Standard molar entropy values

The entropy of a pure crystalline solid at absolute zero is zero. As energy is added, the randomness of the molecular motion increases. Measurements of energy absorbed and calculations are used to determine the absolute entropy or standard molar entropy, and values are then recorded in tables. These molar values are reported as kJ/(mol K). Entropy change, which can also be measured, is defined as the difference between the entropy of the products and the reactants. Therefore, an increase in entropy is represented by a positive value for AS, and a decrease in entropy is represented by a negative value for AS. 

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. 

Table 3.6 Comparison of predictive capacities of various equations in estimating standard molar entropy T = 298.15 K, P = 1 bar). Column I = simple summation of standard molar entropies of constituent oxides. Column II = equation 3.86. Column III = equation 3.86 with procedure of Holland (1989). Column IV = equation 3.85. Values are in J/(mole X K). Lower part of table exchange reactions adopted with equation 3.85 (from Helgeson et al., 1978) and Sj finite differences for structural oxides (Holland, 1989). 

Entropy, which has the symbol 5, is a thermodynamic function that is a measure of the disorder of a system. Entropy, like enthalpy, is a state function. State functions are those quantities whose changed values are determined by their initial and final values. The quantity of entropy of a system depends on the temperature and pressure of the system. The units of entropy are commonly J K" mole". If 5 has a ° (5°), then it is referred to as standard molar entropy and represents the entropy at 298K and 1 atm of pressure for solutions, it would be at a concentration of 1 molar. The larger the value of the entropy, the greater the disorder of the system. 

Entropy changes were estimated with Eq. 4 assuming that V, is equal to the total stationary phase volume existing in the column. Therefore, these values reflect more properly the relative differences in entreaties of transfer instead of the standard molar entropies that would require to use the volume of the active stationary phase. 

The conventional thermodynamic standard state values of the Gibbs energy of formation and standard enthalpy of formation of elements in their standard states are A(G — 0 and ArH = 0. Conventional values of the standard molar Gibbs energy of formation and standard molar enthalpy of formation of the hydrated proton are ArC (H +, aq) = 0 and Ar// (H +, aq) = 0. In addition, the standard molar entropy of the hydrated proton is taken as zero 5 (H+, aq) = 0. This convention produces negative standard entropies for some ions. 

The calculation of the values for the standard molar entropies of hydration of ions requires some groundwork using the data presented in the following sub-section. 

In this section the standard molar entropies of a small selection of cations and anions are tabulated and the manner of their derivation discussed. The values themselves are required in the calculation of entropies of hydration of ions, discussed in Section 2.7.2. 

Work done with electrochemical cells, with particular reference to the temperature dependence of their potentials, has demonstrated that an accurate value for S (H h, aq) is — 20.9 J K mol-1. Table 2.15 gives the absolute molar entropies for the ions under consideration. The values of the absolute standard molar entropies of the ions in Table 2.15 are derived by using the data from Tables 2.13 and 2.14 in equations (2.51) and (2.57). 

If we want to calculate the entropy of a liquid, a gas, or a solid phase other than the most stable phase at T =0, we have to add in the entropy of all phase transitions between T = 0 and the temperature of interest (Fig. 7.11). Those entropies of transition are calculated from Eq. 5 or 6. For instance, if we wanted the entropy of water at 25°C, we would measure the heat capacity of ice from T = 0 (or as close to it as we can get), up to T = 273.15 K, determine the entropy of fusion at that temperature from the enthalpy of fusion, then measure the heat capacity of liquid water from T = 273.15 K up to T = 298.15 K. Table 7.3 gives selected values of the standard molar entropy, 5m°, the molar entropy of the pure substance at 1 bar. Note that all the values in the table refer to 298 K. They are all positive, which is consistent with all substances being more disordered at 298 K than at T = 0. 

We can understand some of the differences in standard molar entropies in terms of differences in structure. For example, let s compare the molar entropy of diamond, 2.4 J-K 1-mol, with the much higher value for lead, 64.8 JKr -mol-1. The low entropy of diamond is what we should expect for a solid that has rigid bonds at room temperature, its atoms are not able to jiggle around as much as the atoms of lead, which have less directional bonds, can. Fead also has much larger atoms... 

Values of S° for some common substances at 25°C are listed in Table 17.1, and additional values are given in Appendix B. Note that the units of S°are joules (not kilojoules) per kelvin mole [J/(K mol)] Standard molar entropies are often called absolute entropies because they are measured with respect to an absolute reference point—the entropy of the perfectly ordered crystalline substance at 0 K [S° = 0 J/(K mol) atT = OK]. 

Standard molar entropies make it possible to compare the entropies of different substances under the same conditions of temperature and pressure. It s apparent from Table 17.1, for example, that the entropies of gaseous substances tend to be larger than those of liquids, which, in turn, tend to be larger than those of solids. Table 17.1 also shows that S° values increase with increasing molecular complexity. Compare, for example, CH3OH, which has S° = 127 J/(K mol), to CH3CH2OH, which has S° = 161 J/(K mol). 

