Third-law molar entropies

Suppose we wish to evaluate the entropy of an amount n of a pure substance at a certain temperature T and a certain pressure. The same substance, in a perfectly-ordered crystal at zero kelvins and the same pressure, has an entropy of zero. The entropy at the temperature 

Consider a reversible isobaric heating process of a pure substance while it exists in a single phase. The definition of heat capacity as q/ 6T (Eq. 3.1.7) allows us to substitute Cp 6.T for q, where Cp is the heat capacity of the phase at constant pressure. 

If the substance in the state of interest is a hquid or gas, or a crystal of a different form than the perfectly-ordered crystal present at zero kelvins, the heating process will include one or more equilibrium phase transitions under conditions where two phases are in equilibrium at the same temperature and pressure (Sec. 2.2.2). For example, a reversible heating process at a pressure above the triple point that transforms the crystal at 0 K to a gas may involve transitions from one crystal form to another, and also melting and vaporization transitions. 

Each such reversible phase transition requires positive heat trs- Because the pressure is constant, the heat is equal to the enthalpy change (Eq. 5.3.8). The ratio trs/n is called the molar heat or molar enthalpy of the transition, AtrsT/ (see Sec. 8.3.1). Because the phase transition is reversible, the entropy change during the transition is given by Atrs5 = trs/ Ttrs where Ttrs is the transition temperature. 

With these considerations, we can write the following expression for the entropy change of the entire heating process  

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The neglect of these two effects results in a practical entropy scale, or conventional entropy scale, on which the crystal that is assigned an entropy of zero has randomly-mixed isotopes and randomly-oriented nuclear spins, but is pure and ordered in other respects. This is the scale that is used for published values of absolute third-law molar entropies. The shift of the zero away from a completely-pure and perfectly-ordered crystal introduces no inaccuracies into the calculated value of AS for any process occurring above 1 K, because the shift is the same in the initial and final states. 

If water or some other compound with a simple molecular structure has been studied, it is possible to combine the entropy of vaporization, A5 = AHyl T, with the third-law calorimetric entropy of the liquid to obtain a thermodynamic value for the entropy of the vapor. The statistical mechanical value of Sg can be calculated using the known molar mass and the spectroscopic parameters for the rotation and vibration of the gas-phase molecule. A comparison of Sg (thermodynamic) with Sg (spectroscopic) provides a test of the validity of the third law of thermodynamics. The case of H2O is particularly interesting, since ice has a nonzero residual entropy at 0 K due to frozen-in disorder in the proton positions. ... 

Entropies calculated using Equ on S 21 (with phase transitions) are called third-law (or) entropies because these values are not measured relative to some reference state. Third-law entropies per mole of material measured at the standard pressure of 1 bar are referred to as standard molar entropies, denoted by S°. Table 8.2 lists standard molar entropies for a variety of inorganic and organic substances— values for many other substances are given in Appendix 2. The units of S° are J mol K , in contrast to A ff values, which are generally given in kJ mol". Entropies of elements and compounds are all positive (that is, 5° > 0) for all T > 0 K. By contrast, the standard enthalpy of formation (AHf) for elements in their stable form is arbitrarily set equal to zero, and for compounds it may be positive or negative. 

Because of the Nemst heat theorem and the third law, standard themrodynamic tables usually do not report entropies of fomiation of compounds instead they report the molar entropy 50 7 for each element and... 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

The third law of thermodynamics establishes a starting point for entropies. At 0 K, any pure perfect crystal is completely constrained and has S = 0 J / K. At any higher temperature, the substance has a positive entropy that depends on the conditions. The molar entropies of many pure substances have been measured at standard thermodynamic conditions, P ° = 1 bar. The same thermodynamic tables that list standard enthalpies of formation usually also list standard molar entropies, designated S °, fbr T — 298 K. Table 14-2 lists representative values of S to give you an idea of the magnitudes of absolute entropies. Appendix D contains a more extensive list. 

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. 

Figure 3.6 shows schematically the molar entropy of a pure substance as a function of temperature. If a structural transformation occurs in the solid state, an additional increase in the molar entropy comes from the heat of the transformations. As shown in the figure, the molar entropy of a pure substance increases with increasing temperature. In chemical handbooks we see the tabulated numerical values of the molar entropy calculated for a number of pure substances in the standard state at temperature 298 K and pressure 101.3 kPa. A few of them will be listed as the standard molar entropy, s , in Table 5.1. Note that the molar entropy thus calculated based on the third law of thermodynamics is occasionally called absolute entropy. 

The pertinent thermodynamic data needed to carry out the calculation of 5g(thermo-dynamic) for water vapor are as follows 5 (1 bar, 298.15 K) = 66.69 J mol which is the calorimetric value obtained by assuming (erroneously) that the third law is valid for ice Cp(l), which is essentially independent of temperature over the range 298-355 K, has the average value 75.36 J mol You may neglect the effect of pressure on the value of the molar entropy of the liquid, an approximation that has very little effect on the calculated thermodynamic value of Sg(p , T ,) for H20(g) at the vapor pressure... 

As mentioned in Sections 1.1 and 2.9, the third law of thermodynamics makes it possible to obtain the standard Gibbs energy of formation of species in aqueous solution from measurements of the heat capacity of the crystalline reactant down to about 10 K, its solubility in water and heat of solution, the heat of combustion, and the enthalpy of solution. According to the third law, the standard molar entropy of a pure crystalline substance at zero Kelvin is equal to zero. Therefore, the standard molar entropy of the crystalline substance at temperature T is given by... 

