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A simplified model is proposed for an easy understanding of the coarse-grained technique and for achieving a first approximation to the behavior of gases. A mole of a gas substance, within a cubic container, is represented by six particles symmetrically moving. The impacts of particles on container walls, the inter-particle collisions, as well as the volume of particles and the inter-particle attractive forces, obeying a Lennard-Jones curve, are taken into account. Thanks to the symmetry, the problem is reduced to the nonlinear dynamic analysis of a SDOF oscillator, which is numerically solved by a step-by-step time integration algorithm. Five applications of proposed model, on Carbon Dioxide, are presented: 1) Ideal gas in STP conditions. 2) Real gas in STP conditions. 3) Condensation for small molar volume. 4) Critical point. 5) Iso-kinetic energy curves and iso-therms in the critical point region. Results of the proposed model are compared with test data and results of the Van der Waals model for real gases.

Recently, in Computational Chemistry, the coarse-grained molecular dynamics technique is often used, by which millions of molecules are represented by a few hundred particles [^{20} molecules. By this technique, the computational handling of chemical problems becomes possible and usually a satisfactory approximation to observed behavior is achieved.

If we consider an amount of a gas substance, represented by a few particles, first in a large container (

1. In the large container of

2. In the large container of

For the above two reasons, for a quite large molar volume, as in

On the contrary, for a small molar volume, as in

J. D. van der Waals [

The kinetic behavior of gases, ideal and real ones, is often described by the Maxwell-Boltzmann stochastic model [

Also, a very coarse-grained model can be used, that is consisting of very few, very large particles. This is similar to the concept of fundamental vibration mode of structural dynamics. Where there exist a lot of high vibration modes (with small periods and usually small amplitudes, too), which are of negligible interest, but complicate the computation and require a very small time steplength. Whereas, the fundamental vibration mode is the simplest mode and, at the same time, the most representative of the dynamic behavior of the structure.

In the present work, such a symmetric deterministic model is proposed for real gases, which is very coarse-grained, that is, it consists of very few - very large particles, and is compared to corresponding test data [

In the recent literature on the coarse-grained technique [

A mole of a gas substance is considered, within a cubic container of side L (

We assume that the six particles move symmetrically. So, by considering the plane Iyz (

The particles are initially provided with equal speeds directed outwards. And they pass successively through three characteristic states: 1) Initial state (

Obviously, all the six particles, as they move symmetrically, they pass simultaneously from the above three characteristic states.

The mutual repulsive forces F between a particle and a container wall (

where D diameter of particle and

that is no force F is developed between the particle and the wall.

On the contrary, if

a mutual repulsive force F is developed between particle and wall, given by the equation (

where _{0} of Lennard-Jones curve is described in the following Section 4.1.

The mutual forces F, attractive or repulsive, between any couple of particles, with a distance

For any distance

where

Thanks to the symmetrical movement of all the six particles, it is enough to study the movement of only one particle, let choose that on right part of axis Iy of _{w}.

At left of _{4} of the four equal forces (attractive or repulsive), by which the points F, B, O, U act on point R is, according to

where

Finally, the left particle L acts on particle R, by an, always attractive, force

So, the horizontal force on particle R, due to inter-particle action, is

and the total horizontal force on particle R, due to inter-particle action and impact on wall is,

and the acceleration of particle R, under consideration, is, at any instant,

It has been described, in the previous Section 2, how the proposed model is reduced, thanks to the symmetric movement of its six particles, to the study of the movement of the single particle R (

For this purpose, the algorithm of trapezoidal rule (or Newmark algorithm of constant average acceleration) is chosen, combined with a predictor-corrector technique, with two corrections per step [

The flow-chart of the proposed algorithm is shown in

First, the constant input data are read: particle mass m and diameter D, force coefficient

The initial conditions are read: position y, temperature T and speed v of the particle R under consideration. The initial speed v results from initial temperature T, by a thermodynamic postulate, which will be described in following section 4.1.

