CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 81, 2020 

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš 

Copyright © 2020, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-79-2; ISSN 2283-9216 

A Modified Alpha Function for Peng-Robinson 

Cubic Equation of State 

Wenying Zhaoa, Li Xiab, Xiaorong Caoa, Rongshan Bib, Shuguang Xiangb,* 

aChemistry and Chemical Engineering, Qilu Normal University Jinan 250200 Shandong China 
bProcess System Engineering Institute, Qingdao University of Science & Technology Qingdao 266042 Shandong China 

 xsg@qust.edu.cn 

A piecewise alpha function was proposed for the Peng-Robinson equation of state (PR EoS) to predict the 

thermodynamic properties of pure compounds. The proposed alpha function and its derivations satisfied the 

requirement of consistency test. The new alpha function with PR EoS was used to predict the vapour pressure 

of 11 kinds of compounds, and the average relative deviation is 0.34 %. The average relative deviations of the 

new alpha function for the prediction of liquid volume, enthalpy of vaporization, and isobaric heat capacity of 7 

kinds of compounds are 7.54 %, 1.37 %, and 6.90 %. The new alpha function is better than Soave and Stryjek-

Vera alpha functions and is as accurate as of the other alpha functions for the prediction of the vapour pressure 

of pure compounds. The results of enthalpy of vaporization and isobaric heat capacity of the new alpha function 

are superior to that predicted with the other alpha functions,  the deviations of liquid volume of the eight alpha 

functions are similar. The new alpha function with PR EoS can accurately predict the vapour pressure and 

enthalpy of vaporization of pure compounds but has great deviations for the prediction of liquid volume and 

isobaric heat capacity. It is a benefit to improve the predictive accuracy for liquid volume and isobaric heat 

capacity by modifying the form of PR EoS. 

1. Introduction

Cubic equation of state (EoS) is one of the most important methods for the prediction of the thermodynamic 

properties of pure compounds and mixture (Rezakazemia et al., 2018). The first cubic equation of state was 

proposed by van der Waals in 1883 for the prediction of the vapour pressure of real gases. Redlich and Kwong 

modified the form of the attractive term of van der Waals EoS and introduced the variable Tr0.5 to the energy 

parameter of the attractive term in 1949. Soave (Soave, 1972) adopts the form of Redlich-Kwong (RK) EoS and 

proposed an alpha function for the attractive term. Peng and Robinson (1976) modified the RK EoS and adopted 

Soave alpha function. Peng-Robinson (PR) EoS can accurately predict the vapour pressure of non-polar and 

weakly polar compounds but has great deviation for polar compounds. 

An alpha function is an important variable in the attractive term of cubic EoS. The alpha function affects the 

predictive accuracy of PR EoS for the vapour pressure of pure compounds. Many alpha functions were modified 

to improve the prediction of vapour pressures for polar compounds. For example, Mathias and Copeman (1983) 

proposed an alpha function to extend the application of PR EoS to highly polar compounds, Stryjek and Vera 

(1986a, b) modified Soave alpha function to improve the prediction of vapour pressure at the low reduced 

tempreture range. Mahmoodi and Sedigh (2017) improved the predictive accuracy of the liquid density of pure 

compounds and the vapour pressure of heavy hydrocarbons with a proposed alpha function. Zhao et al. (2018) 

has concluded the polynomial and piecewise alpha functions based on the PR EoS and analyzed the functional 

forms fit for different kinds of compounds. 

The alpha function as a variable of an attractive form of cubic EoS can weigh the attractive force intermolecular. 

The alpha function should not be negative, or else the alpha function is lack of physical meaning. Guennec et 

al. (2016) concluded and revised the constraints of consistency test of alpha function. Zhao et al. (2019) 

reviewed the development of constraints for alpha function. Among the alpha functions, many piecewise alpha 

function can accurate predict the vapour pressure of pure compounds. However, most cannot satisfy the 

requirement of consistency test. 

