Entropy jump

The star that exploded, SK-202-69, was, as theory required, a massive star. When it lived on the main sequence, it had a mass of 19 3 M . At the time it exploded it had a helium core mass of 6 1 M , a radius 3 1 xlO12 cm, a luminosity 3 to 6 xlO38 erg s 1, and an effective temperature 15,000 to 18,000 K. Further consideration of the stellar models (Woosley 1987 Nomoto, this volume) suggests that the iron core mass at the time of collapse was 1.45 0.15 M0. Adding 0.15 M0 for matter between the iron core and the entropy jump... 

At temperatures above the critical point, the kink in the chemical potentials will disappear. The kink in the chemical potentials causes the entropy, which is -S = 9/u./9r to change abruptly. Ideally, the entropy jumps in the way like a step function. This behavior, in turn makes the heat capacity to show a Dirac delta function peak at the transition point. The situation is shown schematically in Fig. 6.12. 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Observe how in each of these four events, H is zero until, at some critical Ac (which is different for different cases), H abruptly jumps to some higher value and thereafter proceeds relatively smoothly to its final maximum value i max = log2(8) = 3 at A = 7/8. In statistical physics, such abrupt, discontinuous changes in entropy are representative of first-order phase transitions. Interestingly, an examination of a large number of such transition events reveals that there is a small percentage of smooth transitions, which are associated with a second-order phase transition [li90a]. 

Notice from the figure that the effect of temperature on entropy is due almost entirely to phase changes. The slope of the curve is small in regions where only one phase is present. In contrast, there is a large jump in entropy when the solid melts and an even larger one when the liquid vaporizes. This behavior is typical of all substances melting and vaporization are accompanied by relatively large increases in entropy. 

FIGURE 7.12 The entropy of a solid increases as its temperature is raised. The entropy increases sharply when the solid melts to form the more disordered liquid and then gradually increases again up to the boiling point. A second, larger jump in entropy occurs when the liquid vaporizes. 

Since free energy of the conformation F = TS, where S is the entropy of the conformation, it follows, that at given external parameters P and T neither free energy of conformation F nor it s the first derivative upon temperature S do not change in the point c = c, testifying only the hump but their derivatives upon the concentration test the jump. 

Boltzmann s W-function is not monotonic after we perform a velocity inversion of every particle—that is, if we perform time inversion. In contrast, our -function is always monotonic as long as the system is isolated. When a velocity inversion is performed, the 7f-function jumps discontinuously due to the flow of entropy from outside. After this, the 7f-function continues its monotonic decrease [10]. Our -function breaks time symmetry, because At itself breaks time symmetry. 

Accdg to Ref 66, p 135> the usual way of treating shocks is to idealize them to jump discontinuities, in this way taking into account the> effect of the irreversible process caused by friction and heat conditions. It is assumed that the flow involving such a discontinuous process is completely determin ed by the three lawstof conservation of mass, momentum and energy and the condition that the entropy does not decrease in the discontinuous process. Outside of the transition zone the flow is determined by the differential eqs 2.1.1, 2.2.2 2.2.3 listed on p 132 of Ref 66... 

Vq is the frequency of the small oscillation, and AG and AS are, respectively, the difference in Gibbs free energy and entropy of the adatom at the saddle point and the equilibrium adsorption site. Ed is the activation energy of surface diffusion, or the barrier height of the atomic jumps. 

The process of contact adsorption can be viewed in the following way (Fig. 6.90) First, a hole of area of at least 1U —where r, is the radius of the bare ion—is swept free of water molecules in order to make room for the ion. At the same time, the ion strips itself of part of its solvent sheath and then jumps into the hole. During this process, the involved particles—electrode, ion, water molecules—break old attachments and make new ones (change of enthalpy, AH) and also exchange freedoms and restrictions for new freedoms (change of entropy, AS). 

In 1907. Einstein showed that at extremely low temperatures, the atoms of a solid don t have sufficient energy to jump to the first quantized energy level, which is a relatively large jump. The solid, therefore, may be exposed to small increments of heat without any increase in thermal motion. This lowers the ability of the solid to absorb heat, which means that its entropy is also lower. For practically all materials, this quantum effect only occurs at extremely low temperature. A dramatic exception is diamond, which because of this quantum effect resists the absorption of energy even at room temperature (Table 9.2). Diamond is special for many reasons, including its status as a room-temperature "quantum solid"... 

If the barrier to the jump of an interstitial between two sites of differing energy is deformed as indicated in Fig. 8.21, the information given in Fig. 8.25 may be used to derive expressions for the various jump rates that appear in the coefficients of Eq. 8.180. Neglecting small differences in the entropies of activation in the presence and absence of stress, and expanding Boltzmann factors of the form exp[-Ui jf(kT) to first order so that exp[—C/i j7(A T)] = 1 + Ui j / (kT),... 

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