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## Latin American applied research

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*versión impresa* ISSN 0327-0793

### Lat. Am. appl. res. v.34 n.1 Bahía Blanca ene./mar. 2004

**A generalized correlation for the second virial coefficient based upon the Stockmayer potential**

**M. Ramos-Estrada ^{1}, R. Tellez-Morales^{1}, G. A. Iglesias-Silva^{1} and K. R. Hall^{2}**

^{1} *Dto. de Ing. Química, Instituto Tecnológico de Celaya, Celaya, Gto., CP 38010, MEXICO gais@iqcelaya.itc.mx*

^{2}

*Chemical Engineering Department, Texas A&M University, College Station TX. 77843, USA*

krhall@tamu.edu

krhall@tamu.edu

*Abstract* — We have approximated the second virial coefficients obtained from the Stockmayer potential using a two parameter, analytical correlation. The correlation eliminates the need for integration of the potential over nearly the entire range of reduced temperatures. The equation is valid from 0 to 1.5 reduced dipole moment and uses the Boyle temperature as a normalizing variable. Because the equation is a correlation, the parameters do not necessarily have any physical meaning. We have applied the new equation to nonpolar compounds, aromatics, freons, alcohols, ethers, aldehydes, and acids.

*Keywords* — Second Virial Coefficient. Correlation. Stockmayer Potential. Boyle Temperature.

**I. INTRODUCTION**

The precise basis in statistical mechanics of the virial equation is one of its attractions. The virial equation, when truncated after the second virial coefficient, can predict accurate thermodynamic properties in the vapor phase up to about one-half the critical density. Therefore, experiments and correlations have emphasized determination of the second virial coefficient. Correlation use is preferable to experiment if the accuracy in the prediction is close to the experimental error.

Several correlations have appeared in the literature. Among the most popular are those developed by: Pitzer and Curl, (1957); O'Connell and Prausnitz, (1967); Halm and Stiel (1971); Tsonopoulos (1974); and Tanakad and Danner (1977). These correlations, based upon the corresponding states principle, provide generalized expressions for the second virial coefficient. The characteristic parameters for simple molecules in these equations are the acentric factor, the critical pressure and the critical temperature. To predict the second virial coefficient for more complex molecules, the correlations utilize the reduced dipole moment and adjustable parameters. For non-polar and slightly polar substances, prediction of the second virial coefficient is relatively close to experimental values, but the correlations cannot predict strongly polar and associating substances adequately.

Another way to obtain the second virial coefficient is through integration of a pair potential. Hirschfelder *et al.* (1954) give the theoretical expressions for the second and third virial coefficient using different potential functions. The resultant equations can correlate the second virial coefficient fairly well, but they tend to fail at low temperatures.

One of the pair potentials used to represent the second virial coefficient of polar substances is the Stockmayer potential (Hirschfelder *et al.*, 1954). The expression for the second virial coefficient of the Stockmayer fluid is a two characteristic constant equation that requires a numerical integration. Polak and Lu (1972) and Johnson and Eubank (1973) obtained the intermolecular forces constants of alcohols and other polar molecules. They used an iterative least squares method because of the complexity and non-linearity of the expression for the second virial coefficient.

Recently, Tellez-Morales (1998) have developed a simple straight line to correlate the pure and interaction second virial coefficient data and Iglesias-Silva and Hall, (2001) have developed a correlation using the Boyle temperature as a reducing parameter. In this work, we use a new correlation that has the Boyle temperature as a normalizing parameter. This new function can correlate the second virial coefficient from the Stockmayer potential within a maximum error of 3% at low temperatures and within less than 1% at other temperatures. We have generalized the parameters from this correlation in terms of the Stockmayer reduced dipole moment and the Boyle temperature. Also, we use the new equation to represent the second virial coefficient of nonpolars, aromatics, freons, alcohols, ethers, aldehydes, and acids. The force constants used in the new equation agree with the force constants obtained from the Stockmayer expression.

