Helmholtz energies from equations of state and their relative deviations

 

Tej P. Akhouri1, K. Akhtar1, Binay P. Akhouri2

1P. G. Department of Chemistry, Ranchi University, Ranchi-834008, Jharkhand (India)

2Department of Physics, Birsa college, Khunti-835210, Jharkhand (India)

*Corresponding Author E-mail: akhouritej@rediffmail.com

 

ABSTRACT:

Extension of the hard-sphere equations of state to non spherical hard- body systems represents a formidable task. For an equation of state that is applicable to liquid and vapor states (and explicit for pressures), the corresponding characteristic function is the residual energy. The equations of state of Carnahan Starling, Boublik and R J Sadus are used to derive an expression for residual Helmholtz energies. Relative deviations of residual Helmholtz energies for these equations of state have been reviewed. The deviations of compressibility factor against packing fraction have been plotted for these equations of state. The agreement of these plots is found to be good for lower n-alkanes but not for longer n-alkanes. Thus it becomes obvious that the compressibility factor depends considerably on the shape of the studied molecules.

 

KEYWORDS: compressibility factor; residual Hemholtz energy; hard convex body; n-alkanes equation of state

 

 

INTRODUCTION:

The Helmholtz energy of hard body contribution can be derived from equations of state for hard bodies like hard spheres, hard convex bodies and hard convex body chain molecules. Most commonly used equations of state were designed initially for relatively simple molecules but progress [1] is being made increasingly on the accurate description of the properties of large, complicated molecules. The EOS proposed by Carnahan and Starling [2] is at the heart of most theoretically- based equations of state. The hard sphere concept has also found very useful in modeling chain molecules. Typically, a molecular chain can be modeled as a collection of tangent hard spheres. Construction of BACK (Boublik-Alder-Chen-Kreglewski) equation of state [3,4] and BACKONE [5, 6] equation of state were based on equation for hard convex bodies. PC-SAFT [7] equation of state was based on equation for hard chain molecules.

 

Theoretically, five different approaches have been developed namely, PRISM (polymer reference interaction site model integral equation), GFD (generalized Flory dimer), DF (density functional) , W (Wertheim) theory [15-22] and the modified Wertheim theory (MW) as the fifth theory which combines the idea of scaled particle theory and those of the Wertheim theory.  The MW theory uses the EOS proposed by Werthim, but with the nonsphericity parameter replacing the number of spheres of the system [23-30]. The MW theory provides a very accurate description of the equation of state from n-pentane up to n-hexane. The freely jointed hard spheres model [16] has been the subject to which most of the treatments have been applied. This is an interesting model, it is not a good representation of real chain molecules such as n-alkanes, where the bond angles are fixed and the monomer units overlap. Some attempts of extending these theories to hard n-alkanes models have recently appeared [26-32]. A chain of hard convex bodies is likely to be a representation of the geometry of many real chain molecules than the hard sphere chain. Therefore, the properties calculated for the HCB chain are more likely to reflect accurately the actual properties of the molecular fluid. To compare and review the equations of state and Helmholtz energy for hard spheres, hard convex bodies and hard convex body chain is the main purpose of this work.

 

1. Equation of state and Helmholtz energy for hard spheres

1.1 Equation of state of Carnahan Starling

In 1969, Carnahan and Starling [8] (CS) carried out analysis of the reduced virial series and proposed equation of state of the form

 

 

The equation of Carnahan and Starling has been proved to be accurate and has been used by many researchers.

 

1.2 Helmholtz energy using Carnahan Starling EOS

The residual Helmholtz energy can be calculated from compressibility factor Z as following:

 

 

From (1) and (2), we obtain an equation of Helmholtz energy for system of hard spheres:

 

 

2.       Equation of state  and Helmholtz energy for hard chain system

The compressibility factor of the spherical associating molecular system can be obtained by using the SAFT approach of Wertheim [8-11] and its applications by Chapman et al.[12-14]. Expression in terms of compressibility factor of hard spheres and chain contribution is written as:

 

 

The relation between number density of a chain  and number density of hard spheres is . The packing fraction is:

 

The pair correlation function at contact   is related to the compressibility factor as,

 

 

  

2.1                Hard chain equation of state of Carnahan Starling

From Carnahan-Starling equation of state (1) and equation for pair correlation at contact (6) we have:

 

 

The compressibility factor of the spherical associating molecular system from Carnahan- Starling approach can be written as:

 

2.2. Helmholtz energy using Carnahan Starling EOS

Helmholtz energy of a system of hard chains in the WERTHEIM approach can be calculated as:

 

 

From equations (3), (9) ,(10) and (7) we have

 

3.       Equation of state and Helmholtz energy for Hard convex bodies

3.1 Hard convex bodies Equation of State of Boublik

Boublik proposed an equation of state for Hard Convex Bodies [32]:

 

 

Where  is the nonsphericity parameter and defined in terms of three characteristic geometrical functional: the volume  , the surface area  and the radius of the mean curvature. Obviously, for a hard sphere withWhen, , eq.(13) reduces to Carnahan Starling equation. For, other hard convex body .

