The parachor method

The parachor method of McLeod-Sudgen (1923 see, for example, Sudgen, 1930) is one of the most widely 

Reprinted from Quayle (1953), with permission from American Chemical Society. 

A useful approximation of Equation 3.3a for liquids (ignoring the gas density) is  

For liquid mixtures, the surface tension is not always equal to the linear sum of the surface tensions of the individual liquids, i.e. in general  

The difference between the liquid surface tension and the above linear sum is called the surface excess and is usually negative, but can be positive in the case of strong intermolecular interactions. 

---

Infinite dilution activity coefficients are predicted by several methods (4,5,10,27,28, 29,30,31). The most general are the Pierotti-Deal-Derr method (4), the parachor method (27), and the Weimer-Prausnitz method (10), modified by Hellpinstill and Van Winkle (28). Since accuracy is limited in these methods and noninfinite dilution conditions prevail in actual operations, the infinite dilution activity coefficients obtained should only be used for screening purposes. 

The Parachor Method (27). Infinite dilution activity coefficients are obtained according to this method from the following relationship (27) ... 

A comparison between the PDD and the parachor method seems to suggest that the latter is no worse than the former, and often better (27). For the systems considered, the parachor method gives lower maximum deviations in 11 cases, the PDD in 7. Also, the authors of the parachor method claim better accuracy when extrapolation with respect to temperature is required. For example, the case of n-heptane (1) in 1-butanol (2) is cited. Values for y° calculated by extrapolating the PDD constants to temperatures ranging from 114.5°C-171.9°C yield error ranging from 100-200% the errors for the parachor method range between 0.5-3.6%. However, this is the only comparison available (27) and may not always be valid. The parachor values are estimated for different compounds by a group contribution method (32, 33). 

The predictive techniques are rather accurate. However, significant errors have been observed in few cases (4, 13, 27, 40). No direct comparison between the three predictive methods is available. The authors of the parachor method (27) claim that their method yields equal or better results than the PDD method for the cases considered in their study it is believed (42), however, that the latter is more reliable and it is recommended. The Weimer-Prausnitz method probably gives less accuracy than the PDD method, but it is more general. For example, Hanson and Van Winkle (40) report that their data on the hexane-hexene pair were not successfully correlated by the WP method. The Helpinstill-Van Winkle modification is recommended over the WP method. Recently, Null and Palmer (43) have presented a modification of the WP method which provides better accuracy but it is less general. The PDD method should be used cautiously when extrapolation with respect to temperature is used (27). When the GLC method is used, reliable results are expected. Evaluation of infinite dilution relative volatilities is recommended (36). 

Liquid Mixtures Compositions at the liquid-vapor interface are not the same as in the bulk liquid, and so simple (bulk) composition-weighted averages of the pure-fluid values do not provide quantitative estimates of the surface tension at the vapor-liquid interface of a mixture. The behavior of aqueous mixtures is more difficult to correlate and estimate than that of nonpolar mixtures because small amounts of organic material can have a pronounced effect upon the surface concentrations and the resultant surface tension. These effects are usually modeled with thermodynamic methods that account for the activity coefficients. For example, a UNIFAC method [Suarez, J. T. C. Torres-Marchal, and P. Rasmussen, Chem. Eng. Set, 44 (1989) 782] is recommended and illustrated in PGL5. For nonaqueous systems the extension of the parachor method, used above for pure fluids, is a simple and reasonably effective method for estimating a for mixtures. 

From Surface Tension Coefficients Among the different approaches used to calculate the surface tension coefficient, Vj, the most useful seems to be the parachor method [Sugden, 1924 Van Krevelen, 1976 Wu, 1982]. This parameter was defined as [Sugden, 1924] ... 

The example discussed suggests another more realistic formulation of the equilibrium problem. Under the conditions of the previous problem, instead of P, and Pg, we specify the pressure in one phase, say, Pg. The system of Eqs. (8) and (9) is completed by the Laplace equation in the form of Eqs. (23). In the second equation, (23), the dependence /(/ ) is supposed to be known from the geometry of the porous space. The surface tension and the wetting angle are defined as known functions of the thermodynamic conditions (e.g., the surface is assumed to be wet by the condensate and the surface tension is calculated by the parachor method). The volumetric hquid... 

The parachor method is one of the most commonly used correlations for the surface tensions. The method was originally suggested by MacLeod [50] and Sugden [51] for a singlecomponent fluid and was extended to multicomponent systems by Weinaug and Katz [52]. An expression for the surface tension in this method is... 

The parachor method gives, according to some estimates, a good order-of-magnitude value of the surface tension, but the deviations for realistic mixtures may reach a few tens of percent [46,53,54]. That is why several improvements of the method have been developed. The first type of improvement is modification of the value of a in order to improve the agreement with experiments (and, in this case, a may be taken to be variable [53]) or to approach the scaling law characteristic of the surface tension close to the critical point [55]. Another way is the modification of the values of the parachors. The parachors are usually taken from the standard tables but their correlations with the critical properties are also available [46,47]. Different types of the parachor-like expression have also been proposed [55-57] ... 

Figure 8 expresses the dependence of the surface tension, calculated by the parachor method, on the distance from the dew point. The values of surface tension vary significantly with this distance in the region of retrograde condensation and less significantly (but still noticeably) in the region of normal condensation. Variation of the surface tension is due to the fact that... 

The dependence of the Kelvin radius on the relative pressure in the retrograde region is presented in Fig. 16. Unlike the capillary pressure, the Kelvin radius depends monotonously on the dew-point pressure, because it is mostly affected by the values of the surface tension a. These values (estimated by the parachor method) show a high variation 1.579 mN/m for Pj = 80 bar, 0.734 mN/m for Pj = 100 bar, and 0.206 mN/m for Pj = 120 bar. The variation of the ratio V i/Vi is not too strong. 

Example 3.2. Estimation of surface tension using the Parachor method. 

Estimate, using the parachor method of McLeod-Sugden, the surface tension of liquid isobutyric acid at 333 K. The liquid density is 0.912 g cm , the molecular weight is 88.107 g moF and it can be assumed that the liquid density is much higher than the vapour density. 

Problem 3.1 Estimation of the surface tension using the corresponding states method Estimate, using the corresponding states method, the surface tension of liquid ethyl mercaptan at 303 K. The critical temperature is 499 K, the boiling point temperature is 308.2 K and the critical pressure is 54.9 bar. Compare the result to the experimental value (22.68 mN m ) and the estimation using the parachor method (which results in a deviation of 9.1%). 

Problem 3.2 Estimation of the surface tension of liquid mixtures using the parachor method (data from Hammkk and Andrew (1929))... 

Estimate, using the parachor method for mixtures (both the fuU and the approximate method), the surface tension of a liquid mixture having a density of... 

Three other methods have been widely used to estimate polymer melt surface tensions. The oldest of these methods is based on the parachor method first proposed for small molecules by Sugden. ... 

Figures 3 and 4 compare calculated surface tensions of the present theory. Equation (6), with those of the parachor method. Equation (10), the modified solubility parameter method. Equation 11, the corresponding states relation. Equation 12, and with...

---