# Where a a a and b b b are respectively

Where
a, a, a and b, b, b are respectively, repulsive friction constants and attraction friction constants. c represents the quadratic repulsive friction constant and P is critical pressure (bar).

CPA equation of state
CPA model is an EOS that simply combines a cubic EOS (SRK-EOS) and an association term of Wertheim theory (chemical theory) [7], [23], [24]. SRK model [8] calculates interactions between the physical molecules [7]. Internal association term calculates particular interaction of site-site due to hydrogen bonding between similar molecules (self-association) and non-similar molecules (cross-association or solvation) [7]. Association term is based on Wertheim\'s first-order thermodynamic perturbation theory (TPT-1) [7], [23], [24]. In particular, activity of any linked site has been assumed independent from other linked sites on the same molecule. CPA-EOS has a form as follows [7]:
Z is the compressibility factor and ρ is the molar density.
In the key association term parameter, mole fraction of the alcohol molecule (X) is the mole fraction of monomer, meaning that it has not bonded in site of A. ∆ ,related to the association strength between two association sites belonging in two different molecules, is as below [7], [25], [26], [27]:
εAB and β respectively are association Pam3CSK4 Biotin sale parameter (barL/mol) and association volume parameter.
CPA-EOS, which was suggested by Michelsen and Hendriks, can be expressed in form of pressure term as the sum of SRK-EOS and the contribution of association term [28]:
g (), is the constant value of radial distribution function (RDF) for fluid reference (for example fluid of hard sphere, meaning that there is only repulsive forces). Finally, energy parameter of EOS is given by a Soave-type temperature dependency, while b has been shown in accordance with the majority of Equations of state, independent of temperature [7], [30]:
Energy parameter (α) of SRK term [8] in the above equation is the Soave type temperature dependency, Tr is the reduced temperature (T/Tc) and a parameter is in energy term (α) [7]. c parameter could be created for inert compounds such as hydrocarbons, via the acentric factor in classic cubic Equations of state [7]. Associating substances has been estimated with other pure component data parameters (vapour pressure and liquid density) [25], [29]. To calculate the RDF, we use following equations [7], [30]:
y, b, Vm respectively are: reduced density, co-volume parameter (L/mol) and molar volume (L/mol). Huang and Radosz [27] suggested five parameters in CPA-EOS for alcohols, three parameters in physical SRK term (similar EOS) and two parameters in association term (Wertheim) (association energy and association volume) [27].
According to Table 1, two-site (2B) and or three-site (3B) schemes are used for alcohols. In 3B formalism alcohols, oxygen\'s lone pairs, in situations of A and B are similar, while the site C is similar to the hydrogen atom. Due to the asymmetry of the association, fraction of not bonded hydrogen atoms (XC) is not equal to fraction of not bonded lone pairs of (XA or XB). In 2B formalism, the two lone-pair oxygen are considered as a single site [7].
The thermodynamic properties of alcohols and the experimental information used to adjust model parameters are given in Table 2, Table 3, Table 4 and Appendix A, respectively.

Results and discussion
In this work, we used the combination of ƒ-theory and CPA-EOS [7] for modeling viscosity of pure alcohols such as methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 1-pentanol and 1-octanol. Modeling constants are obtained in accordance with CPA EOS for schemes of 2B and 3B by using Terminology of Huang and Radosz (Table 1).
Friction constants calculated by using the least squares method are provided in Table 5, Table 6, Table 7. With replacing friction constants in the relevant equations, the alcohol viscosity can be calculated. Fig. 1, Fig. 2 represent the results of prediction of the CPA ƒ-theory model for schemes of 2B and 3B at atmospheric pressure. As can be seen in these figures, in 2B and 3B schemes, experimental data and the results of prediction of the CPA ƒ-theory model for methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 1-pentanol and 1-octanol create a good overlap for a wide range of temperatures and atmospheric pressure. In the 8a and 8b tables, the results of prediction of the CPA ƒ-theory model for pure alcohol are listed.