Entropy and volume

Figure 4.3 Behavior of thermodynamic variables at T for an idealized phase transition (a) Gibbs free energy and (b) entropy and volume.

With motion along the connodal curve towards the plait point the magnitudes Ui and U2, Si and S2, and ri and r2, approach limits which may be called the energy, entropy, and volume in the critical state. The temperature and pressure similarly tend to limits which may be called the critical temperature and the critical pressure. Hence, in evaporation, the change of volume, the change of. entropy, the external work, and the heat of evaporation per unit mass, all tend to zero as the system approaches the critical state ... 

From Eq. (6) it clearly can be seen that pressure and temperature are related by the changes in entropy and volume for the phase change. This equation also shows that once temperature is chosen, pressure must be fixed accordingly. It is easy to show that the change in entropy for the phase transition is simply the... 

For the metal ions that have been tried [12], this procedure works remarkably well in determining hydration free energies. There has been little [47] detailed attempt to determine hydration entropies and volumes in this way, and those properties are not... 

Equilibrium conditions in terms of internal energy and enthalpy are less applicable since these correspond to systems at constant entropy and volume and at constant entropy and pressure, respectively... 

In open systems consisting of several components the thermodynamic properties of each component depend on the overall composition in addition to T and p. Chemical thermodynamics in such systems relies on the partial molar properties of the components. The partial molar Gibbs energy at constantp, Tand rij (eq. 1.77) has been given a special name due to its great importance the chemical potential. The corresponding partial molar enthalpy, entropy and volume under the same conditions are defined as... 

The variation of the phase transition temperature with pressure can be calculated from the knowledge of the volume and enthalpy change of the transition. Most often both the entropy and volume changes are positive and the transition temperature increases with pressure. In other cases, notably melting of ice, the density of the liquid phase is larger than of the solid, and the transition temperature decreases... 

When 2 is assumed to be independent of temperature and pressure the entropy and volume of the liquid is given as... 

Here again G = G(T) - G(0). Other thermodynamic functions such as enthalpy, entropy and volume may be derived by the thermodynamic relationships discussed in Chapter 1. 

Activation Entropies and Volumes for Carbon monoxide Exchange with cts-[M(CO)2X2r... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

AG is the free energy change of a process with AS and AV as the accompanying changes in entropy and volume. As a first approximation, one can assume AS and AV to be nearly constant within a limited p and T range, and use the integrated equation... 

Agreement is also poor concerning entropy and volume excess terms. Because divalent cations (Mg, Ca, Fe, Mn) occupy only dodecahedral sites whereas octahedral sites are reserved for trivalent cations (Cr, Fe, Al), each cation has only one site at its disposal and permutability is fixed by stoichiometry (cf. section 3.8.1). As regards the occupancy on tetrahedral positions, we have already seen that analyses of natural specimens show silicon deficiencies, compensated by AF ... 

An example of such a system is one in which the internal energy, U, and the volume, V, are constant. If these are the only two constraints on the system then, at thermodynamic equilibrium, the entropy, S, is at a maximum. On the other hand, if entropy and volume are constant for the isolated system then, at thermodynamic equilibrium, the internal energy is at a minimum. See also Closed System Open System... 

A pure substance A exists in three modifications Ai A2 and Ag. At the triple point S j > a2 > o3 and Val>Va2>Vo3,S and V are entropy and volume respectively are there. A suitable P-T diagram on the basis of informations given above and also label the region, line and points constructed ... 

The entropy and volume of -phase are greater that the two other phases A2 andyl3. Therefore phase Ai must be stable in the region of low pressure and high temperature. The entropy and volume of phase 3 are the lowest among all the three phases, hence 3 must be stable in the region of high pressure and low temperature. The phase A2 must be stable for intermediate case. 

Here Cp, a and are the heat capacity, volume thermal expansivity and compressibility respectively. First-order transitions involving discontinuous changes in entropy and volume are depicted in Fig. 4.1. In this figure curves G Gu represent variations in free energies of phases I and II respectively, while // Hu and F, represent variations in... 

In contrast to intrachain changes of entropy and internal energy, which follow from the statistical theory, interchain changes of the internal energy, vibrational entropy and volume can be predicted by the statistical theory only at small strains (X < 1.3). 

The reaction rate is primarily determined by the enthalpy of activation (A// ), which is usually the case in square planar nucleophilic substitution reactions. Of greater importance, so far as a dissociative versus an associative mechanism is concerned, are the entropies and volumes of activation, AS and AY, respectively. Note that the values are negative for both the fct and the steps. The observed decrease in entropy is what we would expect for a mechanism in which two particles come together to give an activated complex. The volume of activation is determined by doing the reaction under high pressure ... 

Whether the phase transition is first- or second-order depends on the relative magnitudes of the coefficients in the Landau expansion, Eq. 17.2. For a first-order transition, the free energy has a discontinuity in its first derivative, as at the temperature Tm in Fig. 17.1a, and higher-order derivative quantities, such as heat capacity, are unbounded. In second-order transitions, the discontinuity occurs in the second-order derivatives of the free energy, while first derivatives such as entropy and volume are continuous at the transition. 

Entropies and volumes of activation,74 though less reliable criteria, are in the range usually found for unimolecular reactions and do not agree with values expected for the A-2 process. Solvent isotope effects also are in agreement with the A-l mechanism.75... 

The anticorrelation between entropy and volume for liquid water has been attributed to the formation of an open hydrogen-bonded network, in which a decrease in orientational entropy is accompanied by a volume increase (Debenedetti, 2003). This network is transient and short-ranged in liquid water (rather than being permanent and long-ranged in ice), and is the microscopic basis for water s negative thermal expansion. This open hydrogen-bonded network has a profound influence on the thermodynamics of liquid water (Debenedetti, 2003). 

Since entropy and volume are extensive properties, they can be combined,... 

According to Ehrenfest, a second-order transition occurs when p and its first derivatives are continuous across the transition region, but the second derivatives, Cp and compressibility, k are discontinuous. This behavior is illustrated in the second column of Figure 13.1. For these transitions, the enthalpy, in addition to the entropy and volume, is continuous across the transition. 

Gas studies are well covered with extensive explanation and interpretation of experimental data, such as steady state calculations, all illustrated by frequent use of worked examples. Solution kinetics are similarly explained, and plenty of practice is given in dealing with the effects of the solvent and non-ideality. Students are given plenty of practice, via worked problems, in handling various types of mechanism found in solution, and in interpreting ionic strength dependences and enthalpies, entropies and volumes of activation. 

In this case the entropy and volume are dependent variables and are functions of the temperature, pressure, and mole numbers of the components. In addition, we have four derivatives that are all equal to the chemical potential, one from each of the four equations, so... 

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