Mechanical and Chemical Equilibrium

The approach outlined here will describe a viewpoint which leads to the standard calculational rules used in various applications to systems in themiodynamic (themial, mechanical and chemical) equilibrium. Some applications to ideal and weakly interacting systems will be made, to illustrate how one needs to think in applying statistical considerations to physical problems. 

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

Hence the thermal, mechanical, and chemical equilibrium conditions in terms of the intensive properties are... 

We now combine all phases and allow thermal, mechanical, and chemical equilibrium to take place throughout the composite system. The temperatures T, pressures P, and mole numbers n now differ from the values that obtained when the phases were isolated. From Eqs. (2.1.2) and (2.1.4) we find for the new set of variables... 

The RISM integral equations in the KH approximation lead to closed analytical expressions for the free energy and its derivatives [29-31]. Likewise, the KHM approximation (7) possesses an exact differential of the free energy. Note that the solvation chemical potential for the MSA or PY closures is not available in a closed form and depends on a path of the thermodynamic integration. With the analytical expressions for the chemical potential and the pressure, the phase coexistence envelope of molecular fluid can be localized directly by solving the mechanical and chemical equilibrium conditions. 

Equations (14.21) express the conditions for thermal, mechanical, and chemical equilibrium of the system, i.e., that temperature, pressure, and the chemical potential of... 

Crossing over from mechanical and chemical equilibrium to kinematics in mechanical sciences and to kinetics in chemistry the analogy is not established as well. Descriptive chemical kinetics is a very different concept than the concept of kinematics. Kinematics is governed by the Hamiltonian equations or other extremal principles. 

Thermod3mamic equilibrium encompasses thermal, mechanical, and chemical equilibrium. Chemical equilibrium, in turn, includes both diffusional and reaction equilibrium. In this section we have considered only nonreacting systems, and so, at this point, we have developed only the criteria for thermal, mechanical, and diffusional equilibrium criteria for reaction equilibrium are given in 7.6. 

With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, Pvap and puq, whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A. 10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. 

Thermodynamic equilibrium means the existence of thermal, mechanical and chemical equilibrium in a system. 

Consider a gas-liquid system where the liquid wets the solid surface. Based on mechanical and chemical equilibrium, one can write... 

A number of algorithms exist for eliminating two of the dependent variables. The method of Hansen (1962) and Turkevich and Mann (1990a) is used, and the method of undetermined multipliers is employed asserting two conditions, one for each phase. The first condition is that of thermal, mechanical, and chemical equilibrium so that each of the... 

We conclude our example with a remark. The simulation boxes posses no physical connection, i.e. there is no continuous migration of molecules from one region in space to another as in a real experiment. All that matters is that thermal, mechanical, and chemical equilibrium is attained. Since the chemical potential is a state function, it is not important which path we choose for this purpose. This implies that unphysical but highly efficient Monte Carlo moves become possible In the present case this is the instantaneous transfer of particles between boxes. 

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