 
 
 
 
 
 
 
 
 
 
                                                                                                                                                                 DOI: 10.3303/CET2081092 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 30/04/2020; Revised: 01/06/2020; Accepted: 05/06/2020 
Please cite this article as: Zhao W., Xia L., Cao X., Bi R., Xiang S., 2020, A Modified Alpha Function for Peng-Robinson Cubic Equation of 
State, Chemical Engineering Transactions, 81, 547-552  DOI:10.3303/CET2081092 
  

547



Polynomial and piecewise alpha functions are important for the prediction of thermodynamic properties of pure 

compounds and mixtures. It is necessary to propose a new piecewise alpha function satisfying the requirement 

of consistency test. The predictive ability of the alpha function for the vapour pressure, liquid volume, isobaric 

heat capacity, and enthalpy of vaporization of pure compounds were also compared. 

2. Equation of state

PR EoS is a common used method for the prediction of thermodynamic properties, such as vapour pressure. 

Based on PR EoS, the new alpha function was proposed and the parameters were fitted. 

2.1  The PR EoS and representative alpha functions 

Peng and Robinson modified the attractive term of RK EoS, adopted Soave alpha function and refitted the 

parameter k with vapour pressures of pure compounds. The PR EoS is presented as Eq(1) to Eq(4). 

𝑃 =
𝑅𝑇

𝑣−𝑏
−

𝑎(𝑇)

𝑣(𝑣+𝑏)+𝑏(𝑣−𝑏)
(1) 

𝑏 = 0.07780
𝑅𝑇𝑐

𝑃𝑐
(2) 

𝑎(𝑇) = [0.45724
𝑅2𝑇𝑐

2

𝑃𝑐
2 ] 𝛼(𝑇) (3) 

𝛼(𝑇𝑟 ) = [1 + 𝑘(1 − √𝑇𝑟 )]
2

(4) 

Since the alpha function proposed by Soave in 1972, there are about 100 different alpha functions. According 

to the evaluation of 20 alpha functions by Young et al. (2016), seven representative alpha functions were chosen 

to be studied in this article. 

Table 1: Seven representative alpha functions selected in this study 

Functional name Reference Functional forms 

Soave (Peng and Robinson, 

1976) 
𝛼(𝑇𝑟 ) = [1 + 𝑘(1 − √𝑇𝑟 )]

2

𝑘 = 0.37464 + 1.54226𝜔 − 0.26992𝜔2 

Mathias-

Copeman 

(Mathias and Copeman, 

1983) 
𝑎(𝑇𝑟 ) = [1 + 𝑐1(1 − √𝑇𝑟 ) + 𝑐2(1 − √𝑇𝑟 )

2
+ 𝑐3(1 − √𝑇𝑟 )

3
]

2

 𝑇𝑟 ≤ 1 

𝛼(𝑇𝑟 ) = [1 + 𝑘(1 − √𝑇𝑟 )]
2

  𝑇𝑟 > 1 

Stryjek and Vera (Stryjek and Vera, 1986a) 𝛼(𝑇𝑟 ) = [1 + 𝑘(1 − √𝑇𝑟 )]
2
 𝑇𝑟 ≤ 0.7 

𝛼(𝑇𝑟 ) = [1 + 𝑘(1 − √𝑇𝑟 )]
2
 𝑇𝑟 > 0.7 

Androulakis-

Kalospiros-

Tassios 

(Androulakis et al., 1989) 
𝑎(𝑇𝑟 ) = 1 + 𝑑1 (1 − 𝑇𝑟

2/3
) + 𝑑2(1 − 𝑇𝑟

2/3
)

2
+  𝑑3(1 − 𝑇𝑟

2/3
)

