Entropy classical meaning

Flory ) and Huggins ) derived a now classical mean-field expression for the configurational entropy and energy of mixing, using a lattice model. The solution, containing moles of solvent and n moles of polymer, is described as a lattice of sites, of which are occupied by solvent and N

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The ordering behaviour in ilmSO, ilm90 and ilmlOO appears to be fully reversible, but the data close to Tc can only be fitted with a critical exponent for the order parameter, /3, which is of the order of 0.1, which does not correspond to any classical mean field Landau-type model. Instead, a modified Bragg-Williams model is required, that describes the free energy phenomenologically in terms of a configurational entropy alongside an enthalpy that contains terms up to... 

The box-like cell model of a PE star can be considered as a generalization of a classical mean-field Flory approach, which was first suggested to describe the swelling of a polymer chain in a good solvent [90], The Flory approach estimates the equilibrium dimensions of a macromolecule, as a function of its parameters, by balancing the free energy of intramolecular (repulsive) interactions with the conformational entropy loss of a swollen chain. Within the box-like approximation, the star is characterized by the radius of its corona, R (end-to-end distanee of the arms), or by the average intramolecular concentration of its monomers ... 

Sanchez [181] used a Taylor expansion of the Flory-Huggins equation for the free energy density, and the Cahn-Hilliard theory with a constant coefficient for the gradient terms. He found the same classical mean field exponents for the temperature dependence of interfacial tension and thickness, but he predicted that, for the symmetric case, both the interfacial tension and the thickness are independent of chain length. Sanchez explained this result to be due to the fact that, in his approach, chain connectivity was only implicitly taken into crmsideration through the entropy of mixing. The theories of Nose [249] and Joanny and Leibler [246] take explicitly into account chain connectivity in various approximations. 

However, solubility, depending as it does on the rather small difference between solvation energy and lattice energy (both large quantities which themselves increase as cation size decreases) and on entropy effects, cannot be simply related to cation radius. No consistent trends are apparent in aqueous, or for that matter nonaqueous, solutions but an empirical distinction can often be made between the lighter cerium lanthanides and the heavier yttrium lanthanides. Thus oxalates, double sulfates and double nitrates of the former are rather less soluble and basic nitrates more soluble than those of the latter. The differences are by no means sharp, but classical separation procedures depended on them. 

For a given uptake and temperature T, dSs/dT = Cp where Cp is the differential molar heat capacity of sorbed fluid. This expression can be approximated by Tm ASs/AT = Cp where Tm is the mean temperature corresponding with the interval AT over which ASs is the entropy change, and where Cp refers to the temperature Tm. For classical oscillators Cp should be 24.9 J/mole/deg, and thus it is interesting to compare Cp calculated as above with this value. A5S/AT did not vary much with amount sorbed, so that Cp found for one uptake is typical. Several values of Cp are given below. All are near but a little below the classical oscillator value. 

FIGURE 11 An entropy function in the sense of fluctuation (i.e., large-deviation) theory, describing how fast the mean magnetization of a spin system gets classical with an increasing number of spins. The figure is based on an approximate calculation for the Curie-Weiss model. The temperature is fixed and has been taken here as one third of the critical (Curie) temperature. Above the Curie temperature the respective entropy Sn ,an would only have one minimum, nameiy, at m = 0. 

Entropy owes its existence to the second law, from wliicli it arises in much the same way as internal energy does from the first law. Equation (5.11) is the ultimate source of all equations that relate the entropy to measurable quantities. It does not represent a definition of entropy there is none in the context of classical themiodynamics. What it provides is tlie means for calculating changes in tills property. Its essential nature is summarized by the following axiom ... 

Now, let us consider chains with N links, not on a lattice but in continuous space (see Fig. 2.3). Then, the number of configurations is infinite. This is not surprising since we are dealing with a classical system. This means that the entropy has to be renormalized by subtraction of an infinite constant. 

The second law specifies that heat will not pass spontaneously from a colder to a hotter body without some change in the system. Or, as Planck himself generalized it in his Ph.D. dissertation at the University of Munich in 1879, that the process of heat conduction cannot be completely reversed by any means. Besides forbidding the construction of perpetual-motion machines, the second law defines what Planck s predecessor Rudolf Clausius named entropy because energy dissipates as heat whenever work is done—heat that cannot be collected back into useful organized form—the universe must slowly run down to randomness. This vision of increasing disorder means that the universe is one-way and not reversible the second law is the expression in physical form of what we call time. But the equations of mechanical physics—of what is now called classical physics—... 

In contrast to other textbooks on thermodynamics, we assume that the readers are familiar with the fundamentals of classical thermodynamics, that means the definitions of quantities like pressure, temperature, internal energy, enthalpy, entropy, and the three laws of thermodynamics, which are very well explained in other textbooks. We therefore restricted ourselves to only a brief introduction and devoted more space to the description of the real behavior of the pure compounds and their mixtures. The ideal gas law is mainly used as a reference state for application examples, the real behavior of gases and liquids is calculated with modern g models, equations of state, and group contribution methods. 

Many of the thermodynamic and transport properties of liquid water can be qualitatively understood if one focusses attention on the statistical properties of the hydrogen bond network [9]. As an example, let us observe the temperature dependence of density and entropy. As temperature decreases, the number of intact bonds increases and the coordination number is closer to the ideal value 4. Because of the large free volume available, this means that the temperature decrease is associated with an increase of the local molecular volume. Of course, this effect superimposes on the classical anharmonic effects, which dominate at high temperature, when the number of intact bonds is smaller. The consequence of both effects is a maximum on the temperature dependence of the liquid density. This maximum is actually at 4°C for normal water and at 11°C for heavy water. Such a large isotopic effect can also be understood because the larger mass of the deuterium makes the hydrogen bonds more stable. 

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