The phase transition of the first order in the critical region of the gas-liquid system

TL;DR

This paper investigates the first-order phase transition in a gas-liquid system near the critical point, using a collective variable approach and mapping to the Ising model, revealing detailed transition dynamics and energy calculations.

Contribution

It introduces a complete integration of the grand partition function in collective variables and models the phase transition as a line where gas and liquid coexist, with explicit energy and density jump calculations.

Findings

Identification of the phase transition line as μ* = 0.

Calculation of work and surface tension during droplet formation.

Quantification of density jumps at the transition.

Abstract

This is a summarising investigation of the events of the phase transition of the first order that occur in the critical region below the liquid-gas critical point. The grand partition function has been completely integrated in the phase-space of the collective variables. The basic density measure is the quartic one. It has the form of the exponent function with the first, second, third and fourth degree of the collective variables. The problem has been reduced to the Ising model in an external field, the role of which is played by the generalised chemical potential $\mu^*$. The line $\mu^*(\eta) =0$, where $\eta$ is the density, is the line of the phase transition. We consider the isothermal compression of the gas till the point where the phase transition on the line $\mu^*(\eta) =0$ is reached. When the path of the pressing reaches the line $\mu^* =0$ in the gas medium, a droplet of liquid springs up. The work for its formation is obtained, the surface-tension energy is calculated. On the line $\mu^* =0$ we have a two-phase system: the gas and the liquid (the droplet) one. The equality of the gas and of the liquid chemical potentials is proved. The process of pressing is going on. But the pressure inside the system has stopped, two fixed densities have arisen: one for the gas-phase $\eta_{g} = \eta_{c} ( 1 - {d}/{2})$ and the other for the liquid-phase $\eta_{l} = \eta_{c} (1 + {d}/{2} )$ (symmetrically to the rectlinear diameter), where $\eta_{c} = 0.13044$ is the critical density. Starting from that moment the external pressure works as a latent work of pressure. Its value is obtained. As a result, the gas-phase disappears and the whole system turns into liquid. The jump of the density is equal to $\eta_{c} d$, where $d = \sqrt{{D}/{2G}} \sim \tau^{\nu/2}$. $D$ and $G$ are coefficients of the Hamiltonian in the last cell connected with the renormalisation-group symmetry.