Single-component systems chemical potential

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

For any single-component system such as a pure gas the molar Gibbs energy is identical to the chemical potential, and the chemical potential for an ideal gas is thus expressed as... 

In this first example, a single-component system consisting of a liquid and a gas phase is considered. If the surface between the two phases is curved, the equilibrium conditions will depart from the situation for a flat surface used in most equilibrium calculations. At equilibrium the chemical potentials in both phases are equal ... 

Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

Does Eq. (8) imply that for a single-component system, the chemical potential is the molar internal energy as well as the molar Gibbs free energy ... 

Chemical potential. In classical thermodynamics, equilibrium states are emphasized. In a state of equilibrium, each component of a material has a chemical potential that takes the same single value at every point in the system considered. A component s chemical potential depends on the pressure and temperature and on the component s concentration. For... 

The thermodynamics of the discontinuity surface can be examined by analyzing how the density of free energy f changes upon transition from one phase to another. From thermodynamics one can establish the relationship between the free energy, F, the isobaric-isothermal potential, f/, and the chemical potential, p, for a single component system ... 

Thus we have normalized the process to the bulk temperature of the gas phase and a single component (1). Importantly, the chemical potentials at this normalized state refer still to an equilibrium state and they will cancel therefore. Note that the chemical potential for a single component system is not dependent on the mol number, or in other words on the size of the phase. 

Similar to the case of a single component system, Eq. (15.2) can be expressed in terms of bulk chemical potentials and interfacial energy as... 

Consider any two phases a and /i of the same substance. Now in a single component system and are each functions of the temperature and pressure only. On the other hand, they are not the same functions and the two phases can co-exist only at such values of the temperature and pressure that the chemical potentials are equal. 

For systems that have more than one chemical component, we will have to give the chemical potential a label (typically a number or a chemical formula) to specify which component. The chemical potential for a single component pcj assumes that only the amount of the /th component, varies, and the amounts of all other components Upj i, remain constant. Equation 4.47 is therefore written... 

Single-component systems are useful for illustrating some of the concepts of equilibrium. Using the concept that the chemical potential of two phases of the same component must be the same if they are to be in equilibrium in the same system, we were able to use thermodynamics to determine first the Clapeyron and then the Clausius-Clapeyron equation. Plots of the pressure and temperature conditions for phase equilibria are the most common form of phase diagram. We use the Gibbs phase rule to determine how many conditions we need to know in order to specify the exact state of our system. 

Chain ends and branches can be thought of as impurities which depress the melting points of polymer crystals. The behaviour can be analysed in terms of the chemical potentials (Section 3.2.3) per mole of the polymer repeat units in the crystalline state and in the pure liquid //H (the standard state). For the pure polymer, which is a single-component system... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

In multicomponent systems, the single diffusivity is replaced by a multicomponent diffusion matrix. By going through similar steps, it can be shown that the [D] matrix must have positive eigenvalues if the phase is stable. In a multicomponent system, the diffusive flux of a component can be up against its chemical potential gradient except for eigencomponents. 

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