According to Boublik et al [32], the hard convex body equation can be extended to hard chain molecules of overlapping hard spheres (0.5 < L < 1) or tangent hard spheres (L=1) with formulations of packing fraction and parameter of nonspherecity as:

 

When hard spheres are tangent, L =1, equations for packing fraction(14) and parameter of nonsphericity (15) become:

 

 

3.2 Helmholtz energy using Boublik EOS

The Helmholtz energy is given by

 

 

4.    Equation of state and Helmholtz energy for Hard convex bodies chain

4.1    Hard convex bodies chain EOS of R J Sadus

We propose that the compressibility factor of a chain composed of m hard convex bodies can be obtained from the hard convex body compressibility factor  and the hard convex body site-site correlation function at contact  via the following equation which is analogous to eq.(10)

                                                                                                               

The site-site correlation function at contact for hard convex bodies is given by [31]:

Where  is the ratio of the actual surface area of the HCB ( ) to the surface area of hard-spheres                      ( ) occupying a diameter equivalent to the HCB diameter. Substituting eq.(18) and (13) into eq.(17) and evaluating the derivative of the logarithm of the HCB site-site correlation function at contact with respect to the packing fraction, we obtain

 

with, where , m and  can only take positive value .

 

 

4.2 Helmholtz energy using R J Sadus EOS

The Helmholtz energy from R J Sadus EOS is given by

 

 

5. Comparison of results derived from equations of state of hard spheres, hard convex bodies and hard convex body chain molecules

In this part we investigate the differences of compressibility factors and residual Helmholtz energies from equations of state of hard spheres of Carnahan Starling, hard chain of Boublik and hard convex body chain of R J Sadus. In these comparisons the values of the parameters of the equations of state are those of n-alkanes. Figure 1 shows the comparisons of compressibility factor derived from Hard sphere Chain of CS and HCB Chain of Boublik for the case L=1 and Comparisons of compressibility factor derived from HCB Chain of R J Sadus and HCB Chain of Boublik for the case L=1;  *( =2.5) (shorter n-alkanes, upper figures) and Δ( =4.5) and □( = 6.5)and o( =8)  (longer n-alkanes, lower figures).

 

Fig. 1 Comparisons of compressibility factor

 

 

Figure 2 shows the comparisons of residual Helmholtz energy derived from Hard sphere Chain of CS and HCB Chain of Boublik for the case L=1 and Comparison of compressibility factor derived from HCB Chain of R J Sadus and HCB Chain of Boublik for the case L=1;  *( =2.5) (shorter n-alkanes, upper figures) and Δ( =4.5) and □( = 6.5)and o( =8)  (longer n-alkanes, lower figures)

 

Fig. 2 Comparisons of residual Helmholtz energy

 

Table 1:

 

m

(nm)               

(nm2)             

(nm3)

Pentane

2.5

2.83803       

81.33965        

56.00749                                                 

1.3738    

3.42

Hexane

3.0

3.08390        

92.90855       

66.61772                  

1.4336                                      

3.20

Heptane

3.5

3.26891       

103.08395      

77.10671                  

1.4567                                      

3.10

Octane

4.0

3.44490       

112.5099        

86.76949                  

1.4889                                      

3.00

Nonane

4.5

3.58805        

121.5036        

97.04800                  

1.4974                                      

2.90

Decane 

5.0

3.73154        

130.2039       

106.9122                  

1.5148                                       

2.80

Undecane

5.5

3.86674        

138.6234        

116.7066                  

1.5309                                      

2.75

Dodecane

6.0

3.99444        

146.7951         

126.4545                 

1.5456                                      

2.71

Tridecane

6.5

4.11607       

154.7462         

136.1399                  

1.5595                                     

2.69

Hexadecane

8.0

4.45049        

177.4761         

164.8902                 

1.5967                                     

2.65

 

 

The non-sphericity of the molecule is modeled in terms of the number  of hard convex bodies. It models a hard molecule as a chain of freely jointed hard convex bodies. Each n-alkane can be represented as a chain of  hard HCB segments.  For determination of value of  for each n-alkane one should understand that there should be some phial basis of our couch of m based on the nature of the HCB.  For example, pentane is CH3CH2CH2CH2CH3, one might decide that the base unit is CH2CH2, which could be approximately modeled as ethane (CH3CH3).  Determining the HCB of ethane and using this basic HCB with m = 2.5 to model pentane.  For decane, using the CH2CH2 as HCB, m = 5.  Using values undefended of the base model for the HCB makes no sense. Using this concept, the value of  determined for each n alkane is tabulated in Table 1.

 

 

CONCLUSION:

Residual Helmhlotz energies and compressibility factors of hard convex bodies EOS of Boublik and hard convex chain EOS of R J Sadus have been reviewed. The results show that the application of hard convex bodies to hard chain systems as presented by Boublik and RJ Sadus is good for shorter n-alkanes but not for longer n-alkanes. Comparision between CS equation of state and HCB Chain equation of R J Sadus shows good agreement both for shorter and longer n-alkanes.

 

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Received on 08.08.2015         Modified on 22.08.2015

Accepted on 31.08.2015         © AJRC All right reserved

Asian J. Research Chem. 8(9): September 2015; Page 574-580

DOI: 10.5958/0974-4150.2015.00092.9