Single-phase, one-component systems

This difference can be evaluated. If we choose to define a standard state of the substance to be the state of the substance at the pressure P and the temperature T0, the difference in the value of the enthalpy at any temperature T and the given pressure P and that of the standard state is 

The differential of the enthalpy at a constant temperature and mole number is 

We may then choose the standard state of the system at the temperature T to be at some arbitrary pressure P0, so that the difference of the enthalpy of the system at the temperature T and pressure P and this standard state is given by 

It may be convenient to define the standard state of the system as the state at an arbitrary temperature, T0, and an arbitrary pressure, P0. The enthalpy of the system in any state defined by the temperature T and the pressure P may then be calculated by a combination of Equations (8.5) and (8.8). Two alternate equations, depending on the path we choose, are obtained. These are 

Exactly the same methods as used above may be applied to the entropy. When the entropy is considered as a function of the temperature, pressure, and mole number, the differential of the entropy is 

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The state of a single-phase, one-component system may be defined in terms of the temperature, pressure, and the number of moles of the component as independent variables. The problem is to determine the difference between the values of the thermodynamic functions for any state of the system and those for the chosen standard state. Because the variables are not separable in the differential expressions for these functions, the integrations cannot be carried out directly to obtain general expressions for the thermodynamic functions without an adequate equation of state. However, each of the thermodynamic functions is a function of the state of the system, and the changes of these functions are independent of the path. The problem can be solved for specific cases by using the method outlined in Section 4.9 and illustrated in Figure 4.1. 

The equations developed in Section 8.1 for single-phase, one-component systems are all applicable to single-phase, multicomponent systems with the condition that the composition of the system is constant. The dependence of the thermodynamic functions on concentration are introduced through the chemical potentials because, for such a system,... 

We shall now explain in greater detail how knowledge of the thermodynamic potential function of a single-phase, one-component system makes it possible to determine the conditions under which the system breaks up into different phases. We start from the stability condition and show the manner in which the thermodynamic potential function of the two-phase equilibrium can be constructed. As an example, let us select energy as the thermodynamic potential function for such a system that can only exchange heat with its surroundings. The respective boundary... 

Thus whenever two phases of the same single substance (one component system) are in equilibrium, at a given temperature and pressure, the molar free energy is the same for each phase. 

This section describes the phase change process for a single component on a molecular level, with both vaporization and condensation occurring simultaneously. Molecules escape from the liquid surface and enter the bulk vapor phase, whereas other molecules leave the bulk vapor phase by becoming attached to the liquid surface. Analytical expressions are developed for the absolute rates of condensation and vaporization in one-component systems. The net rate of phase change, which is defined as the difference between the absolute rates of vaporization and condensation, represents the rate of mass... 

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. 

A sublimation process is controlled primarily by the conditions under which phase equilibria occur in a single-component system, and the phase diagram of a simple one-component system is shown in Figure 15.30 where the sublimation curve is dependent on the vapour pressure of the solid, the vaporisation curve on the vapour pressure of the liquid, and the fusion curve on the effect of pressure on the melting point. The slopes of these three curves can be expressed quantitatively by the Clapeyron equation ... 

The phase behavior of a one-component system wiU illustrate the phase rule. In the pT diagram of Figure 4.6, the single-phase states are in the areas labeled G, L, or S, representing gas, liquid, or solid, respectively. By Equation (4.146), F=C-P + 2 = - + 2 = 2. There are two degrees of freedom, e.g., T and p, or T and p. Specifying two intensive variables completely fixed the intensive state of the phase. 

The derivative of the surface tension with respect to temperature at the interface between condensed phases in binary systems can be either positive, or negative, or even change its sign when the temperature changes, which makes it different from the vapor-liquid interface in a one-component system. Within a certain approximation one may assume that in binary systems, as in single-component ones, the value r = -do/dT is the excess of entropy within the discontinuity surface. Consequently, for the interface between condensed phases, the excess of entropy can not only be positive (as it was with singlecomponent systems), but also negative. This situation is especially typical for the interface between two mutually saturated liquid solutions. 

If a one-component system of n moles consists of a single phase of volume V, then the molar voliune, may be defined by the equation. 

The mesogenic units of liquid crystals have to be anisometric (i.e., non-spherical) objects in order to allow for the essential long-range orientational order. In the case of the thermotropic liquid crystals this precondition can be fulfilled in one-component systems single rod- or disk-like molecules of low molecular weight and also polymers can be suitable. Further possibilities arise if two- or more-component systems are considered and one of the components acts as a solvent for the other if the variation of solvent concentration leads to phase transitions the system is called lyotropic. 

Thermod5mamics Equations of state for density, enthalpy, or any other property for a single- and multi-component system and phase equilibrium that connects a thermod5mamic property in one phase with that in another adjacent phase. 

In a one-component system the chemical potential is equal to the molar Gibbs energy, so that if phases I and II of a single substance are at equilibrium... 

For a one-component system, on a P-T plot, zero degrees of freedom corresponds to a triple point, from which three curves leave. Each of them represents the equilibrium of two phases. The space between the curves represent single phases. 

Further, according to Ricci, although a single substance must behave as a one-component system, unary behaviour does not guarantee that a material is a single substance (9, 165). Finally, the phase rule approach doesn t always make a clear-cut distinction between unary and binary systems or between compound and solution (125). 

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