3
 𝑇𝑟 ≤ 1 

𝑎(𝑇𝑟 ) = exp [𝑑1(1 − 𝑇𝑟
2/3

)] 𝑇𝑟 > 1

Schwartzentruber

-Renon-

Watanasiri 

(Schwartzentruber et al., 

1990) 
𝑎(𝑇𝑟 ) = [1 + 𝑚(1 − √𝑇𝑟 ) − (1 − √𝑇𝑟 )(𝑛1 + 𝑛2𝑇𝑟 + 𝑛3𝑇𝑟

2)]
2
 𝑇𝑟 ≤ 1

𝑎(𝑇𝑟 ) = exp [1 + 𝑚(1 − 𝑇𝑟
𝑛)] 𝑇𝑟 > 1

Almeida-Aznar-

Telles 

(Almeida et al., 1991) 
α(𝑇𝑟 ) = exp [𝑚(1 − 𝑇𝑟 )|1 − 𝑇𝑟 |

(𝛤−1) + 𝑛 (
1

𝑇𝑟
− 1)] 

(Mahmoodi and Sedigh, 

2017) 
𝛼(𝑇𝑟 ) = exp [2𝑐1(1 − √𝑇𝑟 ) − (𝑐2(1 − √𝑇𝑟 ))

2
+

2

3
(𝑐3(1 −

√𝑇𝑟 ))
3

]，|𝑐3| ≤ 1.25|𝑐1|

2.2 A proposed alpha function 

According to the characteristic of polynomial and piecewise alpha function, the parameter k of Soave alpha 

function was re-correlated with a quadratic function of the reduced temperature. To avoid the non-monotonous 

of Soave-type function, Heyen (1980) exponential alpha function was adopted in the supercritical regions. The 

form of the new alpha function was presented as Eq(5): 

𝛼(𝑇𝑟 ) = {
[1 + (𝑚1 + 𝑚2𝑇𝑟 + 𝑚3𝑇𝑟

2)(1 − √𝑇𝑟 )]
2

  𝑇𝑟 ≤ 1

exp[𝑛1(1 − 𝑇𝑟
𝑛2 )]  𝑇𝑟 > 1

 (5) 

The adjusted empirical parameters m1, m2, and m3 of the alpha function in the subcritical regions were fitted 

548



with the experimental vapour pressures of 70 compounds (Zhao et al., 2020). The parameters n1 and n2 of the 

alpha function in supercritical regions were calculated according to the first- and second-order derivations of the 

piecewise function continuous at the critical point. 

3. Results and discussion

The consistency of the proposed and the seven representative alpha functions was checked. The predictive 

accuracies of alpha functions for four kinds of thermodynamic properties were compared. 

3.1 Consistency test of the proposed alpha function 

The changing trends of the proposed and seven representative alpha functions and their derivations with 

increasing reduced temperature were determined with n-butanol in the reduced temperature range of 0.4 to 6. 

The Mathias-Copeman, Stryjek-Vera, Androulakis-Kalospiros-Tassios, and Schwartzentruber-Renon-

Watanasiri piecewise alpha functions are continuous at the dividing point (Figure 1a). However, the first-order 

derivation of Stryjek-Vera and Schwartzentruber-Renon-Watanasiri function are not continuous at Tr = 0.7 and 

Tr = 1 (Figure 1b), the second-order derivation of Mathias-Copeman and Androulakis-Kalospiros-Tassios 

function are discontinuous at Tr = 1 (Figure 1c). Almeida-Aznar-Telles alpha function and its first-order derivation 

are continuous at the critical point. Because of the term |1 − 𝑇𝑟 |
(𝛤−1) in the alpha function, the second- and third-

order derivations does not exist at Tr = 1. Soave alpha function and its derivations are continuous but not 

monotonous in the full temperature range. The proposed piecewise alpha and its second-order derivation are 

positive and monotonically decreasing, the first-order derivation is negative and monotonically increasing, and 

the third-order derivation is negative (Figure 1d). The proposed and Mahmoodi-Sedigh alpha function can satisfy 

the requirement of consistency test. 

Figure 1: The representative and proposed alpha function and their derivatives for n-butanol in the reduced 

temperature region from 0.4 to 6. (a) alpha function; (b) the first-order derivation of alpha function; (c) the 

second-order derivation of alpha function; (d) the third-order derivation of alpha function. 