The subroutine L-J (Lennard-Jones) is called, which, from the initial position y of the particle, determines the initial forces

Within each step of the algorithm, first the steps counter n is increased by 1 and time t by

Then, the prediction is performed, which determines the predicted values

The first correction, by trapezoidal rule, determines the first corrections

The second and final correction finds the final values of

The output, of present step of algorithm, is printed: steps counter n, time t, position y and speed v of the particle, forces

At the end of step of algorithm, three summations are made: The present force

Then, if the first cycle of oscillation has not yet been completed, we continue with the next step of the algorithm.

When the first cycle of oscillation is completed, by returning to the initial state, if we continued the algorithm, everything would be repeated the same, with only a small algorithmic damping. So, the algorithm is interrupted and the global output data are printed, which are:

1) Mean inter-particle force

2) Mean particle-wall impact force

It results

3) Pressure on wall^{2}, which, divided by 101,325 N/m^{2}, turns to atm units.

4) Mean 2nd power of speed

5) Mean total kinetic energy

Based on the step-by-step time integration algorithm, described in the previous section 3.1. and the flow-chart of

The program is written in the version Force 2.0 of Fortran, whose compiler is free available, even in Internet cafés.

The proposed simplified coarse-grained dynamic model for real gases is applied on Carbon Dioxide (CO_{2}), which exhibits a particular behavior in Critical point region, as it condensates for rather high temperatures, slightly lower than

From the next Section 4.1, it is apparent that, in order to calibrate the proposed model on other gases, the following data are required: molar mass, incompressibility limit of molar volume, as well as temperature, pressure and molar volume at the Critical Point.

The numerical values of parameters of proposed model are determined below, which will be used in the following applications:

1) The mass of a particle is

2) The diameter D of a particle is determined on the basis of criterion of in- compressibility of closely-packed equal spherical particles, as shown in ^{3}, which corresponds to a cubic container with side L = 3.684 cm. In

3) The force coefficient _{2},

4) For the side L of cubic container, in STP (standard temperature-pressure) conditions, the value _{2}, values of L ranging from 4.0cm up to 9.0 cm are used, which correspond to container volumes V = L^{3} ranging from 64 cm^{3} up to 729 cm^{3}.

5) The time step-length

where

A maximum stiffness

However, the cost, from using a further shorter time step-length

6) The initial position of the particle R under consideration is

7) Initial temperatures, ranging from

8) The initial speed_{0}, by the thermodynamic postulate:

where R = 8.3144 Joules^{−1}∙K^{−1} is the value of gas constant for ideal gases. However, for small molar volumes, through the oscillation of the particle, the speed v is significantly reduced, which implies mean values of R much smaller than the initial one, as will be shown in the applications.

A mole of Carbon Dioxide is considered, within a cubic container of side L = 28.195 cm, that is volume

In this first application, point particles are assumed, that is with zero volume, and the inter-particle attractive forces are ignored. So, we have an ideal gas. This case is simple, so it will be solved by hand.

Within the first cycle of oscillation, the particle, starting from the position

In

At inter-particle collision and particle-wall impact, the impulse-momentum conservation equation can be written:

Here,

where

The pressure on the wall is

as was expected for an ideal gas in STP conditions.

As the speed is constant, the total kinetic energy is, at any instant,

The potential energy is

and the thermodynamic quantity is

It is observed that

The same input data, of the previous first application, in STP conditions, are again considered, that is a cubic container with side L = 28.195 cm, thus volume

In

The mean inter-particle force is

^{nd} power of speed is

So, the mean (rms) speed is

that is, slightly larger than initial speed.

