Entropy difference

To = Receiver temperature, °R AS = Entropy difference between the source and receiver, Btu/lb °F. 

Gibson and Giauque calculated the entropy difference between the solid and the supercooled liquid at 70 K. They found that at this temperature, the entropy of the supercooled liquid was 23.4 0.4 JK moI"1 larger then the entropy of the solid. Below 140 K, the solid and supercooled liquid have very nearly the same heat capacity. Thus, this entropy difference should not be much different at 0 Kelvin, and the liquid has an Sm.o of approximately 23.4 J K mol l. This, of course, is a reflection of the lack of order in the supercooled liquid. 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

The contribution of this lack of regularity to the entropy of ice is thus R In 3/2 = 0.805 E. U. The observed entropy discrepancy of ice at low temperatures is 0.87 E. U., obtained by subtracting the entropy difference of ice at very low temperatures and water vapor at standard conditions, for which the value 44.23 E. U. has been calculated from thermal data by Giauque and Ashley,7 from the spectroscopic value 45.101 E. U. for the entropy of water vapor given by Gordon.8 The agreement in the experimental and theoretical entropy values provides strong support of the postulated structure of ice.9... 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

One notes in Table 1.2 a uniform increase in the adsorption energies of the alkanes when the microspore size decreases (compare 12-ring-channel zeohte MOR with 10-ring-channel TON). However, at the temperature of hydroisomerization the equilibrium constant for adsorption is less in the narrow-pore zeohte than in the wide-pore system. This difference is due to the more limited mobility of the hydrocarbon in the narrow-pore material. This can be used to compute Eq. (1.22b) with the result that the overall hydroisomerization rate in the narrow-pore material is lower than that in the wide-pore material. This entropy-difference-dominated effect is reflected in a substantially decreased hydrocarbon concentration in the narrow-pore material. 

Since AG° = AH° — TAS° where AH° and AS° are the enthalpy difference and the entropy difference, respectively, associated with the spin-state transition, Eq. (13) may be expressed as ... 

The driving force for the temperature-dependent spin crossover (SCO) is the entropy difference between the HS and the LS isomers which arises mainly from a shift of the vibrational frequencies when passing from the HS to the LS state [97-99]. This frequency shift has been studied by IR- and Raman-spectroscopy and recently also by NIS [23, 39, 87]. The NIS method is isotope ( Fe) selective and, therefore, its focus is on iron-ligand bond-stretching vibrations which exhibit the most prominent contribution to the frequency shift upon SCO [87]. 

The entropy difference A5tot between the HS and the LS states of an iron(II) SCO complex is the driving force for thermally induced spin transition [97], About one quarter of AStot is due to the multiplicity of the HS state, whereas the remaining three quarters are due to a shift of vibrational frequencies upon SCO. The part that arises from the spin multiplicity can easily be calculated. However, the vibrational contribution AS ib is less readily accessible, either experimentally or theoretically, because the vibrational spectrum of a SCO complex, such as [Fe(phen)2(NCS)2] (with 147 normal modes for the free molecule) is rather complex. Therefore, a reasonably complete assignment of modes can be achieved only by a combination of complementary spectroscopic techniques in conjunction with appropriate calculations. 

Fig. 9.38 (a) Measured upper panel) and calculated lower panel) NIS spectra of the HS open circle, dashed line) and the LS filled triangle, solid line) isomer (1 meV = 8.06 cm ) of [Fe (phen)2(NCS)2]. (b) Calculated contributions A5vib(i) of individual modes i to the vibrational entropy difference bars, left axis) and sum I /ASyib (i) of the contributions of modes 1 to i filled circle, right axis). The 15 modes of an idealized octahedron (six Fe-N stretching modes and nine N-Fe-N bending modes) are marked by the letters s and b, respectively. (Taken from [44])... 

Nucleation can occur either homogeneously or heterogeneously. Homogeneous nucleation occurs when random molecular motion in the molten state results in the alignment of a sufficient number of chain segments to form a stable ordered phase, known as a nucleus. The minimum number of unit cells required to form a stable nucleus decreases as the temperature falls. Thus, the rate of nucleation increases as the temperature of the polymer decreases. The rate of homogeneous nucleation also increases as molecular orientation in the molten polymer increases. This is because the entropy difference between the molten and crystalline states diminishes as molecular alignment in the molten state increases. 

The connection between the multiplicative insensitivity of 12 and thermodynamics is actually rather intuitive classically, we are normally only concerned with entropy differences, not absolute entropy values. Along these lines, if we examine Boltzmann s equation, S = kB In 12, where kB is the Boltzmann constant, we see that a multiplicative uncertainty in the density of states translates to an additive uncertainty in the entropy. From a simulation perspective, this implies that we need not converge to an absolute density of states. Typically, however, one implements a heuristic rule which defines the minimum value of the working density of states to be one. 

The three quantities of interest — the free energy difference, AA, the difference in potential energies, AIJq i, and the entropy difference, AS o >i, are connected through a basic thermodynamic relation... 

More accurately, we can determine free energy and entropy differences, since their absolute value remains unspecified. 

The modification factor plays a central role in a WL simulation and has several effects. First, its presence violates microscopic detailed balance because it continuously alters the state probabilities, and hence acceptance criterion. Only for g = 0 do we obtain a true Markov sampling of our system. Furthermore, we obviously cannot resolve entropy differences which are smaller than g, yet we need the modification factor to be large enough to build up the entropy estimate in a reasonable amount of simulation time. Wang and Landau s resolution of these problems was to impose a schedule on g, in which it starts at a modest value on the order of one and decreases in stages until a value very near to zero (typically in the range 10 5-10 8). In this manner, detailed balance is satisfied asymptotically toward the end of the simulation. 

Computing the absolute free energy A is in general a difficult task. For the same reasons the absolute entropy cannot be computed except in special cases. However, entropy differences can be computed. Indeed, using finite differences, we can approximate... 

Entropy can be thought of as describing the fraction of configuration space accessible to a particular energy. Thus, the entropy difference AS between two phase... 

Note that more sampling is needed as the magnitude of the entropy difference (AS < 0) increases, and, much less so, as the g distribution widens. 

This means that, to obtain the least variance in a multistage calculation, the intermediates should be constructed to have equal entropy difference for all stages. This criterion differs from the often used but unjustified rule of thumb that free energy differences should be equal in all stages [22,42], Simulation tests show that the entropy criterion leads to a great improvement in calculation precision compared to its free energy counterpart [26]. The same optimization criterion holds for calculation of entropy and enthalpy differences [44],... 

Now consider the finite sampling systematic error. As discussed in Sect. 6.4.1, the fractional bias error in free energy is related to both the sample size and entropy difference 5e N exp(-AS/kB). With intermediates defined so that the entropy difference for each substage is the same (i.e., AS/n), the sampling length Ni required to reach a prescribed level of accuracy is the same for all stages, and satisfies... 

That is, the optimal number of stages corresponds to unit entropy difference per... 

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

Afj free energy difference per particle, due to depletion, between the floe and the dispersion Ass configurational entropy difference per particle between the floe and the dispersion A depletion thickness... 

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