Entropy of Solids

The entropy is then equal to the entropy of an electromagnetic resonator of frequency v capable of vibrating in all three directions of space. According to Planck s theory this leads to the equation  

For- substances composed of several vibrating systems of different frequency, the summation is to be extended over all the values of v. 

This equation leads us to some important and strikingly simple results, much as we were led to draw conclusions as to the physical behaviour of perfect gases from their entropy equation. In analogy with the term perfect gas, the author proposes the term perfect solid for substances which conform to Einstein s assumptions. 

The entropy of a perfect solid is therefore independent of its volume. From this it follows that if solid solutions can be formed at all, no diffusion can take place in them. 

If we compare the actual behaviour of solid bodies with those of the hypothetical perfect solid, we find that the coefiicient of expansion is very small for all sofids, and appears to approach zero as we diminish the temperature (according to Thiesen, Gruneisen,t and Lindeman J). The variation of compressibifity with temperature is also very small, and appears hkewise to approach zero as the temperature is diminished.  

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We shall assume that Cp = 0 at 7 = 0 K. We wish to obtain the absolute entropy of solid lead at 298 K. Each entry in Table 1-2 leads to a value of CpjT. The... 

Draw the curve of Cp vs. T and Cp/T vs. T from the following heat capacity data for solid chlorine and determine the absolute entropy of solid chlorine at 70.0 K... 

The entropy of solid Agl is a little larger than that of AgCl, namely, 27.1 e.u., as compared with 23.0 e.u. Using (64) we find for the partial molal entropy of Agl in its saturated solution the value... 

For sodium hydroxide the value of AS° may bo obtained by subtracting the entropy of solid NaOH (given in Table 44) from the values given in Table 45 for the Na+ and OH- ions, which are known independently see Sec. 89. 

As mentioned in Sec. 90 we may think of this AS0 as being the difference between the molar entropy of solid 1 j2C03 and the partial molal entropies of one (C03) ion and two Lif ions, each in its hypothetical standard aqueous solution. We propose now to compare the AS of the reaction... 

From Table 44 it will be seen that the molar entropy of solid Li2C03 amounts to 21.60 cal/degree. In Sec. 104 it was pointed out that, if the AS0 for the process (192) is added to the entropy of solid Li2C03, the sum will be equal to the partial molal entropies of the COJ" ion and two Li+ ions. For this sum then we obtain... 

These effects are shown in Figure 17.4, where the entropy of ammonia, NH3> is plotted versus temperature. Note that the entropy of solid ammonia at 0 K is zero. This reflects the fact that molecules are completely ordered in the solid state at this temperature there is no randomness whatsoever. More generally, the third law of thermodynamics tells us that a completely ordered pure crystalline solid has an entropy of zero at 0 K. 

In a procedure similar to that described earlier for N , the entropy at T = 49.44 K point (a) can be calculated from the heat capacities and the enthalpies of transition, by following two different paths. Sm 49 44 was found to be 34.06 J-K-1 mol-1 following the solid III— solid II path and Sm.49.44 = 34.02 JK 1 mol l following the (solid V— solid IV—>solid II) path. The two results agree well within experimental error, which requires that the entropy of solid III and solid V be the same at 0 K, an occurrence that is unlikely unless Sm 0 = 0 for both forms. 

MJ Pikal, DJW Grant. A theoretical treatment of changes in energy and entropy of solids caused by additives or impurities in solid solution. Int J Pharm 39 243-253, 1987. 

So knowing the states of the reactants and products in a chemical reaction should allow us to predict whether the reaction is accompanied by an increase or a decrease in entropy. Consider, for example, the reaction 2Na(s) + Cl fg) -> 2NaCl(s). We know that the entropies of solids are very much smaller than the entropies of gases and, because this reaction results in a decrease in the number of moles of gaseous molecules (from 1 to 0), the entropy will decrease. Similarly, we would predict an increase in entropy for the reaction CaCOjfs) -> CaO(s) + CO g) because there is an increase in the number of moles of gaseous molecules (from 0 to if However, the entropy change for the reaction CaSiOjfs) CaO(s) + SiO s) is difficult to predict because the reactants and products are solids and are likely to have very similar entropy values. All we can say is that the entropy change is likely to be small. 

C(s) < H20(s) < H20(1) < HzO(g), C atoms are more tightly held in the solid state than HzO molecules. The entropy of solid C is thus lower than the entropy of solid H20. The entropy of H20 increases on going from solid to liquid to gas, because the molecules are increasingly disordered. 

The thermal conductivity of solid iodine between 24.4 and 42.9°C has been found to remain practically constant at 0.004581 J/(crnsK) (33). Using the heat capacity data, the standard entropy of solid 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). 

