Entropy volume

Equation (5.52) is the first of our criteria. The subscripts indicate that equation (5.52) applies to the condition of constant entropy, volume, and total moles, with the equality applying to the equilibrium process and the inequality to the spontaneous process. 

Figure 3.1 A reaction profile, showing how the thermodynamic and kinetic quantities are related. X can be any state function (enthalpy, Gibbs energy, entropy, volume, etc.).

Physical State Chemical Potential Enthalpy" Entropy Volume... 

We now distinguish solid state transformations as first-order transitions or lambda transitions. The latter class groups all high-order solid state transformations (second-, third-, and fourth-order transformations see Denbigh, 1971 for exhaustive treatment). We define first-order transitions as all solid state transformations that involve discontinuities in enthalpy, entropy, volume, heat capacity, compressibility, and thermal expansion at the transition point. These transitions require substantial modifications in atomic bonding. An example of first-order transition is the solid state transformation (see also figure 2.6)... 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

Wood B. I (1988). Activity measurements and excess entropy-volume relationships for pyrope-grossular garnets. J. Geol, 96 721-729. 

These equilibrium reactions occur with large decreases in both volume and entropy. Volume changes range from —80 to —300 cm /mol depending on the solute and pressure. These volume changes, A V, are associated with the electrostriction of the solvent around the product anion, Fei(ion), and, to some extent, with a contribution of the partial molar volume of the electron, V(e). Thus ... 

The glass transition is characterized in part by an observed second-order transition distinguished by a discontinuity of the Gibbs free energy with respect to the aforementioned state variables, but by continuity of entropy, volume, and enthalpy. Hence, heat capacity, Cp, as well as the thermal expansion coefficient, a, as defined below, both exhibit a discontinuity at the glass transition temperature (McKenna, 1989). 

In this case both the temperature and pressure are dependent variables and are functions of the entropy, volume, and mole numbers of the components. From Equation (4.23),... 

The meaning and also the limitation of the term possible variations must be considered. For the purposes of discussion, we center our attention on Equation (5.2) and consider a heterogenous, multicomponent system. The independent variables that are used to define the state are the entropy, volume, and mole numbers (i.e., amount of substance or number of moles) of the components. The statements of the condition of equilibrium require these to be constant because of the isolation of the system. Possible variations are then the change of the entropy of two or more of the phases subject to the condition that the entropy of the whole system remains constant, the change of the volume of two or more phases subject to the condition that the volume of the whole system remains constant, or the transfer of matter from one phase to another subject to the condition that the mass of the whole system remains constant. Such variations are virtual or hypothetical,... 

The overall variations of the entropy, volume, and number of moles of each... 

The energy of a single-phase system is a homogenous function of the first degree in the entropy, volume, and the number of moles of each component. Thus, by Euler s theorem2... 

We consider an isolated, homogenous system, and imagine that a part of the single phase is separated from the rest of the phase by a diathermal, nonrigid, permeable wall. By this device we can consider variations of the entropy, volume, and mole numbers of the two parts of the system subject to the conditions of constant entropy, volume, and mole numbers of the... 

The specific requirements are determined more easily when the quadratic form of Equation (5.105) is changed to a sum of squared terms by a suitable change of variables. The general method is to introduce, in turn, a new independent variable in terms of the old independent variables. The coefficients in the resultant equations are simplified in terms of the new variables by a standard mathematical method. First, the entropy is eliminated by taking the temperature as a function of the entropy, volume, and mole numbers, so... 

Every coefficient in Equation (5.112) except that of (ST)2 can be expressed more simply as a second derivative of the Helmholtz energy. We take only the coefficient of (SV)2 as an example. Both (dE/dV)Sn and (cM/5K)r are equal to — P. The differential of (dE/dV)Sn is expressed in terms of the entropy, volume, and mole numbers and the differential of (dA/dV)Tmole numbers, so that at constant mole numbers... 

Figure 5.5. The projection of the equilibrium energy surface on the entropy-volume plane.

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

We recognize from our previous experience that pt is a function of the entropy, volume, temperature, or pressure in appropriate combinations and the composition variables. The splitting of into these two terms is not an operational definition, but its justification is obtained from experiment. The quantity pt is the quantity that is measured experimentally, relative to some standard state, whereas the electrical potential of a phase cannot be determined. Neither can the difference between the electrical potentials of two phases alone at the same temperature and pressure generally be measured. Only if the two phases have identical composition can this be done. If the two phases are designated by primes,... 

The quantity y is usually called the surface tension for liquid-gas interfaces and the interfacial tension for liquid-liquid interfaces. We see from Equation (13.2) that y da is the differential quantity of work that must be done reversibly on the system to increase the area of the system by the differential amount da at constant entropy, volume, and mole numbers. 

Equation (14.35) shows that the energy of the system is a function of the entropy, volume, and mole numbers, as before, but with one addition. Either from Equation (14.4) or from the fact that... 

We then see that the state of a system is defined by assigning values to the entropy, volume, mole numbers, and the position of a single homogenous region in the field. However, in so doing we need also to have a knowledge of the cross section of the system. 

We choose the total system to be the condenser and the entire dielectric medium. The condenser is immersed in the medium which, for purposes of this discussion, is taken to be a single-phase, multicomponent system. The pressure on the system is the pressure exerted by the surroundings on a surface of the dielectric. In setting up the thermodynamic equations we omit the properties of the metal plates, because these remain constant except for a change of temperature. The differential change of energy of the system is expressed as a function of the entropy, volume, and mole numbers, but with the addition of the new work term. Thus,... 

When we assume that the mole numbers of the materials that compose the solenoid are constant, the energy of the total system is a function of the entropy, volume, mole numbers of the material within the solenoid, and the magnetic induction. Thus, we have for the differential of the energy of the system... 

The number of water molecules bound by an ion, the hydration number, is not at all well defined. Also according to different methods e.g. transference, mobility, entropy, volume change on solution, specific heat etc. diverging results are obtained. 

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