Entropy scale

The third law of thermodynamics states that the entropy of any crystalline, perfectly ordered substance must approach zero as the temperature approaches 0 K, and at T = 0 K entropy is exactly zero. Based on this, it is possible to establish a quantitative, absolute entropy scale for any substance as... 

The entropy change in a process is given by eqn. (14) and it follows that entropies can be assigned to individual substances. As entropy is a state function, its value will depend on the state of the substance and, with the aid of eqn. (14), the entropy difference between any two states can be calculated. The third law enables a zero to be fixed for the entropy scale and there are tables [5—9] which give the entropies of many substances in their standard states at the reference temperature of 298 K. As long as there is no change of phase, the entropy at any other temperature can be calculated using... 

Figure 9. Free energy, enthalpy, and entropy profiles of the azurin (I/II)-Fe(CN)64 /3 system. Reference state 1M reactants, 0.1M potassium phosphate, pH 7.0, I = 0.22. Free energy given for 298°K. For this temperature the entropy scale is equivalent to the energy scales.

Larger Homonuclear Clusters. As one might expect, the scarcity of data on homonuclear polymers increases with cluster size. To date we have only been able to locate published entropy data on four pentamers 85, 805, and Te5 from Mills compilation (7) and C5 from the JANAF tables (6). Furthermore, only data on the hexa-mers and heptamers of 8 and 8e exist (7). All of these data have been estimated by various means. These entropy data are plotted In Fig. 5 against the appropriate form of the logarithmic term In Eq. 3. Please note that the dimensionless entropy scale Is shifted for each polymer type to facilitate comparison. Also plotted In Fig.5 are the least squares fits of unit slope from the corresponding dimensional model tests for the MX ] species (Table 1). These are shown as the solid lines In Fig. 5 and represent the results for MX4 on the pentamer plot, MX5 on the hexamer plot, and MXg on the heptamer plot. 

The Third Law of Thermodynamics establishes the zero of the entropy scale. 

As with electrode potentials themselves, confusion can occur regarding alternative entropy scales for inorganic ions and redox couples. In addition to values of AS obtained from nonisothermal cells, it is common to encounter reaction entropies for redox couples that actually refer to the entropy of reaction, AS, of a complete cell containing a hydrogen electrode. These latter quantities will, therefore, differ from AS by an amount equal to the reaction entropy of the hydrogen electrode half-cell it follows that AS° SJ, + 85 n J deg mol. A scale of ionic and redox reaction entropies is also established by arbitrarily assigning the entropy of the hydrogen ion, SJ,., a value... 

Diffusion-entropy scaling relation the Rosenfeld relation... 

T. Goel, C.N. Patra, T. Mukheqee, and C. Chakravarty, Excess entropy scaling of transport properties of Lennard-Jones chains. J. Chem. Phys., 129 (2008), 164904-164909. 

The simple model in the previous paragraph demonstrates that as the confinement radius a is reduced, the confinement entropy increases dramatically. Furthermore, the confinement entropy scales linearly with polymer size, as defined by the number of Kuhn segments N, resulting in considerable... 

In his original works, Rosenfeld considered hard spheres, soft spheres, Lennard-Jones system, and one-component plasma [52,53]. Thereafter, the excess entropy scaling was applied to many different systems, including core-softened liquids [17,18,51,54,55], liquid metals [56,57], binary mixtures [58,59], ionic liquids [60,61], network-forming liquids [54,60], water [62], chain fluids [63], and bounded potentials [51,64,65]. 

The neglect of these two effects results in a practical entropy scale, or conventional entropy scale, on which the crystal that is assigned an entropy of zero has randomly-mixed isotopes and randomly-oriented nuclear spins, but is pure and ordered in other respects. This is the scale that is used for published values of absolute third-law molar entropies. The shift of the zero away from a completely-pure and perfectly-ordered crystal introduces no inaccuracies into the calculated value of AS for any process occurring above 1 K, because the shift is the same in the initial and final states. 

Statistical mechanical theory applied to spectroscopic measurements provides an accurate means of evaluating the molar entropy of a pure ideal gas from experimental molecular properties. This is often the preferred method of evaluating Sm for a gas. The zero of entropy is the same as the practical entropy scale—that is, isotope mixing and nuclear spin interactions are ignored. Intermolecular interactions are also ignored, which is why the results apply only to an ideal gas. 

This entropy scales with the size of the system (the number N) but is not proportional with the number of defects ny. 

In general, polymeric mixtures are far less miscible than mixtures of small molecule materials. This effect results from the fact that the driving force for mixing is usually entropy, not interaction energy. In other words, miscible materials usually form a solution not because their interaction with each other is more favorable than their self-interaction, but because of an increase in entropy and hence free energy associated with increasing the amount of volume available to each component. This increase in entropy scales with the number of particles (or moles) being mixed. 

P , and A(U), is the van der Waals surface area of the residue i side chain in isolation, i.e., the residue alone with the rest of the protein removed. The parameter a. is taken from the entropy scale of Pickett and Sternberg [27]. It is an estimate of the side-chain folding entropy of residue i or the entropic burden incurred when residue i is taken from an exposed state at the surface of the protein and buried within the protein. Pickett and Sternberg [27] estimated the a. values for all 20 amino acids by analyzing the torsional preferences of surface exposed side-chains relative to buried side chains. This analysis was performed over a set of 50 non-homologous protein crystal structures which included HIV-1 protease. 

In technical calculations it is an advantage to assign an absolute standard entropy 5 298 to elements and compounds similarly to the introduction of the standard enthalpy iJfgg in section 3.9. Determination of a zero point for this absolute entropy scale can be made on the basis of the third law of thermodynamics. 

The mathematical structure of thermodynamics is based on two laws. The third law, also called Nernst s heat theorem, properly belongs to statistical theory. Its main use in thermodynamics is in establishing an entropy scale. In 1931 Fowler raised the postulate regarding the existence of thermal equilibrium to the status of the zeroth law of thermodynamics. We need discuss neither the zeroth or the third laws here. 

The entropy that is lost upon confinement of a polymer chain to nanoscale dimensions depends on both chain size and confinement size. deGennes argued that this confinement entropy scales with the length of the chain and inversely with the confinement size [85] ... 

---