Calorimetric entropy

Robinson G. R. Jr. and Haas I L. Jr. (1983). Heat capacity, relative enthalpy and calorimetric entropy of silicate minerals An empirical method of prediction. Amer. Mineral, 68 541-553. 

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. ... 

The supposed differences between chemical constants determined from vapour pressures (7/) and from chemical equilibrium data from the Nernst Heat Theorem Jn found by Eucken, Karwat, and Fried, are illusory, and due to inaccuracies in the experiments made in Eucken s laboratory, The supposed constant differences between the calorimetric and optical values (from moments of inertia found from band spectra. or otherwise) listed, with a theoretical explanation, by Eucken, Karwat, and Fried, are also illusory for the same reason. The relation between the spectroscopic and calorimetric entropies, as found from experimental results meriting confidence, is fully discussed in 70.11 and 26-29.IV. Apart from the incorrect data used by Eucken, Karwat, and Fried, 11 their method of extrapolation of specific heats of gases to 0°K. is much cruder than they suspected. 12... 

A calorimetric entropy of 60.17 cal K mol" at 226.48 K was obtained by Koehler and Giauque ( ) for the ideal gas. If the lattice doesn t discriminate between F and 0 atoms then at 0 K a residual entropy of 2.75 cal k" mol exists and the third law entropy would be 62.92 cal k" raol". This value may be compared with 62.64 obtained statistically. The difference 0.3 cal K" mol" is within the range found for other molecules and explained on the basis of some discrimination in the lattice, see Koehler and Giauque, (9) for references. 

The calorimetric entropy is calculated from the thermodynamic equation... 

Adding (13 75) and (13 77) we obtain a quantity which is usually denoted called the calorimetric entropy of the gas at the... 

Let it be supposed that the substance in question passes into a crystalline form (and not a glass) on cooling. Its calorimetric entropy can be calculated at any temperature T by use of equation (13 78) provided that heat-capacity measurements are available at close... 

Using the following data, which are from Landolt-Bomstein, compare the calorimetric entropy of mercury vapour at 343.9 K with a value calculated from the Sackur-Tetrode equation, assuming that the vapour is entirely monatomic. 

Dimethylacetylene.—A discrepancy of 1.7 JK mol between the calculated and calorimetric entropy for dimethylacetylene was revealed long ago. It cannot arise from hindered internal rotation since this would increase the discrepancy. The vibrational assignment is well established but there is a shift of the lowest fundamental from 213 cm for the liquid to 194 cm" for the vapour state, and use of the latter wavenumber gives excellent agreement between calculated and measured values. 

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). 

As seen in previous sections, the standard entropy AS of a chemical reaction can be detemiined from the equilibrium constant K and its temperature derivative, or equivalently from the temperature derivative of the standard emf of a reversible electrochemical cell. As in the previous case, calorimetric measurements on the separate reactants and products, plus the usual extrapolation, will... 

Since the saturated solutions of AgT and AgCl are both very dilute, it is of interest to examine their partial molal entropies, to see whether we can make a comparison between the values of the unitary terms. As mentioned above, the heat of precipitation of silver iodide was found by calorimetric measurement to be 1.16 electron-volts per ion pair, or 26,710 cal/mole. Dividing this by the temperature, we find for the entropy of solution of the crystal in the saturated solution the value... 

Conventional Partial Molal Entropy of (H30)+ and (OH)-. Let us now consider the partial molal entropy for the (1I30)+ ion and the (OH)- ion. If we wish to add an (HsO)+ ion to water, this may be done in two steps we first add an H2O molecule to the liquid, and then add a proton to this molecule. The entropy of liquid water at 25°C is 16.75 cal/deg/mole. This value may be obtained (1) from the low temperature calorimetric data of Giauque and Stout,1 combined with the zero point entropy predicted by Pauling, or (2) from the spectroscopic entropy of steam loss the entropy of vaporization. 2 Values obtained by the two methods agree within 0.01 cal/deg. 

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... 

Polymerization thermodynamics has been reviewed by Allen and Patrick,323 lvin,JM [vin and Busfield,325 Sawada326 and Busfield/27 In most radical polymerizations, the propagation steps are facile (kp typically > 102 M 1 s l -Section 4.5.2) and highly exothermic. Heats of polymerization (A//,) for addition polymerizations may be measured by analyzing the equilibrium between monomer and polymer or from calorimetric data using standard thermochemical techniques. Data for polymerization of some common monomers are collected in Table 4.10. Entropy of polymerization ( SP) data are more scarce. The scatter in experimental numbers for AHp obtained by different methods appears quite large and direct comparisons are often complicated by effects of the physical state of the monomei-and polymers (i.e whether for solid, liquid or solution, degree of crystallinity of the polymer). 

