** **

** The free energy of formation of a substance can be obtained from the following thermodynamic relationship: **

** The â–³H _{298.15} ^{Î˜} and â–³ _{298.15} Î˜ values â€‹â€‹of each substance at 298.15K can be obtained from the literature. When the temperature T is not 298.15, the Î”H _{T} ^{Î˜} and Î”S _{T} ^{Î˜} can be obtained by the formula (1) and 2) Find, so that the following thermodynamic relationship can be obtained: **

** (1) **

** (2) **

** (3) **

** Therefore, the most useful function for predicting the thermodynamic properties of electrolyte solutions above 298.15 K is the molar heat capacity Cp å‡½æ•° as a function of temperature. When no phase change occurs, the Cp ä¸€ç§ of a substance can usually be assumed to be constant over a temperature range of 100K. However, the Cp Î˜ value of a substance in an aqueous solution generally changes rapidly with temperature, and therefore, the temperature range in which the aqueous solution is assumed to be constant by Cp åº” should be set smaller. **

** If the Cp Î˜ value of a substance varies significantly from 298.15K to T, but is not particularly large, the average value Î”Cp between the two temperatures may be taken. Calculate and get **

** (4) **

** The Î”G _{423} ^{T} value of each substance calculated at the 423K using the above formula **

** In equation (4) used to calculate the free energy of formation of matter at different temperatures, the latter two are related to Î”C _{p} ^{Î˜} . They will cancel each other, such that when approaching the temperature of 298K â–³ G ^{[Theta]} is less sensitive to reaction Î˜C _{p} ^{Î˜.} When dealing with the multiphase equilibrium between pure substances at high temperatures, since the heat capacity of these substances is small, the change with temperature is slow and predictable, and can be accurately estimated if necessary. Therefore, equation (4) can be used to estimate Changes in Î”G ^{Î˜} over a range of degrees above a thousand degrees. However, aqueous solutions, especially those in electrolyte solutions, do not. Their C _{p} ^{Î˜} values â€‹â€‹can be very large negative numbers and change rapidly with temperature. Most of the solutes that can be found have a negative heat capacity at 298 K, rise to a maximum at 320 to 373 K, and then drop rapidly at higher temperatures, especially at low molar concentrations. At 573K, the heat capacity value may be more negative than -1000 J âˆ• (KÂ·mol). Since there is a maximum value on the curve, it is possible to obtain a satisfactory result by estimating the Î”G ^{Î˜} of the solute using the heat capacity value at 298 K, at least until 423 K. However, if the data is available, it is better to use an average between 298K and the temperature considered. **

** If you use equation (4) for calculations, you also need to know the entropy of ions and other substances. The ion entropy at high temperatures can be used to derive the partial molar heat capacity of almost all materials. In the 1950s, some degree of success was achieved in correlating the partial molar ion entropy of a substance with its structural parameters such as the charge, scale, mass, and geometry of ions. Although this association is not strictly based on theory, it can still estimate the entropy of a considerable number of ions at 25 Â° C with sufficient accuracy, causing only a small error in the calculation of free energy. **

** Criss and Cobble noted that the functional relationship between the high temperature entropy values â€‹â€‹obtained by the laboratory is the same as that observed at 25 Â°C, and their corresponding principles are proposed. They suggested that it is not necessary to know the relationship of these functions, but it should be known that the entropy of ions without internal degrees of freedom at temperature T is determined by the ionic charge z, the solvent dielectric constant Îµ _{T} , the mass m and the ionic radius r and others. Dependent on a certain function of the parameter **

** Some of these functions are largely independent of the nature of the ions, but only depend on the choice of standard state, solvent, and temperature. Therefore, the equation of S _{T} ^{å¯ä»¥} can be written **

** The third cross term is included in the above formula because some parameters may affect each other. Assume that equation (5) can be expanded on the reference entropy STÎ˜ at temperature T1, then **

** (5) **

** (6) **

** This is the overall correspondence, indicating that the first two terms of the equation are sufficient to express the entropy data for various ions in water below 200 Â°C. **

** Ions may be divided into four categories: simple cations, simple anions (including OH ^{-} included), oxyanion XO _{n} ^{m} ^{-} _{l} ^{m,} and the female is higher than the oxygen acid XO _{n} (OH) ^{-.} For each selected temperature, 60 Â° C, 80 Â° C and 150 Â° C, the entropy of H ^{+} (aq) at each temperature is fixed, and a set of ion entropy values â€‹â€‹can be obtained. The entropy of H ^{+} (aq) is as follows. **

** Temperature âˆ•Â°C 25 60 100 150 **

** S ^{Î˜} âˆ• (JÂ·mol ^{-1} Â·K ^{-1} ) -20.92 10.46 8.37 27.20 **

** They are selected from the experimentally determined entropy values â€‹â€‹of a particular ion using the first two terms of equation (6) to obtain a linear relationship between S _{298.15} ^{Î˜} and S _{T2} ^{Î˜} . A linear relationship cannot be obtained with other values. Since the values â€‹â€‹specified at 25 Â°C are within the range of "absolute" ion entropy values â€‹â€‹(-8 to -26.4 JÂ·mol ^{-1} Â· K ^{-1} ) of H ^{+} (aq) proposed by many researchers, Criss and Cobble consider them The ion entropy value used should be considered to be expressed on an "absolute" scale. Their scale must be distinguished from the scale used for thermionic calculation. **