**II. DEVELOPMENT**

Many correlations for the second virial coefficient are based upon the corresponding states principle. The reduced second virial coefficient is:

(1) |

where *B* is the second virial coefficient, *T* is the absolute temperature, *P _{c}* is the critical pressure,

*T*is the critical temperature, ω is the acentric factor, and μ is the reduced dipole moment. In Eqn. (1),

_{c}*np*,

*ns*,

*s*, and

*p*stand for non-polar, non-spherical, spherical and polar, respectively.

A different approach to obtain the second virial coefficient is to use the statistical mechanical definition of the second virial coefficient

(2) |

or

(3) |

for angle dependent potentials. In the above Eqns. (2) and (3), *u* is the pair potential function, *r* is the intermolecular distance, *k* is the Boltzmann constant. For the Stockmayer potential, *u* is

(4) | |

with

Integration over the angles gives,

(5) | |

with

where *T*^{*} is the dimensionless temperature and *m* is the reduced dipole moment defined as . In this work, we have solved numerically Eqn. (5) using a 1/3-Simpson method with 30 steps for each interval. In the multiple integration, we first apply the Simpson rule in the first dimension with each value of the other dimensions held constant. Then, we apply the method to integrate the second dimension keeping constant the remaining dimensions and so on until having completed all the dimensions.

Several expressions have available to calculate the second virial coefficient. Recently, Tellez-Morales (1998) and Iglesias-Silva and Hall (2001) have suggested simple correlative functions for the second virial coefficient. They use the Boyle and the critical temperatures as reducing variables. We propose a simple function that uses these variables,

(6) |

where is the reduced Boyle temperature and *m*, *l*, *b*_{1}, and *n* are characteristic parameters.

**III. RESULTS AND DISCUSSION**

We have used Eqn. (6) to correlate the reduced second virial coefficient from the Stockmayer potential. We have used the reduced Stockmayer dipole moment, *t*^{*} = 8^{-0.5}(μ^{*})^{2}, and solved Eqn. (5) over the interval 0 ≤ *t*^{*} ≤ 1.5. We have used a nonlinear least squares method to obtain the parameters of Eqn. (6) using the reduced Stockmayer second virial coefficient. Table 1 contains the characteristic parameters of Eqn. (6). The average percentage error for the second virial coefficient appears in Table 2. We have observed that the reduced Stockmayer Boyle temperature is a function of the reduced dipole moment, *t*^{*}.

**Table 1.** Characteristic parameters used in Eqn. (6).

Figure 1 shows this behavior and can be represented accurately by

(7) | |

Equation 7 has an accuracy of ±0.0001 around the Boyle temperature. Figure 1 shows the behavior of Eqn. (7).

**Figure 1.** Behavior of the reduced Boyle temperature as a function of reduced dipole moment.

The characteristic parameters, *m*, *l*, *b*_{1}, and *n* are functions of the reduced Boyle temperature:

(8) | |

(9) | |

(10) | |

(11) |

**Table 2.** Average Percentage deviation of Eqn. (6) and Eqn. (6) plus Eqns. (7)-(11).

Figures 2-5 show the behavior of these parameters together with Eqns. (8)-(11). Upon imposing Eqns. (7)-(11), the average percentage deviation of these generalized correlations from the calculated reduced Stockmayer second virial coefficient is within 1.3 %. Practically, no degradation of the fit exists when using the generalized correlation to calculate the reduced virial coefficient, as shown in Table 2, therefore the force constants obtained from Eqn. (6) are equivalent to the constants obtained from Eqn. (5). Figure 6 shows the residual second virial coefficient for *t*^{*}, which is equivalent to the second virial coefficient obtained from the Lennard-Jones potential. To prove the validity of Eqns. (6)-(11), we have correlated the second virial coefficients of aromatics, freons, ethers, aldehydes, alcohols, and acids.

