-2.78 1.0-81.0 10.0-80.0

0.33 (5.68) 0.01 (0.11) 0.42 (0.86)

2.09 0.04 0.60

G L SC

350-465 80-145 520-600

0.1-3.4 1.0-81.0 10.0-100.0

0.33 (3.99) 0.01 (0.11) 0.17 (0.35)

1.33 0.06 0.25

G

180-280

0.10-3.08

0.33 (2.85)

1.17

G L SC

AC C

O2

G L SC

G L SC

H2

Ne

1.02

450-640 290-600 750-950

EP

H2O

0.76 (1.46)

SC

CH3OH

5.1-30.1

M AN U

CH4

440-560

TE D

Kr

SC

ACCEPTED MANUSCRIPT

L SC

1.0-201.0 20.0-680.0

0.01 (0.33) 0.12 (0.71)

0.10 0.48

SC

RI PT

180-280 300-700

Table 7

M AN U

The experimental and the calculated common bulk modulus point and common compression point densities ( ρOZ and ρOB ).

∆T (K )

Ar

110-150

N2

80-100

CH4

110-150

ρ OB

exp

35.0744

exp

cal

42.8962

42.8153

30.1122

30.1666

36.6466

36.6366

26.5666

26.5222

33.2816

33.2334

EP AC C

cal

ρ OZ

35.2534

TE D

Fluid

ACCEPTED MANUSCRIPT

Table 8 Comparison of the absolute percent deviations of densities,

( ∆ρ ρ )

which were

RI PT

calculated using the new, BGH, and PSP equations of state. ρ cal − ρ exp

C7H16

supercritical

gas

AC C

H2O

liquid

H2O

liquid

H2O

Supercritical

CH4

gas

∆ρ (mol.L-1 )

ρ exp

× 100

This work

BGH

PSP

0.3641 0.3452 0.3533 0.9666 2.9555 5.7111

0.3783 0.3871 0.4694 1.2058 1.4325 7.0385

0.3785 0.3954 0.4243 0.4516 0.8764 3.8754

0.10-0.43 0.10-0.77 0.10-1.29 0.10-1.76 0.10-0.218 0.10-2.72

0.028-0.141 0.026-0.260 0.025-0.469 0.024-0.715 0.031-0.072 0.022-1.875

200 250 300 350 450 500

1.00-31.00 1.00-96.00 1.00-96.00 1.00-96.00 1.00-96.00 1.00-96.00

7.602-7.745 7.188-7.679 6.775-7.392 6.343-7.123 5.292-6.625 4.473-6.394

0.0029 0.0101 0.0124 0.0212 0.0301 0.0653

0.0030 0.0137 0.0182 0.0276 0.0864 0.1210

0.0032 0.0132 0.0118 0.0214 0.0354 0.1710

550 560 570 580 590 600

5.10-95.10 5.10-95.10 5.10-95.10 5.10-95.10 5.10-95.10 5.10-95.10

3.907-6.165 3.648-6.122 3.351-6.079 3.017-3.037 2.659-5.995 2.317-5.954

0.0789 0.1210 0.2326 0.3905 1.0421 0.9515

0.3736 0.2605 0.3421 0.4863 3.2631 4.3521

0.2478 7.5263 7.1368 6.0887 1.9876 2.3214

450 550 580 600 640

0.24-0.93 0.60-6.11 0.60-9.44 0.60-12.34 1.10-20.26

0.067-0.267 0.133-1.747 0.126-2.872 0.121-4.043 0.210-9.833

0.3288 1.1587 1.4916 2.5555 5.4234

0.3764 0.3783 0.5755 1.2604 4.7777

0.4523 0.4466 0.5432 0.6471 3.2312

290 350 400 500 600

1.00-201.00 1.00-201.00 1.00-201.00 1.00-201.00 1.00-201.00

55.465-59.709 54.072-58.112 52.059-56.476 49.426-54.602 46.144-52.523

0.0324 0.0275 0.0549 0.0520 0.0999

0.0114 0.0237 0.0232 0.0182 0.0185

0.0182 0.0129 0.0126 0.0180 0.0321

750 800 850 900 950

10.31-42.52 10.31-42.52 10.31-42.52 10.31-42.52 10.31-42.52

7.085-39.710 5.854-36.844 5.124-34.003 4.615-31.279 4.230-28.763

1.3296 1.9532 2.4232 2.0985 2.9742

1.1111 1.4342 1.6532 1.5493 1.7851

0.6428 0.3542 0.4224 0.3514 0.8753

140 150

0.10-0.64 0.10-1.03

0.087-0.632 0.081-1.017

0.0954 0.0222

0.4571 0.4687

0.4342 0.4876

SC

400 430 460 490 510 540

M AN U

C7H16

gas

∆P (M Pa )

TE D

C7H16

T (K)

State

EP

Fluid

ACCEPTED MANUSCRIPT

supercritical

0.076-1.582 0.071-2.429 0.067-3.825

0.0341 0.0211 0.0222

0.4833 1.2233 2.4272

0.4463 0.4896 0.8213

110 125 140 155 170

1.00-81.00 1.00-151.00 1.00-201.00 1.00-201.00 1.00-201.00

26.52-29.42 25.12-30.32 23.53-30.66 21.66-30.07 19.35-29.51

0.0143 0.0246 0.0321 0.0355 0.0387

0.0086 0.0166 0.0129 0.1777 0.1911

0.1077 0.0246 0.0289 0.0352 0.1529

300 350 400 450

20.10-200.10 20.10-200.10 20.10-200.10 20.10-200.10

9.73-25.28 9.73-25.28 9.73-25.28 9.73-25.28

0.0995 0.1496 0.2225 0.2708

1.0994 1.7342 2.8431 3.6842

0.2044 0.3433 2.2631 0.5244

AC C

EP

TE D

M AN U

SC

CH4

liquid

0.10-1.59 0.10-2.32 0.10-3.28

RI PT

CH4

160 170 180

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT