Temperature molar entropy

The partial molar entropy of adsorption AI2 may be determined from q j or qsi through Eq. XVII-118, and hence is obtainable either from calorimetric heats plus an adsorption isotherm or from adsorption isotherms at more than one temperature. The integral entropy of adsorption can be obtained from isotherm data at more than one temperature, through Eqs. XVII-110 and XVII-119, in which case complete isotherms are needed. Alternatively, AS2 can be obtained from the calorimetric plus a single complete adsorption isotherm, using Eq. XVII-115. This last approach has been recommended by Jura and Hill [121] as giving more accurate integral entropy values (see also Ref. 124). 

Molar entropy of ammonia as a function of temperature. Note the large increases in entropy upon fusion (melting) and vaporization. 

The entropy of a substance, unlike its enthalpy, can be evaluated directly. The details of how this is done are beyond the level of this text, but Figure 17.4 shows the results for one substance, ammonia. From such a plot you can read off the standard molar entropy at 1 atm pressure and any given temperature, most often 25°C. This quantity is given the symbol S° and has the units of joules per mole per kelvin (J/mol-K). From Figure 17.4, it appears that... 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

TABLE 7.2 Standard Molar Entropy of Water at Various Temperatures... 

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. 

Oxygen gas has many applications, from welders torches to respirators. The gas is sold commercially in pressurized steel tanks. One such tank contains O2 at p = 6.50 bar and T — 298 K. Using standard thermodynamic data, compute the molar entropy of the gas in the tank at 6.50 bar and the change in entropy of a 0.155-mol sample of gas withdrawn from the tank at 1.10 bar and constant temperature. 

P is pressure T is temperature V is the molar volume S is molar entropy... 

Here /g,hq and y ,ss are the activity coefficients of component B in the liquid and solid solutions at infinite dilution with pure solid and liquid taken as reference states. A fus A" is the standard molar entropy of fusion of component A at its fusion temperature Tfus A and AfusGg is the standard molar Gibbs energy of fusion of component B with the same crystal structure as component A at the melting temperature of component A. 

Here A vac. S and A vacH are the molar entropy and enthalpy of formation of the defects. Using a pure metal like aluminium as an example, the fractional number of defects, heat capacity and enthalpy due to defect formation close to the fusion temperature are 5-10-4, 0.3 J K-1 mol-1 and 30 J mol-1 [30],... 

For an ideal gas we will show later that the molar entropy is a function of the independent variables, molar volume Vm and temperature T. The total differential dSra is given by the equation... 

Whether obtained from an actual experimentally feasible process or from a thought process, As i Gg, which is obtained from Eq. (2.9) by re-arrangement, pertains to the solvation of the solute and expresses the totality of the solute-solvent interactions. It is a thermodynamic function of state, and so are its derivatives with respect to the temperature (the standard molar entropy of solvation) or pressure. This means that it is immaterial how the process is carried out, and only the initial state (the ideal gaseous solute B and the pure liquid solvent) and the final state (the dilute solution of B in the liquid) must be specified. 

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. 

Besides equilibriumconstants, additional thermodynamic data were included, if available, although little emphasis was put on their completeness. The data for primary master species comprise the standard molar thermodynamic properties of formation from the elements (AfG standard molar Gibbs energy of formation AfH°m standard molar enthalpy of formation ApSm- standard molar entropy of formation), the standard molar entropy (5m), the standard molar isobaric heat capacity (Cp.m), the coefficients Afa, Afb, and Afc for the temperature-dependent molar isobaric heat capacity equation... 

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). 

An analytical method for applying Polanyi s theory at temperatures near the critical temperature of the adsorbate is described. The procedure involves the Cohen-Kisarov equation for the characteristic curve as well as extrapolated values from the physical properties of the liquid. This method was adequate for adsorption on various molecular sieves. The range of temperature, where this method is valid, is discussed. The Dubinin-Rad/ush-kevich equation was a limiting case of the Cohen-Kisarov s equation. From the value of the integral molar entropy of adsorption, the adsorbed phase appears to have less freedom than the compressed phase of same density. 

Adsorbed Phase Entropy. Since Equations 7 and 8 can accurately describe the relationship between q, T, and p, we may use them to calculate the integral molar entropy of the adsorbed phase. At temperatures significantly lower than critical for the adsorbate, the entropy of the adsorbed phase is usually compared with the entropy of the liquid at same temperature in order to compare the freedom of each phase. Because our experimental domain was higher, we shall make this comparison with the gaseous phase compressed to the same density p as determined by Equation 8. 

Let Sg° be the standard entropy of the gas at the standard pressure (p0 = 1 atm) and at the experimental temperature. The partial molar entropy Ss and the integral entropy Ss of the adsorbed phase are given by ... 

The entropy of vaporization, ASvap, is the change in molar entropy when a substance changes from a liquid to a vapor. The heat required to vaporize the liquid at constant pressure is the enthalpy of vaporization (AHvap, Section 6.12). It then follows from Eq. 1, by setting qiev = AHvap, that the entropy of vaporization at the boiling temperature is... 

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... 

Standard molar entropies increase as the complexity of a substance increases. The standard molar entropies of gases are higher than those of comparable solids and liquids at the same temperature. 

We can see from Table 7.2 that at 0°C the molar entropy of liquid water is 22.0 J-K -mol 1 higher than that of ice at the same temperature. This difference makes sense, because the molecules in liquid water are more disordered than in ice. It follows that when water freezes at 0°C, its entropy decreases by 22.0 J-K -mol-1. Entropy changes do not vary much with temperature so just below 0°C, we can expect almost the same decrease. Yet we know from everyday experience that water freezes spontaneously below 0°C. Clearly, the surroundings must be playing a deciding role if we can show that their entropy increases by more than 22.0 J-K -mol 1 when water freezes, then the total entropy change will be positive and freezing will be spontaneous. 

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