Once we have values for standard molar entropies, it s easy to calculate the entropy change for a chemical reaction. The standard entropy of reaction, AS°, can be obtained simply by subtracting the standard molar entropies of all the reactants from the standard molar entropies of all the products ... 

To determine the sign of AStotai = ASsys + ASsurr/ we need to calculate the values of ASsys and ASsurr. The entropy change in the system equals the standard entropy of reaction and can be calculated using the standard molar entropies in Table 17.1. To obtain ASsurr = —AH°/T, first calculate AH° for the reaction from standard enthalpies of formation (Section 8.10). 

We have now answered the fundamental question posed at the beginning of this chapter What determines the value of the equilibrium constant—that is, what properties of nature determine the direction and extent of a particular chemical reaction The answer is that the value of the equilibrium constant is determined by the standard free-energy change, A G°, for the reaction, which depends, in turn, on the standard heats of formation and the standard molar entropies of the reactants and products. 

The behavior of hA in real micellar systems is more complex as seen in Fig. 2.12. Similar data have been obtained for several other amphiphiles148,149). The deviations in hA from the standard value at infinite dilution appear clearly below the CMC, but at these concentrations one has a compensating change in the partial molar entropy. This effect might be due to a repulsive interaction between the hydrophobically hydrated alkyl chains leading to a breakdown of the water structure with a concomitant increase in entropy. 

Table 16-1 compiles some data for S°, the molar entropy, and AGp the free energy of formation from the elements. All values in Table 16-1 are presented at 25°C and at standard states. Notice that the units of entropy and free energy are stated per mole, mol-1. This means that the moles used to balance a chemical reaction are included by the multiplication of the coefficient (mol in balanced equation) and the value from the table so that unit mol cancels. This is also the way in which we handled calculations involving AH values. 

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. 

In electrochemistry we make it a rule that the standard chemical potential ju. of hydrogen ions is set zero as the level of reference for the chemical potentials of all other hydrated ions. The standard chemical potentials of various hydrated ions tabulated in electrochemical handbooks are thus relative to the standard chemical potential of hydrogen ions at unit activity in aqueous solutions. Table 9.3 shows the numerical values of the standard chemical potential, the standard partial molar enthalpy h°, and the standard partial molar entropy. 5 ,° for a few of hydrated ions. 

Absolute standard molar entropy values, >9 (298 K) can be provided. They are tabulated at 25 °C and for P° = 1 bar pressure. It should be noted from equation (16.4) that the values are absolute entropy values (in contrast to values of AfH° and AfG° values which are quoted as differences (i.e. relative values) in thermochemical tables (Frame 11, section 11.2)). 

As shown in the previous section a common feature of all systems in the liquid state is their molar entropy of evaporation at similar particle densities at pressures with an order of magnitude of one bar. Taking this into account a reference temperature, Tr, will be selected for systems at a standard pressure, p° = 105 Pa = 1 bar, having the same molar entropy as for the pressure unit, p = 1 Pa at T = 2.98058 K. As can easily be verified, the same value of molar entropy and consequently the same degree of disorder results at p if a one hundred-fold value of the above T-value is used in Eq. (6-14). This value denoted as Tw = 298.058 K = Tr is used as the temperature reference value for the following model for diffusion coefficients. The coincidence of Tw with the standard temperature T = 298.15 K is pure chance. 

Like enthalpy, entropy can t be measured directly. It is possible to measure changes in the enthalpy of a system, which allows you to better understand the entropy of a system under specific conditions. The entropy values for one mole of a substance are known as standard molar entropies, 5°. The entropy change in a chemical reaction can be calculated using the equation ... 

The second indicator of kosmotropicity is the standard molar entropy of hydration. For all ions it is highly negative the higher its absolute value, the more water is ordered upon ionic hydration, and the higher the electrolyte kosmotropicity [2,21]. 

Table 8.1 contains values of the standard entropies of a number of important chemical compounds. These are the molar entropies of the real substances, corrected in the case of gases for gas imperfections, at a pressure of 1 atm. and temperature of 25 °C. 

Standard molar entropies S° are tabulated for a number of elements and compounds in Appendix D. If Cp is measured in J moP, then the entropy S° will have the same units. For dissolved ions, the arbitrary convention S° H (aq)) = 0 is applied (just as for the standard enthalpy of formation of discussed in Section 12.3). For this reason, some S° values are negative for aqueous ions—an impossibility for substances. 

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. 

Lists of values of experimental standard molar entropies at 25°C for many substances standard = pure substance at 1 bar)... 

The value of A5° is obtained from the standard molar entropy values as follows ... 

In order to avoid the possibility of misunderstanding, it may be stated here that the partial molar free energy and partial molar entropy of the solute in the standard state are entirely different from the values in the infinitely dilute solution (see Exercise 24). 

Show that the partial molar entropy SX of a solute at high dilution is related to the value in the standard state /SJ by the expression — 5J = — g In m, where is the molality of the dilute solution. Hence prove that the partial molar entropy of a solute approaches an infinitely large value at infinite dilution. 

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