A eiei is the number of atoms of element i in the crystalline substance and (j m (298.I5 is the standard molar entropy of element i in its thermodynamic reference state. This equation makes it possible to calculate Af5 ° for a species when Sm ° has been determined by the third law method. Then Af G° for the species in dilute aqueous solution can be calculated using equation 15.3-2. Measurements of pATs, pA gS, and enthalpies of dissociation make it possible to calculate Af G° and Af//° for the other species of a reactant that are significant in the pH range of interest (usually pH 5 to 9). When this can be done, the species properties of solutes in aqueous solution are obtained with respect to the elements in their reference states, just like other species in the NBS Tables (3). 

The heat capacity of thiazole was determined by adiabatic calorimetry from 5 to 340°K by Goursot and Westrum (295,296). A glass-type transition occurs between 145 and 175 K. Melting occurs at 239.53°K (-33.62°C) with an enthalpy increment of 2292 cal mole" and an entropy increment of 9.57 cal mole" -"K". Table 1-44 summarizes the variations as a function of temperature of the most important thermodynamic properties of thiazole molar heat capacity Cp, standard entropy S°, and Gibbs function -(G°-The variation of Cp for crystalline thiazole between 145 and 175°K reveals a marked inflection that has been attributed to a gain in molecular freedom within the crystal lattice. The heat capacity of the liquid phase varies nearly linearly with temperature to 310°K, at which temperature it rises more rapidly. This thermal behavior, which is not uncommon for nitrogen compounds, has been attributed to weak intermolecular association. The remarkable agreement of the third-law ideal-gas entropy at... 

The third law of thermodynamics states that the entropy of any pure substance in equilibrium approaches zero at the absolute zero of temperature. Consequently, the entropy of every pure substance has a fixed value at each temperature and pressure, which can be calculated by starting with the low-temperature values and adding the results of all phase transitions that occur at intervening temperatures. This leads to tabulations of standard molar entropy S° at 298.15 K and 1 atm pressure, which can be used to calculate entropy changes for chemical reactions in which the reactants and products are in these standard states. 

This table lists standard enthalpies of formation AH°, standard third-law entropies S°, standard free energies of formation AG°, and molar heat capacities at constant pressure, Cp, for a variety of substances, all at 25 C (298.15 K) and 1 atm. The table proceeds from the left side to the right side of the periodic table. Binary compounds are listed under the element that occurs to the left in the periodic table, except that binary oxides and hydrides are listed with the other element. Thus, KCl is listed with potassium and its compounds, but CIO2 is listed with chlorine and its compounds. 

As the temperature of a substance increases, the particles vibrate more vigorously, so the entropy increases (Figure 15-14). Further heat input causes either increased temperature (still higher entropy) or phase transitions (melting, sublimation, or boiling) that also result in higher entropy. The entropy of a substance at any condition is its absolute entropy, also called standard molar entropy. Consider the absolute entropies at 298 K listed in Table 15-5. At 298 K, any substance is more disordered than if it were in a perfect crystalline state at absolute zero, so tabulated values for compounds and elements are always positive. Notice especially that g of an element, unlike its A// , is not equal to zero. The reference state for absolute entropy is specified by the Third Law of Ther-... 

Understand the meaning of entropy (5) in terms of the number of microstates over which a system s energy is dispersed describe how the second law provides the criterion for spontaneity, how the third law allows us to find absolute values of standard molar entropies (5°), and how conditions and properties of substances influence 5° ( 20.1) (SP 20.1) (EPs 20.4-20.7, 20.10-20.23)... 

Since any impure crystal has at least the entropy of mixing at the absolute zero, its entropy cannot be zero such a substance does not follow the third law of thermodynamics. Some substances that are chemically pure do not fulfill the requirement that the crystal be perfectly ordered at the absolute zero of temperature. Carbon monoxide, CO, and nitric oxide, NO, are classic examples. In the crystals of CO and NO, some molecules are oriented differently than others. In a perfect crystal of CO, all the molecules should be lined up with the oxygen pointing north and the carbon pointing south, for example. In the actual crystal, the two ends of the molecule are oriented randomly it is as if two kinds of carbon monoxide were mixed, half and half. The molar entropy of mixing would be... 

This equation, which can be derived from Eq. 4.2, is strictly valid if a small decrease of the entropy change with a temperature increase is neglected. As can be seen from Eq. 6.17, equality of the molar enthalpies determined by the second- and third-law methods = E ) can be reached if Analysis of the data listed in Table 6.1 yields the following conclusions. 

Because of the conventimi S = 0 at T = 0 for ideally crystallized solids (third law of thermodynamics), it is possible to determine not only differences but also absolute values of entropies and therefore absolute molar entropies as well. This determination is not only possible for substances in the state stable at 0 K but also for... 

SECTION 19.4 The third law allows us to assign entropy values for substances at different temperatures. Under standard conditions the entropy of a mole of a substance is called its standard molar entropy, denoted S°. From tabulated values of S°, we can calculate the entropy change for any process under standard conditions. For an isothermal process, the entropy change in the surroundings is equal to —AH/T. 

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