And the mean kinetic energy is

that is, it slightly deviates from the corresponding value of ideal gas. The mean kinetic energy is noted on the diagram K.E.-t of

A small cubic container with side L = 4.55 cm, thus molar volume

The application run by the proposed step-by-step algorithm, with

In Figures 12(b)-(e), the variations, with respect to time t, of four quantities, are presented: b) position y of the particle. c) speed v. d) inter-particle force

In the present application, because of the low initial temperature, thus low initial speed and kinetic energy, too, the inter-particle attractive forces

Because of the zero particle-wall forces,

The mean 2^{nd} power of speed results

thus the mean (rms) speed is

which is noted on the diagram K.E.-t of

The same small cubic container of side L = 4.55 cm of previous application is considered, which implies a volume

The application run by the proposed step-by-step algorithm, with

In Figures 13(b)-(f), the variations, with respect to time t, of five quantities, are presented: b. position y of the particle. c. speed v. d. inter-particle force

The mean inter-particle force results

It is observed that _{i} − t, F_{w} − t of

The pressure on the wall is

close to the experimental critical pressure

The mean 2^{nd} power of speed is

Thus, the mean (rms) speed results

much smaller than the initial speed.

The mean kinetic energy results

which is noted on the diagram K.E.-t of

The above mean kinetic energy ^{−1}∙K^{−1}, as obtained by equating

The present application is adapted to the Critical point by its initial temperature_{2} will be determined in two different ways: By the group of iso- kinetic energy curves of

For side of cubic container ranging from L = 4.0 cm up to 9.0 cm, with a step^{3} ranging from 64 cm^{3} up to 729 cm^{3}. And for initial temperature ranging from

_{0}, the corresponding pressure P and mean kinetic energy

(V, P) was placed on the volume-pressure plane, with the corresponding

Then, by linear interpolation between successive values of

It is observed that, under the iso-kinetic energy curve of 1100 Joules, a Liquid phase exists, with zero pressures. Between the curve of 1100 Joules and the Critical curve of 1300 Joules, a Vapor phase exists with low pressures. And above the Critical curve, a Gas phase exists, with high pressures. That is, the Critical iso-

By placing the above iso-kinetic energy curves of proposed model on the same P-V (pressure-molar volume) plane, together with the corresponding iso-therms of test data of Eastman-Rollefson [

It is observed that the gas constant R exhibits values, in Critical point region, ranging from 2.85 Joules mole^{−1}∙K^{−1} up to 5.0, much smaller than the well- known value 8.3144, which is approximately valid for large molar volumes and accurately valid for ideal gases. The obtained variation of R values is described by the graph of

With the help of this graph, the iso-kinetic energy curves of proposed model, of

The above iso-therms of proposed model are compared with corresponding ones of the test data of Eastman-Rollefson [

It is observed, in the

A simplified coarse-grained dynamic model, for real gases, is proposed. Five applications of this model, on Carbon Dioxide, are presented:

1) In STP conditions, by ignoring particle volume and inter-particle attractive forces, the proposed model accurately represents the behavior of an ideal gas.

2) Again in STP conditions, but taking into account the particles volume and inter-particle attractive forces, the proposed model slightly deviates from the behavior of an ideal gas, as was expected.

3) For a small molar volume and a low initial temperature, the inter-particle attractive forces overcome the initial kinetic energy of particles and prevent them from reaching at impact with container wall. So, a Liquid phase exists, with zero pressure.

4) At the Critical point of Carbon Dioxide, the proposed model closely predicts the values of critical molar volume, temperature and pressure, known from experiments [

5) Iso-kinetic energy curves have been determined, by the proposed model, in the Critical point region of Carbon Dioxide. By comparing these iso-kinetic energy curves to corresponding iso-therms of test data by Eastman-Rollefson [^{−1}∙K^{−1}. With the help of this variation of values of R, the iso-kinetic energy curves of proposed model are transformed to iso-therms, which are compared to corresponding ones of test data by Eastman-Rollefson, as well as to iso-therms of Van der Waals model. And a better agreement is achieved between the proposed model and the Van der Waals model, as shown in

The above five numerical experiments show that the proposed simplified model can approximate the observed behavior of real gases.

In the present work, in order to achieve simplicity, the accuracy is reduced. However, if a refined version of the proposed model, with more particles, is developed, the accuracy can be improved.

Papadopoulos, P.G., Koutitas, C.G., Dimitropoulos, Y.N. and Aifantis, E.C. (2017) Simplified Coarse- Grained Dynamic Model for Real Gases. Open Journal of Physical Chemistry, 7, 50-71. https://doi.org/10.4236/ojpc.2017.72005