** The corresponding principle of use can be described by the following general relationship **

** (7) **

** Where a _{T2} and b _{T2} are constants related to the ion type (simple cation, simple anion, oxyanion, and oxyacid anion) and the temperature considered; S _{T2} ^{Î˜} (abs) represents an 'absolute' scale Partial molar ion entropy. In fact, the absolute scale is not absolute. Consider the selected standard state at 298K, then **

** (8) **

** The constants a and b in the formula (7) are listed in Table 1. Note that these values â€‹â€‹are related to the entropy value expressed by cal âˆ• (molÂ·K), and the entropy value expressed by J/(molÂ·K) in use should be converted into cal (1cal=4.184J), and the obtained S _{T2} ^{Î˜} Convert to J again. **

** Table 1 Entropy constant in equation (7) (calâˆ•(molÂ·K) **

Temperature âˆ• Â°C | Simple cation | Simple anion OH | Oxygenated anion | Oxyacid anion | ||||

a | b | a | b | a | b | a | b | |

25 | 0 | 1.000 | 0 | 1.000 | 0 | 1.000 | 0 | 1.000 |

60 | 3.9 | 0.955 | -5.1 | 0.969 | -14.0 | 1.217 | -13.5 | 1.380 |

100 | 10.3 | 0.876 | -13.0 | 1.000 | -31.0 | 1.476 | -30.3 | 1.894 |

150 | 16.2 | 0.792 | -21.3 | 0.989 | -46.4 | 1.687 |

** If known at 298K of C _{p} ^{Î˜,} S ^{Î˜} and SÎ˜ values of ^{â–³} G Î˜ and the temperature T, the formula (4) heat capacity between the two average temperature T of 298K used can be calculated. Since S _{T} ^{Î˜} can now be calculated using the corresponding principle, the average ion heat capacity between 25 Â° C and 60 Â° C, 100 Â° C and 150 Â° C can be calculated: **

** (9) **

** It can be concluded from equations (7) and (8) **

** (10) **

** The above formula can be written as **

** (11) **

** Where Î± _{T} =Î± _{T} âˆ•ln(Tâˆ•298) **

** And Î² _{T} =-(1.000-b _{T} )âˆ•ln(T/298) **

** It should be emphasized that the S ^{Î˜} values â€‹â€‹used in equations (7) and (8)-(11) are expressed on the â€œabsoluteâ€ scale and are considered from equation (9) before being used for thermodynamic calculations. The S ^{Î˜} value of H ^{+} (aq) at temperature is converted to the conventional scale. The heat capacity constants Î± _{T} and Î² _{T} corresponding to the entropy constants of Table 1 are listed in Table 2. **

** Table 2 Entropy constant in equation (9) [calâˆ•(molÂ·K)] **

Temperature âˆ• Â°C | Simple cation | Simple anion OH | Oxygenated anion | Oxyacid anion | ||||

Î± | Î² | Î± | Î² | Î± | Î² | Î± | Î² | |

60 | 35 | -0.41 | -46 | -0.28 | -127 | 1.96 | -122 | 3.44 |

100 | 46 | -0.55 | -58 | 0.00 | -138 | 2.24 | -135 | 3.97 |

150 | 46 | -0.59 | -61 | -0.03 | -133 | 2.27 | - | - |

** The literature provides partial molar ion heat capacity in the temperature range of 25 ~ 60 Â° C, 25 ~ 100 Â° C, 25 ~ 150 Â° C and 25 ~ 200 Â° C, the entropy constant and heat capacity parameters at 200 Â° C are also included in the table, similar here Table 1 and Table 2. However, due to the aqueous solution of electrolyte at a temperature of 200 â„ƒ range of domestic value of ^{C} _{p} Î˜ rapidly increasing, these data are unlikely to be considered a reliable guide to behavioral reaction of an aqueous system at 200 â„ƒ's. **

** If the half-cell reaction force is used to plot the dominant zone map, even if the ^{Î”G Î˜} value at this temperature as obtained above is used, an incorrect result will still be obtained. The reason for this problem is that the free energy of generation of electrons contained in the equation in the half-cell reaction is not considered. For example, when dealing with the following balance **

** The value of Î”G _{e} ^{å§‘} is assumed to be 0. And E obtained from the ^{Î”G of} the reaction Fe = -0.409V is taken as the standard potential expressed on the hydrogen scale. Because the reaction can be written as **

** Wherein P _{H2} =1 atm. Reduction of Fe ^{2} ^{+} electronics required by H _{2,} which itself is oxidized to 2H ^{+.} Therefore, Farreira recommends that a plot of the dominant area at high temperatures be used for calculations. A full-cell reaction containing hydrogen should be used, which is now widely used. **

** The potential obtained is a value expressed on a hydrogen scale. In determining this point, the formation of H ^{+} ions ^{Î”G Î˜} and Î”S ^{Î˜} are taken as 0 at all temperatures, and the E ^{Î˜} value of the standard hydrogen electrode is considered to be 0 at all temperatures. However, H _{2} has an entropy S _{298.15} ^{Î˜} = 130.684 kJ âˆ• (molÂ·K), and both H ^{+} and H _{2} have a heat capacity. The correct dominant area map can be obtained by using the thermodynamic data of the H ^{+} and H _{2} corrected for the temperature considered. **

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