3.2 The prediction of thermodynamic properties 

The prediction of alpha functions for the vapor pressure, liquid volume, enthalpy of vaporization, and heat 

capacity were discussed in the following sections. 

3.2.1 Vapour pressure 

To compare the predictive ability of the alpha function in different forms, the parameters of alpha functions, 

except for Soave and Stryjek-Vera, are refitted with the vapour pressure of 70 compounds. The parameter k0 of 

Stryjek-Vera alpha function was calculated with the correlation of acentric factor, and the parameter k1 was also 

refitted. The parameter m of Soave (1976) in PR EoS was also calculated with the correlation of acentric factor. 

The average relative deviation (ARD) was calculated by the Eq (6). 

(a) (b) 

(d) (c) 

549



𝐴𝑅𝐷 =
1

𝑁
∑

|𝑃𝑐𝑎𝑙
𝑠𝑎𝑡−𝑃𝑒𝑥𝑝

𝑠𝑎𝑡 |

𝑃𝑒𝑥𝑝
𝑠𝑎𝑡 × 100 %

𝑁
1  (6) 

Where 𝑃𝑐𝑎𝑙
𝑠𝑎𝑡 is the calculated value of saturated vapour pressure, and 𝑃𝑒𝑥𝑝

𝑠𝑎𝑡 is the experimental saturated vapour

pressure. N is the number of experimental data. 

Comparing the ARD of Stryjek-Vera and the proposed alpha function for the prediction of vapour pressure, it is 

shown that the proposed alpha function is more accurate than Stryjek-Vera alpha function in the reduced 

temperature range between 0.7 and 1.0. The ARDs of Stryjek-Vera is close to that of the proposed alpha function 

for the prediction of the vapour pressure of ketones, esters, hetercycles and water, but less accurate than the 

new alpha function for the other kinds of compounds. According to the mean ARD of 11 kinds of compounds 

(Table 2), the proposed alpha function (0.34 %) is more accurate than the generalized one-parameter Soave 

alpha function (4.50 %), especially for polar compounds and heavy hydrocarbons, such as normal alcohols, 

acids, and water. The mean ARD of semi-generalized two-parameter alpha function is 0.87 %. The mean ARDs 

of the Mathias-Copeman, Androulakis-Kalospiros-Tassios, Schwartzentruber-Renon-Watanasiri, Almeida-

Aznar-Telles, and Mahmoodi-Sedigh alpha functions are 0.34 %, 0.33 %, 0.34 %, 0.33 %, and 0.38 %. The 

proposed alpha function is slightly more accurate than Mahmoodi-Sedigh alpha function and is as accurate as 

Mathias-Copeman, Androulakis-Kalospiros-Tassios, Schwartzentruber-Renon-Watanasiri, and Almeida-Aznar-

Telles. The generalized and semi-generalized alpha functions with one or two parameters are not accurate as 

of the new alpha function for the prediction of vapour pressure, especially for the heavy hydrocarbons and polar 

compounds. The three-parameter alpha functions have similar predictive capability. 

Table 2: The average relative deviation of eight alpha functions for vapour pressures of 11 kinds of 

