Thanks to the professor Pedro L. Contreras E. for bringing up this interesting topic,

In fact, what guides us in accepting theories; it is the interpretation they give about what is happening. The more the theory can offer comprehensive explanations, the more popular it is. The number of repeatable data and the accuracy of their measurement are very important in rejecting and accepting theories. Therefore, measurement tools are also influential in the acceptance of theories. As measurement tools became more precise, so did theories to explain high-precision data. Of course, the measuring tools used by scientists in 200 years ago are different from the measuring tools that are used today to determine the density, volume, and pressure of a gas. In the days when it was debated whether molecules exist or not, the emergence of a theory that can explain pressure-volume-temperature changes for dilute gases using the kinetic theory of particles is nothing short of a miracle. Having the least knowledge about the nature of matter, it is as if a miracle has occurred in human knowledge.

By finding any answer to one of the questions that has occupied the scientists mind for a long time, it brings many questions to their. All the theories that have been presented so far are a function of today's knowledge, and the characteristic of science is the evolution of the chain of consciousness. Whatever it seems to each scientist depends on the awareness of his era and the power of his imagination regarding the interpretation of data results. Sometimes even the imagination of a scientist has superiority over the data obtained from experiments and experiments. It is like trying to climb a mountain slope, the higher we go; the more we see a vast landscape in front of us. And among them there are people whose power of imagination goes higher than our short steps and like birds that inform about spectacular sights that are not visible yet for us.

In my recent studies (A general model for isochoric heat capacity of matter in different states by introducing thermodynamic dimension concept, DOI: 10.1016/j.fluid.2021.113355& Molecular cages in supercritical fluids at high pressures, DOI:10.1016/j.fluid.2022.113564), I realized that it is possible to look at the material world from the point of view of fractal geometry. This new vision helps to understand the integrity of matter. In other words, the different states of matter in this perspective depend upon the thermodynamic dimension. The lower the temperature of the fluid and the higher its pressure, the thermodynamic dimension of the fluid increases, so that along such changes, the ideal gas first becomes a dense gas, then a liquid, and finally a solid. This view is rooted in the effective intermolecular potential (A new analytical modeling for the determination of thermodynamic quantities of refrigerants, DOI: 10.1002/aic.16293). As the intermolecular potential of the particles deepens, the particles tend to spend more time together.

Actually, quantum effects are observed in all cases, what differentiates it is our ability to measure quantum effects between ideal gas particles. That is, intermolecular interactions are so weak that we ignore them. In fact, there is no ideal gas in the universe. Even in the most dilute gases, it can still be claimed that there are intermolecular interactions, but there is no precise instrument that can reveal their existence.

Dear Prof. Ali Ghandili

Thank you so much for your great answer. First, I should apologize for the late reply.

Yes, the "real gases subject", & in general the subject of "dense gases, liquids and solids" is a wonderful subject, that with the development of fast speed computers, and new algorithms plus visualization tools is being explored intensively using different approaches, and where ab initio methods play a significant role.

Kind Regards.

The study of the radial distribution functions g(r) in the case of liquids is an interesting subject of study.

Despite the great number of works; for now, it looks like only numerically can be treated.

Please check the following textbook that gives a nice advanced introduction to the subject:

• "Statistical Thermodynamics: An Engineering Approach" by John W. Daily. Cambridge University Press, 2019.

Kind Regards.

Dear thread readers,

For dense gases, the practical use of several real equations of state such as the Redlich–Kwong & the Peng–Robinson equations (useful for liquids as well) are widely discussed in the textbook by Prof. J. W. Daily

"Statistical Thermodynamics: An Engineering Approach" by John W. Daily. Cambridge University Press, 2019.

Best Regards.

Dear Doctor

Go To

Thermodynamics of Real Gases

Chapter 14. Statistical study of a fluid

"In a gas, the potential energy of interactions between molecules is small compared to their kinetic energy. At high temperatures and low densities, the regime where the ideal gas approximation is valid, it is even negligible. However as the temperature decreases and/or the density increases the contribution of the interaction energy to the total energy becomes more important and the properties of the gas start to deviate from that of an ideal gas. Finally, when the interaction energy becomes comparable to the kinetic energy, the gas condenses and becomes a liquid. The Van der Waals model allows to understand this transition from gas to liquid, even though it relies on a rather crude mean field theory and should only be expected to yield qualitative results. This is because it possesses the necessary properties to describe the physical phenomena at play in real gases and liquid-gas phase transitions."

Thank you for your post, Dear Prof. Sundus F Hantoosh

Indeed, the dense gases can be described using real equations of state such as the Van der Waals, but there are others as well that account for the gas-liquid transition as the Peng–Robinson real equation of state.

In real gases is important to have a method to evaluate the configuration integral that accounts for the potential term.

A textbook with the topic checking cutting-edge advancements is:

Statistical Thermodynamics: An Engineering Approach" by John W. Daily. Cambridge University Press, 2019.

Dear Readers & Followers,

To evaluate the configuration integral in dense (real) gases, it is needed to know an expression for the potential energy function.

There are 4 functions used for this goal:

• The rigid sphere potential.
• The square well potential.
• The Sutherland weakly attractive sphere.
• The Lennard-Jones potential.

A concise explanation is given by Prof. J. W. Daily in his textbook:

Statistical Thermodynamics: An Engineering Approach" by John W. Daily. Cambridge University Press, 2019

Best Regards