Sohds entropy

Tables 2,3, and 4 outline many of the physical and thermodynamic properties ofpara- and normal hydrogen in the sohd, hquid, and gaseous states, respectively. Extensive tabulations of all the thermodynamic and transport properties hsted in these tables from the triple point to 3000 K and at 0.01—100 MPa (1—14,500 psi) are available (5,39). Additional properties, including accommodation coefficients, thermal diffusivity, virial coefficients, index of refraction, Joule-Thorns on coefficients, Prandti numbers, vapor pressures, infrared absorption, and heat transfer and thermal transpiration parameters are also available (5,40). Thermodynamic properties for hydrogen at 300—20,000 K and 10 Pa to 10.4 MPa (lO " -103 atm) (41) and transport properties at 1,000—30,000 K and 0.1—3.0 MPa (1—30 atm) (42) have been compiled. Enthalpy—entropy tabulations for hydrogen over the range 3—100,000 K and 0.001—101.3 MPa (0.01—1000 atm) have been made (43). Many physical properties for the other isotopes of hydrogen (deuterium and tritium) have also been compiled (44).

The entropy value of gaseous HCl is a sum of contributions from the various transitions summarized in Table 4. Independent calculations based on the spectroscopic data of H Cl and H Cl separately, show the entropy of HCl at 298 K to be 186.686 and 187.372 J/(mol K) (44.619 and 44.783 cal/(mol K), respectively. The low temperature (rhombic) phase is ferroelectric (6). SoHd hydrogen chloride consists of hydrogen-bonded molecular crystals consisting of zigzag chains having an angle of 93.5° (6). Proton nmr studies at low temperatures have also shown the existence of a dimer (HC1)2 (7). 

The thermal conductivity of soHd iodine between 24.4 and 42.9°C has been found to remain practically constant at 0.004581 J/(cm-s-K) (33). Using the heat capacity data, the standard entropy of soHd iodine at 25°C has been evaluated as 116.81 J/ (mol-K), and that of the gaseous iodine at 25°C as 62.25 J/(mol-K), which compares satisfactorily with the 61.81 value calculated by statistical mechanics (34,35). 

It should be noted that the methodology for a and b results in a ASf s associated with the phase change from a sohd at 0 K to the liquid at Tmit No entropy changes resulting from solid transitions are taken into account, and ASfus for a substance that undergoes such a transition will be overestimated by this technique. 

To calculate the entropy changes, it is necessary to consider a series of reversible steps leading from liquid water at —10°C to sohd ice at —10°C. One such series might be (1) Heat supercooled water at —10°C very slowly (reversibly) to 0°C, (2) convert the water at 0°C very slowly (reversibly) to ice at 0°C, and (3) cool the ice very slowly (reversibly) from 0°C to —10°C. As each of these steps is reversible, the entropy changes can be calculated by the methods discussed previously. As S is a thermodynamic property, the sum of these entropy changes is equal to AS for the process indicated by Equation (6.97). The necessary calculations are summarized in Table 6.2, in which T2 represents 0°C and Ti represents 10°C. 

To calculate the entropy of a substance at a temperamre at which it is no longer a sohd, it is necessary to add the entropy of transformation to a hquid or gas and the subsequent entropies of warming. The same procedure would apply to a solid that exists in different crystalline forms as the temperature is increased. The procedure can be illustrated by some sample calculations. 

J. S. Chickos, W. E. Acree, Jr. and J. F. Liebman, J. Phys. Chem. Ref. Data, 28, 1535 (1999) for estimating sohd-hquid phase change enthalpies and entropies. 

A typical profile of a curve of Cp/ T plotted versus T for a substance remaining sohd and undergoing no phase transitions (changes in crystalline form) or phase changes (to a liquid, for example) is shown in Figure 16.1 and the area under this curve will be equal to the standard entropy, SrfH at 298 K. (Frame 2, section 2.3 and Frame 10, section 10.4). 

In many (and probably most) cases, phase changes may occur between T = 0 K and 298 K, as for example in the case of a gas at 298 K and having therefore undergone at least one or more sohd — liquid and liquid —gas phase transitions between OK and 298 K. In such a case the absolute standard entropy of the gas will be given (adapting equation (15.22),... 

Mezzasalma, S.A. (1999). Hysteresis and isotherm equations in gas—sohd adsorption from maximum entropy production. /. Phys. Chem. B, 103, 7542—50. 

