ABSTRACT
The properties of physical systems are determined by the interactions between their constituting particles: from the expression of the Hamiltonian and in particular the potential energy term, follows the calculation of the (grand) partition function and finally the estimation of the thermodynamic quantities (see Chapter 4). The ideal gas approximation is quite justified in the case of dilute gases, but to understand the behaviour of a real fluid, gas or liquid, interactions must be taken into account. First, to obtain a more realistic gas model, whose predictions are in better agreement with the experimental data. Secondly and more importantly, because the interactions between molecules generate collective behaviours that manifest at human scale by the emergence of the different states of matter and phase transitions. Thus, the existence of liquid, an intermediate phase between solid and gas, is not obvious, as it will be discussed in this chapter.
The simple classical fluid model (Section 5.1) falls within a sufficiently general framework to study interaction effects and the thermodynamic properties of real fluids (at least to a first approximation). Despite its simplicity, the (grand) partition function cannot be calculated exactly and very common approximations in physics must be used: series expansion, called virial expansion (Section 5.2) and the mean field approximation, the simplest method to handle a system of interacting particles (Section 5.3). The van der Waals equation of state, the first model proposed to study phase transition, will be then derived (Section 5.3.2) and the universal behaviour of the thermodynamic quantities at the critical point, that is independent of the microscopic details specific to a fluid, will be highlighted (Section 5.3.3). In Section 5.4, it will be shown how to describe the microscopic structure of a fluid in terms of the pair correlation function, https://www.w3.org/1998/Math/MathML"> g ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_22.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , which gives the probability to find a particle at a distance https://www.w3.org/1998/Math/MathML"> r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_23.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of a given particle. The main thermodynamic quantities will be expressed as functions of https://www.w3.org/1998/Math/MathML"> g ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_24.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