Liquid helium presents an interesting case leading to further understanding of the third law. When liquid 4He, the abundant isotope of helium, is cooled at pressures of < 25 bar, a second-order transition takes place at approximately 2 K to form liquid Hell. On further cooling Hell remains liquid to the lowest observed temperature at 10 5 K. Hell does become solid at pressures greater than about 25 bar. The slope of the equilibrium line between liquid and solid helium apparently becomes zero at temperatures below approximately 1 K. Thus, dP/dT becomes zero for these temperatures and therefore AS, the difference between the molar entropies of liquid Hell and solid helium, is zero because AV remains finite. We may assume that liquid Hell remains liquid as 0 K is approached at pressures below 25 bar. Then, if the value of the entropy function for sol 4 helium becomes zero at 0 K, so must the value for liquid Hell. Liquid 3He apparently does not have the second-order transition, but like 4He it appears to remain liquid as the temperature is lowered at pressures of less than approximately 30 bar. The slope of the equilibrium line between solid and liquid 3He appears to become zero as the temperature approaches 0 K. If, then, the slope is zero at 0 K, the value of the entropy function of liquid 3He is zero at 0 K if we assume that the entropy of solid 3He is zero at 0 K. Helium is the only known substance that apparently remains liquid as absolute zero is approached under appropriate pressures. Here we have evidence that the third law is applicable to liquid helium and is not restricted to crystalline phases. 

It seems to be sufficiently justified because the change, A S°, in the entropy of solid-state formation of the ordered phases is relatively small. Hence, at low temperatures the term TA S° in the equation... 

Even though the Third Law of Thermodynamics, as expressed in equation (16.1) may not hold precisely, the entropy of solid materials close to T = 0 K becomes extremely small and the use of clever extrapolation techniques, (Frame 3, section 3.2) ensures that reasonably accurate values of S°T can usually be obtained. 

As a good approximation it is assumed, that the adsorbed species are vibrating in resonance with the lattice phonon vibrations of the solid stationary phase. The phonon frequency can be evaluated from phonon spectra, from the standard entropy of solid metals, from the Debye temperatures or from the Lindemann equation [9]. 

If the heat capacities and entropies of solids are simply the weighted sums of those of their elemental constituents, then the entropy change should be zero for symmetrical reactions such as. 

Drozin, N.N., Application of the Berthelot principle in the calculation of the standard entropies of solid inorganic compounds. Russ. J. Phys. Chem., 35, 879-881 (1961). 

Caloric Data. Enthalpy and entropy of solid ammonia are given in [32]. Enthalpy and entropy may be calculated by the equations in [31]. Further data are given in [37], [50] and [32]-[37], An enthalpy logp diagram can be found in [37]. 

Examination of table 8.1 shows that the standard entropies of solids are all of roughly the same magnitude so that for reactions between them 298(9 /9 ) is of the order of thousands of calories per mole, while heats of reaction between solids are usually of the order of tens of thousands of calories. It is thus usually justifiable to employ (9.16) as a rough approximation. It should be noted however, that (9.16) cannot be used for solid reactions such as allotropic changes where the heat of reaction is small. 

Determine the entropy of solid chlorine at its melting point, 172.12 K. The entropy contribution at temperatures below 14 K should be obtained by assuming the Debye equation to be applicable. 

Problem The standard heat of solution of 1 g. atom of potassium in dilute acid is found to be — 60.15 kcal. at 25 C. The standard oxidation potential of this metal is 2.924 volt at the same temperature the standard entropy of solid potassium is 15.2 e.u. g. atom and of gaseous hydrogen it is 31.21 e.u. mole. Calculate the standard entropy of the potassium ion in solution at 25 C. 

The solubility of silver chloride in pure water is 1.314 X 10 molal, and the mean ionic activity coefficient is then 0.9985 [Neuman, J. Am. Chem. Soc., 54, 2195 (1932)]. The heat of solution of the salt is 15,740 cal. mole". Taking the entropy of solid silver chloride as 22.97 e.u. mole ", and using the results of the preceding exercise, calculate the standard free energy and heat of formation and the entropy of the Cl" ion at 25 C. 

The value of (Se02, cr, 298.15 K) calculated from the selected values of the entropies of solid and gaseous Se02 is ... 

S02(g) and using auxiliary data different from those used by the review. The results were therefore re-evaluated using the selected properties of Se2(g), the heat capacity and entropy of In2Se(g) calculated by the review from the molecular parameters obtained by Erkoc, Katircioglu, and Yilmaz [2001ERK/KAT], and the heat capacity, transition enthalpies, and entropy of solid 102803 given in Section V.8.4.1.5, yielding A //° (298.15 K) = 549 kJ-mol and A 5° (298.15 K) = 332 J K mor from the second law and A //° (298.15 K) = 596 kJ-mol from the third law. 

The standard molar entropy of solid NiB407 was estimated by the linear dependence of entropies on the radius of the compound s cation constituent. The extrapolation to r(cation) —> 0 led to the borate contribution of the standard molar entropy depending on the cationic charge. 

Here U, is an internal energy of the skeleton, S, -entropy of solid phase, <50 - external stream of heat, SQ >0 - uncompensated heat, JA " -elementary work of the internal surface strength. 

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