The entropy change experienced by the water is negative, which is not surprising because it turns completely from liquid into sohd at the same temperature (section 5.2). 

This therefore corresponds to a highly negative entropy change for an all-solid reaction, and this in conjnnction with the uncertainties in the formation of sohd solutions in the solid phases, snggests that the reaction studied may not have been that given in (X.2), or there were other experimental errors in the study of this reaction. These data of [1971KUS/1MO] are therefore not considered further in this review. 

The small value of the entropy change of this all-sohd reaction gives confidence in the experimental results. 

In conclusion, we should note that the first statement of the third law of thermodynamics was made by Nernst in 1906, the Nernst heat theorem, which states that in any chemical reaction involving only pure, crystalline sohds the change in entropy is zero at 0 K. This form is less restrictive than the statement of Planck. 

For a sohd to liquid phase transition (melting) the entropy always increases (AS > 0) and the reaction is always endothermic (AH > 0). 

Fig. 9. Differential molar entropy of the Xe adlayer on Ih(lll) as a function of Xe coverage. Data points (circles) and theoretical curves assuming a completely mobile 2D ideal gas (sohd line), a completely mobile Volmer gas (dashed line) and a localized adlayer (dash-dotted line) [88K1]. The coverage is given in substrate...

A typical entropy-elastic material is cross-linked natural rubber, ds-poly(l-methyl-1-butenylene) or cts-l,4-polyisoprene, as summarized in Fig. 5.166 (see also Fig. 1.15). Its extensibility is 500 to 1,000%, in contrast to the 1% of typical energy-elastic sohds. Natural rubber has a molar mass of perhaps 350,000 Da (about 5,000 isoprene monomers or 20,000 carbon backbone bonds) and is then vulcanized to have about 1% cross-links (see Fig. 3.50). A rubber with a Young s modulus of 10 Pa (depending on cross-link density) must be compared to its bulk modulus (= 1/p,... 

Consider a pme, perfect crystalUne soHd. At absolute zero, the individual atoms or molecules in the lattice would be perfectly ordered in position. Because none of them would have thermal motion, there is only one possible microstate. As a result. Equation 19.5 becomes S = fc In W = fc In 1 = 0. As the temperature is increased from absolute zero, the atoms or molecules in the crystal gcun energy in the form of vibrational motion about their lattice positions. This meeuis that the degrees of freedom and the entropy both increase. What happens to the entropy, however, as we continue to heat the crystal We consider this important question in the next section. 

Nonadsorbing polymers between two plates also show a decrease in entropy when the two plates move toward one another. As the available space is reduced, the random excursions that the polymer chain was taking begin to see restrictions and the entropy decreases. However, another factor comes into play. In the nonadsorbing system, the polymer does not fike the sohd siuface, and when, in addition, its entropy drops, it has no reason to stay in the gap, and the reservoir becomes a more favored place. There is another somce of entropy that has to be considered. Under ideal solution theory in dilute systems, the chemical potential of the polymer is written approximately as + kT n(f, where the first... 

Many metals form sohd carbonates, such as CaC03. When carbonates are heated, gaseous CO2 can be driven off, leaving behind a metal oxide. What is the sign of the entropy change in the system for this type of chemical reaction Explain your answer. 

These conclusions make sense, given that gases invariably have greater entropy than liquids and sohds. Eor reactions involving only liqnids and sohds, predicting the sign of AS° is more difhcnlt, but in many such cases an increase in the total number of molecules and/or ions is accompanied by an increase in entropy. 

The separation of the solids is usually expressed as mass recovery or total efficiency (in filtration this is also known as retention ) as dealt with in depth in chapter 3, whilst the separation of the liquid is usually characterized by the moisture content of the cake or concentration of solids in the underflow. Separation efficiencies of the solids and the liquid are best considered separately because different applications place different emphasis on the two in thickening, for example, the emphasis is on the high efficiency for the hquid (i.e. high sohds content in cakes or underflows), whilst in recovery or clarification, high efficiency for the solids is required. If the emphasis placed on the two efficiencies is equal then they can be combined in one criterion, the entropy index, discussed in Part II, chapter 18. 

In order to be able to evaluate the relative reduction in entropy during or after the separation process, we have to be able to calculate the entropy of the suspension as a function of the sohds concentration. The necessary relationship may be derived in analogy with a molecular model for an ideal solution as used in chemical thermodynamics, summarized in the following. A very small sohd particle is treated simply as if it were a large molecule. 

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