Comparison and agreement with the calorimetric value verifies the assumption that So = 0. For example, we showed earlier that the entropy of ideal N2 gas at the normal boiling point as calculated by the Third Law procedure had a value of 152.8 0.4 J-K mol. The statistical calculation gives a value of 152.37 J K -mol-1, which is in agreement within experimental error. For PH3, the Third Law and statistical values at p 101.33 kPa and T— 185.41 K are 194.1 0.4 J K, mol 1 and 194.10 J-K 1-mol 1 respectively, an agreement that is fortuitously close. Similar comparisons have been made for a large number of compounds and agreement between the calorimetric (Third Law) and statistical value is obtained, all of which is verification of the Third Law. For example, Table 4.1 shows these comparisons for a number of substances. 

Use the Third Law to calculate the standard entropy, S°nV of quinoline (g) p — 0.101325 MPa) at T= 298,15 K. (You may assume that the effects of pressure on all of the condensed phases are negligible, and that the vapor may be treated as an ideal gas at a pressure of 0.0112 kPa, the vapor pressure of quinoline at 298.15 K.) (c) Statistical mechanical calculations have been performed on this molecule and yield a value for 5 of quinoline gas at 298.15 K of 344 J K l mol 1. Assuming an uncertainty of about 1 j K 1-mol 1 for both your calculation in part (b) and the statistical calculation, discuss the agreement of the calorimetric value with the statistical... 

R. H. Sherman and W. F. Giauque, "Arsine. Vapor Pressure, Heat Capacity, Heats of Transition, Fusion, and Vaporization. The Entropy from Calorimetric and from Molecular Data", J. Am. Chem. Soc., 77, 2154-2160 (1955). 

It should be pointed out that a finite residual entropy calculated for a substance from experimental data obtained at temperatures extending down to a certain temperature, with extrapolation below that point, may arise either from failure of the experimenter to obtain thermodynamic equilibrium in his calorimetric measurements or from error in the extrapolation. Measurements made under ideal conditions and extended to sufficiently... 

The chemical potential difference —ju may be resolved into its heat and entropy components in either of two ways the partial molar heat of dilution may be measured directly by calorimetric methods and the entropy of dilution calculated from the relationship A i = (AHi —AFi)/T where AFi=/xi —/x or the temperature coefficient of the activity (hence the temperature coefficient of the chemical potential) may be determined, and from it the heat and entropy of dilution can be calculated using the standard relationships... 

Fig. 113.—Comparison of observed entropies of dilution (points and solid lines with results calculated for ASi according to Eq. (28) (broken line). Data for polydimethyl-siloxane, M =3850, in benzene, A (Newing ), obtained from measured activities and calorimetric heats of dilution. Entropies for polystyrene (Bawn et in methyl ethyl ketone,, and in toluene, O, were calculated from the temperature coefficient of the activity. The smoothed results for benzene solutions of rubber, represented by the solid curve without points, were obtained similarly.

Papisov et al. (1974) performed calorimetric and potentiometric experiments to determine the thermodynamic parameters of the complex formation of PMAA and PAA with PEG. They investigated how temperature and the nature of the solvent affected the complex stability. They found that in aqueous media the enthalpy and entropy associated with the formation of the PMAA/PEG complex are positive while in an aqueous mixture of methanol both of the thermodynamic quantities become negative. The exact values are shown in Table II. The viscosities of aqueous solutions containing complexes of PMAA and PEG increase with decreasing temperature as a result of a breakdown of the complexes. 

Gorman-Lewis D, Fein JB, Jensen MP (2006) Enthalpies and entropies of proton and cadmium adsorption onto Bacillus subtilis bacterial cells from calorimetric measurements. Geochim Cosmochim Acta 70 4862-4873... 

Calorimetric studies indicate that the enthalpies of complexation tend to show related trends to the observed stability constants and display selectivity peaks, although there is not necessarily a coincidence between the two sets of peaks. Complexation is characterized by the entropy becoming progressively less positive (less favourable) as the cation size decreases. This is illustrated in Figure 6.5 for the complexation of 2.2.1 with the alkali metals. 

Enthalpies of interaction of Ca2+ with several aldopentoses and aldohexoses have been determined calorimetrically (652,653,656-658). Complex formation is generally characterized by the balance between a fairly large favorable enthalpy term and an almost equal but unfavorable entropy term - for ribose both AH and TAS° are — 24 kJ mol-1 a value of +5 cm3 mol 1 for AV° can be understood in terms of electrostriction (658). 

Carell and Olin (58) were the first to derive thermodynamic functions relating to beryllium hydrolysis. They determined the enthalpy and entropy of formation of the species Be2(OH)3+ and Be3(OH)3+. Subsequently, Mesmer and Baes determined the enthalpies for these two species from the temperature variation of the respective equilibrium constants. They also determined a value for the species Be5(OH) + (66). Ishiguro and Ohtaki measured the enthalpies of formation of Be2(OH)3+ and Be3(OH)3+ calorimetrically in solution in water and water/dioxan mixtures (99). The agreement between the values is satisfactory considering the fact that they were obtained with different chemical models and ionic media. 

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