INTERNAL ENTROPY

Vibrational energy states are too well separated to contribute much to the entropy or the energy of small molecules at ordinary temperatures, but for higher temperatures this may not be so, and both internal entropy and energy changes may occur due to changes in vibrational levels on adsoiption. From a somewhat different point of view, it is clear that even in physical adsorption, adsorbate molecules should be polarized on the surface (see Section VI-8), and in chemisorption more drastic perturbations should occur. Thus internal bond energies of adsorbed molecules may be affected. 

It is interesting to note that although the first examples of template effects were observed in nitrogen macrocycles (see chapter 2) no template effect appears to operate in the synthesis of 72. Richman and Atkins note this in their original report . The authors replaced the sodium cation with tetramethylammonium cations and still obtained greater than 50% yield of tetra-N-tosyl-72. Shaw considered this problem and suggested that because of the bulky N-tosyl groups, .. . the loss of internal entropy on cyclization is small He offered this as an explanation for the apparent lack of a template effect in the cyclization. 

The internal entropy production this represents the time-related entropy growth generated within the system (djS/df). The internal entropy production is the most important quantity in the thermodynamics of irreversible systems and reaches its maximum when the system is in a stationary state. The equation for the entropy production is then ... 

Fig. 9.3 Entropy-time diagram of an evolution process. If a negative fluctuation of the internal entropy production o occurs in a system, the controlling stationary state is terminated. An instability occurs, starting from which a new stable state is taken up. The change in the internal entropy is always negative (5Si < 0). The new stationary state has a lower entropy, i.e., the order of the system is increased (Eigen, 1971a, b)...

The second law of thermodynamics asserts that the total entropy 5 of a system may change in time because of exchanges with its environment and internal entropy production which is vanishing at equilibrium and positive out of equilibrium [5]... 

Entropy effects. The replacement of the coordination shell of the cation by a multidentate ligand has also the very important effect of decreasing the free energy of the system by the increase in translational entropy of the displaced water molecules. If there were no variation in solvation and internal entropies of the free ligand and of the complexes, the increase in translational entropy would amount to about 8 e.u., where x is the number of displaced solvent molecules minus one (38). This estimate is, however, very inaccurate large deviations are expected, especially in the case of complicated multidentate ligands for which complex formation may produce appreciable internal and solvation entropy changes. 

The summation replaces the product in going from Equation (53) to Equation (54) since we are dealing with logarithms in the latter. Note that the configurational entropy refers explicitly to the entropy associated with the mixture itself the internal entropy of the molecules themselves is clearly not included. Next a series of mathematical manipulations will transform Equation (54) into a more useful form ... 

Several attempts have been made to explain the variations in efficiency of ion exchanger catalysts for different esters and reaction media. Hammett et al. [366,479,488] suggested that the difference in efficiency for different esters arises from a difference in the magnitude of the loss in internal entropy of the ester molecule which accompanies its fixation on the resin catalyst in the formation of the transition state. It can be shown that the ratio of efficiencies for two esters, 1 and 2, is given by... 

The entropy of a molecule is composed of the sum of its translational, rotational, and internal entropies. The translational and rotational entropies may be precisely calculated for the molecule in the gas phase from its mass and geometry. The entropy of the vibrations may be calculated from their frequencies, and the entropy of the internal rotations from the energy barriers to rotation. 

The combining of two molecules to form one leads to the loss of one set of rotational and translational entropies. The rotational and translational entropies of the adduct of the two molecules are only slightly larger than those of one of the original molecules, since these entropies increase only slightly with size (Table 2.4). The entropy loss is up to 190 J/deg/mol (45 cal/deg/mol) or 55 to 59 kJ/mol (13 to 14 kcal/mol) at 25°C for the small molecules. This may be offset somewhat by an increase in internal entropy due to new modes of internal rotation and vibration (Figure 2.6). 

Fig. 1. Open systems Internal entropy production. d-,S a 0. dcS is the exchange of entropy with the environment.

For the elements and small, roughly spherical molecules, the entropy of melting is primarily due to expansional and positional entropy. This is because no rotational entropy is gained upon melting and there is no internal entropy contribution. Using Equation (19), with a o value of 100 ... 

AS, is the internal entropy generation of the process, and D, = Ta AS, is the exergetic loss associated with the temperature Ta. 

There are n(n -1)/2 independent diffusivities Ay, which are also the coefficients in a positive definite quadratic form, since according to the second law of thermodynamics, the internal entropy of a single process never decreases. In terms of these symmetric diffusivities, the mass flow becomes... 

From the general entropy balance equation dS= dJS+ dxS, we conclude that for an incompressible and isothermal process, we have deS d,S. This relation shows the equality between the dissipated heat flow and internal entropy production and hence the loss of power is q = Eloss. Therefore, Eq. (b) becomes... 

Thus, it is seen that the effect described by Schwarzenbach has precise thermodynamic meaning—the change in the entropy of translation that accompanies metal chelate ring formation. The entropy effects estimated by Schwarzenbach, up to 2.0 log K units, agree quite well with the value obtained with the thermodynamic approximation. Experimentally, one would expect wide deviations from this value (7.9 entropy units per chelate ring) because of the variations in solvation and internal entropies of complexes and ligands that occur in the displacement reaction. 

Irreversible processes of phase transfer and chemical reaction within a closed system, whether homogeneous (a single phase) or heterogeneous (more than one phase), lead to T djS > 0. At equilibrium, T djS = 0. For fixed S and V constraints, dE = —T djS. A reversible process corresponds to zero internal entropy change and a minimum in dE. 

For constant V and S ys, the external work variation, dw is dw < —dE. For a reversible process, the external work is —dE, the maximum external woric available with dV = 0 and dS y. For an irreversible process, the available work is less than the maximum. In terms of the internal entropy production, equation 26 becomes... 

Again comparing with equation 25a, we note that E /, drii = —T djS. The internal entropy production is a result of change in the composition of the system. At equilibrium, the drii vanish, as does the entropy production within the system. For a fixed temperature and pressure system, dG = E /t, drii = -TdiS. 

To describe the state of a reaction in a phase, we need to know the stoichiometric coefficients, j, and the chemical potential, pi, for each species in the reaction. For reaction equilibrium, the quantity AG = E Vi pi = 0 (as is T diS). For a possible, or spontaneous, reaction, AG < 0. For multireaction systems, complete equilibrium corresponds to dG = 0 for the system, that is, the Gibbs energy of the phase is a minimum. The total internal entropy production must vanish for the entire system. Similar consideration apply to multiphase systems. An expression analogous to equation 39 for dE, but for fixed T and p conditions, is ... 

Entropy change for the system is q/T under reversible conditions. Therefore, A y = q/T = 24,050/298.15 = -1-80.67 J K" mol". Maximum work, w is obtained under reversible conditions for the entire system (e.g., chemical reaction and an external circuit for electric work), when the internal entropy change is zero. Then, w = -AG = -(A - TA5sys) = 79.88 kJ mol... 

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