compounds 

Compounds type Soave 
Mathias-

Copeman 

Stryjek-

Vera 

Androulakis-

Kalospiros-

Tassios 

Schwartzen

truber-

Renon-

Watanasiri 

Almeida-

Aznar-

Telles 

Mahmoodi-

Sedigh 

The 

proposed 

alpha 

function 

Normal alcohols 17.38 0.83 1.92 0.82 0.81 0.87 1.06 0.84 

Normal alkanes 6.27 0.55 0.87 0.54 0.57 0.52 0.70 0.55 

Aromatic 

hydrocarbons 
2.11 0.33 0.91 0.30 0.34 0.36 0.38 0.34 

Halogenated 

hydrocarbons 
2.58 0.23 1.53 0.12 0.13 0.11 0.13 0.12 

Gases 1.72 0.34 0.90 0.37 0.36 0.28 0.33 0.33 

Acids 6.41 0.11 0.42 0.20 0.13 0.13 0.10 0.11 

Ethers 2.22 0.14 0.76 0.16 0.14 0.10 0.14 0.13 

Ketones 0.80 0.05 0.23 0.03 0.10 0.04 0.06 0.06 

Esters 3.27 0.52 0.61 0.48 0.53 0.51 0.53 0.50 

Heterocycles 2.21 0.60 0.85 0.60 0.60 0.65 0.61 0.64 

Water 4.58 0.10 0.60 0.05 0.07 0.02 0.12 0.15 

Average 4.50 0.34 0.87 0.33 0.34 0.33 0.38 0.34 

3.2.2 Liquid volume 

Because halogenated hydrocarbons and heterocycles are lack of the experimental data of liquid volume, 

enthalpy of vaporization, and isobaric heat capacity, and the strong associating compounds of acids and water 

were not suitable be predicted with PR EoS, the four kinds of compounds were not discussion in the following 

sections. The ARDs of eight alpha functions for the prediction of liquid volume, enthalpy of vaporization, and 

isobaric heat capacity of seven kinds of compounds were presented in Figure 2. The mean ARD of the proposed 

alpha function is 7.54 % for the liquid volume of seven kinds of comounds (Figure 2a). The ARDs of the new 

function for the liquid volume of normal alcohols, normal alkanes, aromatic hydrocarbons, ethers, and esters 

are 6.68 %, 8.56 %, 3.31 %, 3.30 %, and 4.69 %, and the ARDs for gases and ketones are 12.0 % and 14.2 %. 

The mean ARD of Soave is 7.84 %, the mean ARDs of Mathias-Copeman, Stryjek-Vera, Androulakis-

Kalospiros-Tassios, Schwartzentruber-Renon-Watanasiri, Almeida-Aznar-Telles, and Mahmoodi-Sedigh are 

between 7.51 % and 7.56 %. The proposed alpha function is more accurate than Soave, and is slightly better 

than Mathias-Copeman, Androulakis-Kalospiros-Tassios, Mahmoodi-Sedigh. The PR EoS with eight alpha 

function has similar results for the prediction of liquid volume of seven kinds of compounds. According to the 

results of liquid volume, PR EoS is not suitable for the prediction of liquid volume of pure compounds. 

550



3.2.3 Enthalpy of vaporization 

The mean ARDs of the proposed alpha function is 1.37 % (Figure 2b) for enthalpy of vaporization of seven kinds 

of compounds. The new alpha function is more accurate than Soave (2.48 %) and Stryjek-Vera (1.80 %) alpha 

functions, is better than Mathias-Copeman (1.40 %), Androulakis-Kalospiros-Tassios (1.39 %), Almeida-Aznar-

Telles (1.40 %), and Mahmoodi-Sedigh (1.46 %), and is as accurate as Schwartzentruber-Renon-Watanasiri 

(1.37 %). The ARDs of eight alpha functions, except Soave and Stryjek-Vera, for enthalpy of vaporization of 

normal alcohols, is between 2.5 % and 3 % and is less than 1.5 % for normal alkanes, aromatic hydrocarons, 

gases, ethers, ketones, esters. The prediction of PR EoS with an alpha function for enthalpy of vaporization is 

merely affected by the first-order derivation and the relative ratio of the compression factors of liquid and vapour. 

The form of PR EoS has little effect on the prediction of the enthalpy of vaporization. The predictions of enthalpy 

of vaporization are accurate. 

