The following set of equations can be used for reproducing the thermodynamic profile:

where two physical properties (R, γ ) along with the model parameter η can be evaluated for any real fluid by assuming any model fluid. It is observed that the above thermodynamic properties are highly sensitive to the choice of hard sphere (cavity) radii. Normally a real fluid can be represented by a minimum of two parameters such as the size of the molecule and its binding energy. Hence the equation of state containing a single parameter has limited success. A different size of cavity (model parameter) is required to reproduce a different thermodynamic property. Because the hard sphere model, which is used for real fluids, is not at all sufficient and significant to reproduce density, volume expansion coefficient, and velocity data simultaneously.

The physical significance of R is the work per degree per mole. It may be expressed in any set of units representing work or energy (such as joules), other units representing degrees of temperature (such as degrees Celsius or Fahrenheit), and any system of units designating a mole or a similar pure number that allows an equation of macroscopic mass and fundamental particle numbers in a system, such as the ideal gas. In order to take deviation or check the gas constant in a liquid state for any model equation, let R→ ζ .R, then the above set of equations becomes

The above three equations are the basic equation of state and the partial derivatives with respect to temperature and volume. This set of equations can be used to determine (i) deviation parameter ζ for scaling thermodynamic variables such as pressure, volume, temperature, and gas constant; (ii) γ associated with the ratio of specific heat; (iii) model parameter η . Hence the solution set is a triplet (ζ , γ , η ). The present calculation shows that the ratio of specific heat does not have a logical domain [0 < γ ≤ 1.6667] corresponding to ζ = 1 (no deviation) and vice versa for an assumed model function Z, which naturally depends on η

2.2. Two parameters model

Hence it is necessary to use the equation of state, which contains a minimum of two model parameters (σ , ε ), and there should be one-to-one correspondence between model variables and thermodynamic variables, i.e., (σ , ε ) → (V, T),

It is a simple observation that all thermodynamic potentials are primarily determined by any two thermodynamic variables such as volume and temperature. For a given EOS, introducing scaling parameter ζ for existing thermodynamic variables such as P, V, T or a constant such as R along with model parameters such as regular molecular cavity diameter σ and binding energy ε , then the above set of equations becomes

One can evaluate model variables (η , τ ) in terms of model parameters (σ , ε ). Hence the inter consistency demands a set of solutions in terms of unique triplet [ζ , η , τ ]. Here the model variable again depends on scaling parameter ζ , which means [η (ζ ), τ (ζ )] and [σ (ζ ) , ε (ζ )]. Hence we conclude that model variables (η , τ ) depend on ζ , which is to be decided by the set of three equations. If we accept the variation of ζ , which means that it has to be related to some measurable physical properties, then the gas constant is a universal constant only when η → 0, τ → ∞ , (point object !). In this case, we obtain

i.e., we obtain the standard relations which are independent of the model function. It is not a surprising result, because every model equation must satisfy the following set of conditions:

Hence we conclude that if η ≠ 0 , τ ≠ 0 then R must be related to a physical property. It is observed that η is a reduced thermodynamic variable as the volume is related to hard sphere diameter σ , which can be related to the potential energy. Similarly, reduced temperature τ is related to the binding energy and hence related to the kinetic energy, i.e.,

The compressibility factor Z for a fluid of Lennard– Jones molecules^{[15– 17]} can be represented as Z(η , τ )), which contains the physical condition in terms of relation λ = g(η , f)

where

For graphical recreations, the above set of equations becomes

We first assume a liquid state γ = 1 at 273.16  K, and evaluate the model parameters η and τ from the following combined equations:

In this way, we can easily obtain the following model parameters:

Substituting these model parameters in the basic equations of state, we can easily evaluate thermodynamic scalar variable ζ ,

However, the deviation parameter ζ does not come out to unity as we expect. Hence, we should declare that the equation is mathematically exact, but physically fails to accommodate the thermodynamic data profile. We check the basic formulation or domain of such equations of state and relax or modify any physical property relating conditions so that the equations of state have some additional parameter, which explains or throws some light on the problem, then we could say some aspects are more within logical extensions. With the help of λ , we can explore the situation by evaluating the internal parameter or variable such as f. It is observed that when we select f suitably, we can use the equations to reproduce the computer simulation result in the neighborhood of the critical temperature. Note that f = 0 represents an original boundary condition.

The density data profile for a liquid^{[18, 19]} is represented in the following equation:

The ultrasonic wave velocity data profile for the liquid is represented in the following equation:

where temperature t= (T− 273.16). For T= 273.16  K and M= 41.05(1− hexanol), we have

We assume the following functional form of λ :

which is already tested in the light of the computer simulation result for the equation of state for Lennard– Jones fluids. If we assume other forms of function, such as

then our maple code returns no solution, even if we run the program for 75000  s. If we search the solution under constraints such as η 1, there is no solution. The form

does yield a solution, but it has poor computer simulation reproducibility in the vicinity of the critical temperature. With standard values ζ = 1, γ ≈ 1 and solving a set of equations for η , f, and τ

we have the following form of solution:

It is better to represent the numerical representation of the solution in Maple format than a conventional table format

We select a few real solutions by rejecting complex solutions in the first instant for simple physical interpretation, see Table  1. The corresponding critical constants are

and the reduced parameters are

due to the input

Table 1.

Table 1.

Table 1. State of a fluid corresponding to real solution S[s]. S. No. η τ fn State of fluid → S[s] 1 0.102773259 ± (3.736226) − 2.0246005 gas 2 0.235806403 ± (1.223700) 1.87529739 gas liquid (supported implicit graph) 3 0.621029512 ± (1.237404) 2.17313770 liquid state 1 4 0.669601118 ± (0.826764) 1.63532971 liquid state 2 (supported by implicit graph)

Table 1. State of a fluid corresponding to real solution S[s].

For visualization, we show Z(η , τ ) by red, Y(η , τ ) by blue, and X(η , τ ) by green. Looking at the intersections, we feels that they might represent two eigen states, as shown in Fig.  1.

Fig.  1.

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	Fig.  1. Graphical representation of solutions at 273.15  K for a real fluid. The red, blue, and green lines represent the solutions to Eqs.(17a)– (17c).

The equation of state with the first set of critical constants (η = 0.07423292188, τ = 1.772491374) is

which is closer to the experimental expectation. This is justified by the intersection of red, blue, and green curves. Another set of critical constants η =0.3633095199, τ =1.307124675 leads to

The second set of critical constants should be rejected, as supported by the following plot. Observing the intersection of (blue, green) and (green, red) curves and using a graphical language, we have

Fig.  2.

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	Fig.  2. Graphical representation of solutions at critical temperature for a real fluid. The red, blue, and green lines represent the solutions to Eqs.(18a)– (18c).