**Figure 2.** Relation between characteristic parameter *l* and reduced Boyle Temperature.

We have used the Thermodynamics Research Center (TRC) SOURCE Database (1998) as our data source and, we have used a nonlinear least squares method developed by SAS (1991) to fit the curves. The values of the dipole moment come from Poling *et al.* (2001) and TRC Freons Database (1999).

**Figure 3.** Relation between characteristic parameter *m* and reduced Boyle temperature.

**Figure 4**. Relation between characteristic parameter *b*_{1} and reduced Boyle temperature.

**Figure 5.** Relation between characteristic parameter *n* and reduced Boyle temperature.

**Figure 6.** Relative deviation of measured reduced second virial coefficient from Eqn. (6) + Eqns. (7)-(11).

The experimental data have been weighted using the uncertainties supplied by the TRC database. In Table 3, we show the characteristic parameters, σ and ε/k, obtained from the fitting procedure together with the asymptotic standard error associated with them. Also, we present the reduced Stockmayer dipole moment, *t*^{*}. The difference between the values presented in Table 3 and the values presented earlier (Hirschfelder *et al.*, 1954) is that the earlier results appear only at round numbers such as 0.1, 0.2, 0.3, *etc*. while now we can have intermediate values. Table 4 compares the intermolecular force constants obtained by different authors (Hirschfelder *et al.*, 1954 and Polak and Lu, 1971) and the present work. In general, the values are within about 10% of each other except for molecules that can associate. The larger discrepancy for those molecules is to be expected because the Stockmayer potential does not include associative effects. Intermolecular parameters for *t*^{*} = 0 from this work are compared to those obtained by Hirschfelder *et al.* (1954). Table 5 shows the absolute average deviation and the average of the absolute deviation divided by the uncertainty using Eqn. (6) and the Tsonopolous equation (Poling *et al.*, 2001). In general, our values of Δ*B*/σ are more nearly constant and smaller than those from Tsonopolous for each class of compounds although the values are similar for many compounds. Determination of the Boyle Temperature is not accurate when extrapolating the Stockmayer potential. When experimental data near the Boyle temperature exist, we have compared the Boyle temperatures obtained to those reported by Iglesias-Silva and Hall, (2001). Table 6 contains these results. In this work, we have not included a specific term for associating substances and consider only the effect of the dipole moment. We have correlated some substances (such as ethanol, hydrogen cyanide, acetonitrile) outside the range of validity, but we still obtain good agreement with the experimental data. Because of the large number of references, we present only the number of references and the number of points used in the calculation of the force constants.

**Table 3.** Characteristic Parameters Used in Eqns. (6)-(11).

**Table 4.** Comparison of intermolecular force constants.

**Table 5.** Deviations of experimental second virial coefficients from this work and from Tsonopolous (1974).

**Table 6.** Comparison of Boyle temperatures obtained in this work with those from Iglesias-Silva and Hall (2001).

The references are available through TRC and the authors. Figure 7 shows typical absolute deviations from the new equation for different substances.

**Figure 7.** Absolute deviation of measured second virial coefficients from Eqns. (6)-(11) for different substances.

**Cross Second Virial Coefficients**

Although few experimental data exist for polar systems with which to do a complete evaluation of the equations, we have estimated the cross second virial coefficients of several systems using Eqns. (6)-(11). The combining rules we have used for σ_{12}, ε_{12}, are

and

. |

Table 7 contains the absolute average deviation of the data from the prediction using Eqns (6)-(11). Experimental data come from Warowny and Stecki (1979). Equations (6)-(11) predict the behavior for polar + polar and polar + nonpolar systems reasonably, however it is difficult to assign an accuracy estimate to the prediction because the experimental data themselves are uncertain.

