Standard Molar Entropies and the Third Law

Both entropy and enthalpy are state functions, but the nature of their values differs in a fundamental way. Recall that we cannot determine absolute enthalpies because we have no easily measurable starting point, no baseline value for the enthalpy of a substance. Therefore, we measure only enthalpy changes. 

To obtain a value for 5 at a given temperature, we first cool a crystalline sample of the substance as close to 0 K as possible. Then we heat it in small increments, dividing q by T to get the increase in 5 for each increment, and add up all the entropy increases to the temperature of interest, usually 298 K. The entropy of a substance at a given temperature is therefore an absolute value that is equal to the entropy increase obtained when the substance is heated from 0 K to that temperature. 

As with other thermodynamic variables, we usually compare entropy values for substances in their standard states at the temperature of interest 1 atm for gases, I M for solutions, and the pure substance in its most stable form for solids or liciuids. Because entropy is an extensive property, that is, one that depends on the amount of substance, we are interested in the standard molar entropy (S°) in units of J/moEK (or J mol -K ). The S° values at 298 K for many elements, compounds, and ions appear, with other thermodynamic variables, in Appendix B. 

Predicting Relative S° Values of a System Based on an understanding of systems at the molecular level and the effects of heat absorbed, we can often predict how the entropy of a substance is affected by temperature, physical state, dissolution, and atomic or molecular complexity. (All S° values in the following discussion have units of J/mohK and, unless stated otherwise, refer to the system at 298 K.) 

Temperature changes. For a given substance, S° increases as the temperature rises. Consider these typical values for copper metal  

---

The Second Law of Thermodynamics Predicting Spontaneous Change Lrnitations of the First Law The Sign of AH and Spontaneous Change Freedom of Motion and Dispersal of Energy Entropy and the Number of Microstates Entropy and the Second Law Standard Molar Entropies and the Third Law... 

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... 

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... 

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-... 

The standard molar entropy of hydration of an ion is A 5" = 5 - 5, , the difference between its standard molar entropy in the aqueous solution (Table 2.8) and the standard molar entropy of the isolated ion in the ideal gas phase (Table 2.3). The latter, S°, are calculated from the third law of thermodynamics and spectroscopic data without invoking any extra-thermodynamic assumptions. The former, do involve the assumptions leading to A5 (H+, aq)=-22.2 2J K" mol" for the hydrogen ion (Section 2.3.1.2). With 5°(H% g)=108.9J K" mol", the standard molar entropy of hydration of the hydrogen ion is then A,5"(H ) = -22.2 2-108.9 =-131.1 2 J-K" mor. The standard molar entropies of hydration of ions are shown in Table 4.1, derived from A 5,°° = S -S,° but also obtainable from the conventional values by use of the absolute value of the hydrogen ion. They are related to the effect that ions have on the structure of water according to various approaches. This aspect is fully dealt with in Section 5.1.1.7. 

The Third-Law entropy, which is commonly called simply the entropy , at any temperature, S(T), is based on setting S(0) = 0. The entropy of a substance depends on the pressure we therefore select a standard pressure (1 bar) and report the standard molar entropy, S, the molar entropy of a substance in its standard state at the temperature of interest. Some values at 298.15 K (the conventional temperature for reporting data) are given in Table 2.2. 

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. 

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. 

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. 

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)... 

Because entropy is a state function and because the third law allows us to obtain a value for the standard molar entropy of any substance, we can derive a useful equation for the entropy change in a reaction. Figure 10.5 shows how the entropy change in a reaction may be determined by a method that is reminiscent of the way we used heats of formation and Hess s law in Chapter 9. 

A note on the third law of thermodynamics There is a point of lowest possible temperature, close to zero degrees kelvin, Tq 0 K, and for every solid body that forms a perfectly regular crystal the entropy at Tq equals zero. Water can crystallize so at 0 K the entropy of water is zero (or nearly so). It is perhaps a little stretch to assume that Josiah Gibbs would make a perfect crystal, even at 0 K, but we will not be too pedantic about this point right now. So what is the entropy of water at, say, T = 309.8 K It equals the difference between the entropy of water at 309.8 K and the entropy of water at 0 K, which is zero. We can calculate this or we can take the listed value it is determined for room temperature, T = 298.15 K, and is listed as the absolute entropy or standard molar entropy. For water at room temperature, S298 = 69.9 J K mol True, T = 298.2 K is not the same as T = 309.8 K, but we should not worry about this detail now the error is likely to be less than 1%. The entropy of water will change significantly between 273 and 274 K, when ice (solid) melts into... 

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. 

As in the case of energy or enthalpy, we are usually interested in differences in entropy rather than absolute values. However, in the case of entropy, it is possible to assign absolute values. This is a consequence of the third law of thermodynamics, which states that the entropy of perfect crystals of all pure elements and compounds is zero at the absolute zero of temperature (OK), Consequently, the absolute entropy of a substance at any temperature T is given by the change in the entropy of the substance in moving from OK to T. The absolute entropies of many substances (generally at 25°C and 1 atm - indicated by 5 for the molar absolute entropy under standard conditions) are... 

Since G. N. Lewis avoided entropy in his early works, it seems appropriate to conclude with a couple of passages from the influential text that he co-authored with Merle Randall in 1923. Lewis and Randall took considerable care to explain entropy, devoting separate chapters to its numerical calculation (both thermodynamic and statistical mechanical), the third law understood entropically, and atomic entropies (standard molar entropy values), areas to which Lewis made important experimental contributions. And they wrote ... 

The standard molar entropy of methyl chloride, CH3CI, is plotted at various temperatures from 0 to 298.15 K, with the phases noted. The vertical segment between the solid and liquid phases corresponds to AfusS the other vertical segment, to A apS. By the third law of thermodynamics, an entropy of zero is expected atO K. Experimental methods cannot be carried to that temperature, however, so an extrapolation is required. 

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. 

We cannot calculate AjG from the standard molar Gibbs energies themselves because these quantities are not known. One practical approach is to calculate the standard reaction enthalpy from standard enthalpies of formation (Section 1.11), the standard reaction entropy from Third-Law entropies (Section 2.5), and then to combine the two quantities by using... 

---