Figure 2: The ARDs of thermodynamic properties predicted with eight different alpha functions. (a) liquid volume; 

(b) enthalpy of vaporization; (c) isobaric heat capacity 

3.2.4 Isobaric heat capacity 

The mean ARD of the proposed alpha function is 6.90 % for the prediction of isobaric heat capacity (Figure 2c), 

the mean ARDs of Soave, Mathias-Copeman, Stryjek-Vera, Androulakis-Kalospiros-Tassios, Schwartzentruber-

Renon-Watanasiri, Almeida-Aznar-Telles, and Mahmoodi-Sedigh are 10.21 %, 7.38 %, 8.59 %, 7.10 %, 7.60 

%, 7.40 %, and 7.72 %. The new alpha function is better than the other seven alpha functions for the prediction 

of isobaric heat capacity. However, the ARDs of new alpha function for normal alcohols and gases are great, 

being 14.37 % and 9.86 %, and the ARDs for normal alkanes, aromatic hydrocarbons, ethers, ketones, esters 

are between 3 % and 6 %. The prediction of isobarc heat capacity is not only related to the second-order alpha 

function but also related to the functional form of the attractive and repulsive term of cubic EoS. Seeing from the 

perspective of molecular, the hydrogen bond forms between associating compounds, the strong induced force 

exists between the strong polar compounds, and the dispersion force mainly presents among the non-polar 

compounds. The intermolecular force is quite different between non-polar and polar compounds. PR EoS can 

accurately describe the intermolecular force among non-polar compounds, but it is difficult to describe that 

among polar compounds. The predicted deviation of isobaric heat capacity is great. 

4. Conclusion

A piecewise alpha function for PR EoS was proposed for the prediction of the vapour pressure, liquid volume, 

enthalpy of vaporization, and isobaric heat capacity of non-polar, weakly polar, and polar compounds. The 

proposed alpha function and its first- and second-order derivations are monotonous and continuous in the full 

temperature range, and the value of third-order derivation is negative, which meets the requirement of 

consistency test. The proposed alpha function is superior to Soave, Stryjek-Vera, Mathias-Copeman, 

Androulakis-Kalospiros-Tassios, Schwartzentruber-Renon-Watanasiri, Almeida-Aznar-Telles alpha functions in 

the aspect of consistency test. The new proposed alpha function can accurately predict the vapour pressure 

and enthalpy of vaporization of pure compounds. The ARD of the new alpha function for vapour pressures of 

11 kinds of compounds is 0.34 %, and the ARDs for enthalpy of vaporization, liquid volume, and isobaric heat 

capacity of seven kinds of compounds are 1.37 %, 7.54 %, and 6.90 %. The proposed alpha function for the 

prediction of vapour pressures is obviously better than Soave and Stryjek-Vera alpha function, is slightly more 

accurate than Mahmoodi-Sedigh alpha function and is as accurate as Mathias-Copeman, Androulakis-

Kalospiros-Tassios, Schwartzentruber-Renon-Watanasiri, and Almeida-Aznar-Telles alpha functions. The new 

(a) (b) (c) 

551



alpha function are most accurate than the other seven alpha functions for the prediction of enthalpy of 

vaporization and isobaric heat capacity, but has similar precision with for the prediction of liquid volume. The 

proposed alpha improved the prediction of enthalpy of vaporization and isobaric heat capacity, and has good 

accuracy for the estimation of vapour pressures. To further improve the predictive accuracies of isobaric heat 

capacity and liquid volume of pure compounds, the form of PR EoS should be modified. 

Acknowledgement 

This research was supported by Major Science and Technology Innovation Projects of Shandong Province (No. 

2018CXGC1102). 

Nomenclature 

ARD average relative deviation Superscripts/subscripts 

a the parameter of attractive term c critical properties 

b volume of hard core cal calculated 

c, d, k, 

m, n 
parameters in alpha functions exp experimental 

EoS equation of state sat saturation 

PR Peng-Robinson  r reduced state 

P vapour pressure, Pa Greek letters 

R universal gas constant, J/(mol∙K) α 
temperature dependence of the attractive 

term in PR EoS 

T tempeture, K 𝛤 a parameter in the alpha function 

v volume, m3/mol ω acentric factor 

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