**Table 7.** Absolute and percentage deviations of experimental second cross virial coefficients from Eqns. (6)-(11).

**IV. CONCLUSIONS**

We have correlated the second virial coefficient from the Stockmayer potential using an analytical expression. The new equation uses the reduced dipole moment and the reduced Boyle temperature as correlating parameters. This generalized correlation allows calculation of the second virial coefficient for polar molecules as well as the Stockmayer potential but without using a numerical integration. The new equation compares favorably with the Tsonopoulos correlation. We show that the reduced Boyle temperature obtained from the Stockmayer second virial coefficient can be accurately correlated using the reduced dipole moment, *t*^{*}. Also, we have obtained the force constants for a large number of substances using one of the most complete databases for second virial coefficients. The new parameters agree well with existing values in the literature. We believe this work presents the most recent force constants for the second virial coefficient of the Stockmayer potential.

**REFERENCES**

1. Halm, R.L. and L. Stiel, "Second Virial Coefficient of Polar Fluids and Mixtures", *AIChE J.*, **17**, 259-265 (1971). [ Links ]

2. Hirschfelder J.O., C.F. Curtiss, and R.B. Bird, *Molecular Theory of Gases and Liquid*, John Wiley and Sons, Inc., USA (1954). [ Links ]

3. Iglesias-Silva, G.A. and K.R. Hall, "An Equation for the Prediction and Correlation of Second Virial Coefficients", *Ind. & Eng. Chem. Res.*, **8**, 1968-1974 (2001). [ Links ]

4. Johnson, J.R. and P.T. Eubank, "Intermolecular Interactions of Highly Polar Gases", *Ind. & Eng. Chem. Fundam.*, **12**, 156-165 (1972). [ Links ]

5. O'Connell, J.P. and J.M. Prausnitz, "Empirical Correlation of second Virial Coefficients for Vapor-Liquid Equilibrium Calculations", *Ind. Eng. Chem. Process Des. Develop.*, **6**, 245-250 (1967). [ Links ]

6. Pitzer, K.S. and R.F. Curl, "The Volumetric and Thermodynamic Properties of Fluids. III. Empirical Equation for the Second Virial Coefficient", *J. Am. Chem. Soc.*, **79**, 2369-2370, (1957). [ Links ]

7. Polak, J. and B. C-Y. Lu, "Second Virial Coefficients of Polar Gases- A Correlation with Stockmayer Potential Function", *Can. J. Chem. Eng.*, **50**, 553-556 (1971). [ Links ]

8. Poling, B.E., J.M. Prausnitz, and J.P. O'Connell, *The Properties of Gases and Liquids*, McGraw Hill Co., USA (2001). [ Links ]

9. Tanakad, R.R. and R.P. Danner, "An Improved Corresponding States Method for Polar Fluids: Correlation of Second Virial Coefficients", *AIChE J.*, **23**, 685-695 (1977). [ Links ]

10. Tellez-Morales, R., *Correlations for the Second and Third Virial Coefficients*. M.S. Thesis, Instituto Tecnológico de Celaya, Guanajuato, México (1998). [ Links ]

11. TRC, *Databases for Chemistry and Engineering-SOURCE Database Version 1998-2,* Thermodynamics Research Center, Texas A&M University System, College Station, TX. (1998). [ Links ]

12. TRC, *Databases for Chemistry and Engineering-Freons Database*, Thermodynamics Research Center, Texas A&M University System, College Station, TX. (1999). [ Links ]

13. Tsonopoulos, C. "An Empirical Correlation of Second Virial Coefficients", *AIChE J.*, **20**, 263-272 (1974). [ Links ]

14. SAS, *The SAS System for Windows*, SAS Institute Inc., Cary N.C. 27513, USA (1991). [ Links ]

15. Warowny, W. and J. Stecki, *The Second Cross Virial Coeffcients of Gaseous Mixtures*, Polish Academy of Sciences, Institute of Physical Chemistry, Polish Scientific Publishers, (1979). [ Links ]

**Received: September 25, 2001. Accepted for publication: October 7, 2002. Recommended by Subject Editor Gregorio Meira.**