Conditional entropy fraction

Next, the probability function Ptj for the maximum and minimum values of 1(0, R) is discussed mathematically. The self-entropy H(C) in Eq. (2.38) is decided only by the fraction of each component in the feed, and the value does not change through the mixing process. Then, the maximum and minimum values of the mutual information entropy are determined by the value of the conditional entropy H(C/R). Since the range of the variable j is fixed as l[Pg.70]

Figure 9.6 (a) Molar entropy of mixing of ideal polymer solutions for r = 10, 100 and 1000 plotted as a function of the mole fraction of polymer compared with the entropy of mixing of two atoms of similar size, r = 1. (b) Activity of the two components for the same conditions. 

These considerations allow us to link the time required for humification (always directed to an increase in entropy) to the type of chemical transformations in humic system, which best suit this demand The system of NOM and HS should unavoidably evolve toward molecular compositions with the maximum number of isomers. Given that the overwhelming part of humic matter is being formed under oxic conditions, these structures are represented by low-molecular-weight aromatic and alicyclic acids. This suggests that under the same environmental constraints, the humification of NOM should lead to the formation of structures with an increased content of aromatic structures (or more precisely, the amount of DBE) and with a decrease in size similar to what was revealed by the results of data analysis on size-fractionated samples of humic materials shown in Figures 13.14A-D. 

The determination of the mole fraction of the fcth component in its standard state for the entropy follows the same argument that was used for the chemical potential. By the use of Equation (8.77) and the condition that [7, P, x] must equal Se[T, P] for the standard state, we find that... 

For fractioned PMCS-6 samples, isotropization heat and entropy are much higher and the tempera-ture range is much narrower, which may be induced by the solvent applied and conditions of sample precipitation dining fractionation. 

This equation describes the pressure difference because of the mass fraction difference when there is no temperature difference. This is called the osmotic pressure. This effect is reversible because AT - 0,, /2 = 0. and at stationary state J = 0. Therefore, Eq. (7.244) yields Jq = 0, and the rate of entropy production is zero. The stationary state under these conditions represents an equilibrium state. Equation (7.263) does not contain heats of transport, which is a characteristic quantity for describing nonequilibrium phenomena. 

On the other hand the nature of the retractive forces in the yield and post-yield regions has been the subject of much controversy. Bull (1945), Woods (1946a,b), Astbury (1947), Elod and Zahn (1949a), and Breuer (1962) have concluded from the effects of temperature on the retractive forces that entropy contributes very little to retractive forces at strains up to 30 %. Meyer and Haselbach (1949) and Meyer et al. (1952), however, consider that the fibers must reach an equilibrium condition before measurements are made and conclude that the forces are entirely entropic. There can be no doubt that after stress relaxation at high temperatures the residual force is largely entropic (Feughelman and Mitchell, 1959), but this force is only a fraction of the initial force. 

Liquid water at 325 K and 8000 kPa flows into a boiler at tlie rate of 10 kg s and is vaporized, producing saturated vapor at 8000 kPa. What is tlie maximum fraction of the heat added to the water in the boiler that can be converted into work in a process whose product is water at initial conditions, if Ftr = 300 K What happens to the rest of the heat What is the rate of entropy change in the surroundings as a result of the work-producing process In the system Total ... 

Strictly speaking, the equation K =S is an extension of Boltzmann s theory, in so far as we have ascribed a definite value to the entropy constant. According to Boltzmann, the probabihty contains an undetermined factor, which cannot be evaluated without the introduction of new hypotheses. Boltzmann and Clausius suppose that the entropy may assume any positive or negative value, and that the change in entropy alone can be determined by experiment. Of late, however, Planck, in connection with Nemst s heat theorem, has stated the hypothesis that the entropy has always a finite positive value, which is characteristic of the chemical behaviour of the substance. The probabihty must then always be greater than unity, since its logarithm is a positive quantity. The thermodynamical probabihty is therefore proportional to, but not identical with, the mathematical probabihty, which is always a proper fraction. The definition of the quantity w on p. 15 satisfies these conditions, but so far it has not been shown that this definition is sufficient under all circumstances to enable us to calculate the entropy. 

The model leads to a variation of structure as the relative fractions of each block within the copolymer molecules changes. This can be seen heuristically as follows. If the in vidual blocks have different relaxed radii of gyration (set by the maximrim entropy constraint imposed on each block species on its own), the existence of a bond that fuses the moieties in the copolymer imposes the condition that assemblies of the copolymers cannot form without perturbation of the original configmations (Fig. 4.23(a),(b)). 

Thg necessary conditions for miscibility are that G < 0 a that d G/d iji < 0, where mole fraction of the i component. In equation (1), the combinatorial entropy of mixing depends on the number of molecules present according to... 

The change in entropy during collapse controls the fraction of the dripped protons with respect to that of the heavy nuclei and this influences the overall neutrino spectrum received on earth as the spectrum of neutrinos generated by electron capture on protons are different from captures on heavy nuclei. The received neutrino spectrum depends not only upon the initial conditions from which the collapse started, but also on the details of the electron capture properties of the stellar matter. Properties of nuclei at finite temperatures and density during this phase of the collapse, where shell and pairing corrections are relevant were computed in [94] and utilized to evolve self-consistently with the electron capture physics and the consequent changes in nuclear and thermodynamic variables. 

---