Thermodynamic functions, calculated

Characteristic bands occur in the 1300-1000 cm region for 3,4- and 3,5-disubstituted isoxazoles (7i PMh(4)265, p. 330), while bands below 1000 cm contain modes for most substitution patterns (71PMh(4)265, p. 332). Total assignments for isoxazole and isoxazole-d have been made (63SA1145, 7lPMH(4)265,p. 325) and some of the thermodynamic functions calculated (68SA(A)361, 71PMH(4)265,p.330). 

IV, D. Dissociation energy-bond length relations have been proposed which take account of the atomic number of the atoms (4). The dissociation energies of Group VA monofluorides have been calculated using a Hulbert-Hirschfelder function (23) and thermodynamic functions calculated for di- and trifluorides (2). Another calculation of D0(Ft) has been made (20). 

In some works including textbooks, it is proposed to compare the values of the thermodynamic functions calculated per certain amount of a diffusing element. In such a case, however, it would be necessary first to determine this element. Furthermore, in the compounds of a given multiphase system, enriched in component A, diffusion of the A atoms often prevails, whereas in those enriched in component B diffusion of the B atoms is dominant. This makes any comparison of such values of the thermodynamic functions quite meaningless. 

The principal thermodynamic functions calculated from the spectral data are collected elsewhere21. The Ge—H chemical shift in the 1II NMR spectrum is at 7.6 ppm and has a half-width of ca. 12 Hz. This can be taken as evidence for exchange in the presence... 

Upper values data from most extensive calculations lower values best experimental data available, thermodynamic functions calculated from partition functions by means of the Sackur-Tetrode equation... 

Lithium vapor contains an appreciable amount of dimer, whose enthalpy of dissociation has been selected by Evans (5), from spectroscopic and molecular beam measuresments to be 25.76 0.10 kcal mol at 0 K. This enthalpy of dissociation, together with the thermodynamic functions calculated in this work, has been used to find the partial pressures of Li(g) and Li2(g) from the measured total vapor pressures. Hartmann and Schneider (6), report values from 1204 to 1353 K while Mancherat (7) reports effusion measurements from 735 to 915 K. Mancherat s (7 ) pressures are calculated on the assumption of monatomic vapor and have been recalculated to fine the true total pressure. Effusion measurements by Lewis (8) and Bogros (9) have been disregarded. Mancherat (7) considers them to be inaccurate because of impurities In the lithium used, and Lewis (S used a doubtful calibration method. Enthalpy of sublimation to monatomic vapor calculated from the vapor pressures of Hartmann and Schneider (6) and of Mancherat (7) agree to within 2% and the average value has been adopted. The enthalpy of sublimation of the dimer was then calculated using this value. 

The thermodynamic functions calculated from computer simulations can be fitted to experimental values. 

It follows from the discussion in this paragraph that only standard differential thermodynamic functions can be calculated from any chromatographic distribution constant defined in whatever way. Also, it is necessary to always specify the choice of the standard states for the solute in both phases of the system. Without specifying the standard states the data on the thermodynamic functions calculated from chromatographic retention data lack any sense. When choosing certain standard states it may happen that the standard differential Gibbs function is identical with another form of the differential Gibbs function, or includes such a form situations described by equations 46 and 49 may serve as examples. The same also holds true for standard differential volumes, entropies and enthalpies (compare Section 1.8.3). However, every particular situation requires a special treatment. 

The summation for unsymmetrical molecules is over all /, and for symmetrical molecules is over even or odd values of J for para- or ortho-species. Since qxot cannot be expressed in closed form, its evaluation as a sum with the thermodynamic functions calculated from equations (3)—(5), (7), and (9)—(11) is necessary, for example for hydrogen at moderate temperatures. However, if is small, then the summation can be replaced by an integral and evaluated as (07) where a is the symmetry number taking the value 2 or 1 for symmetrical or unsymmetrical molecules, respectively. The same result appears as the leading term in a power series for rot which may be derived ... 

Data were obtained using the thermodynamic functions calculated by the method of Mayer and Goeppert-Mayer with inclusion of the electronic excitation energies for LaQ. The use of the thermodynamic functions calculated by the direct summation method changes the enthalpy value by 0.5-0.8 kJ/moL Same calculation method but with inclusion of the electronic excitation energies of the free La+ ion. 

We have adopted a vialue of AHS(CF2,g,298) = -44.6 kcal/mol from the data of Modica and LeGraff (16,17) and of Carlson (19). This yields values of the equilibrium constant for reaction t ) with in a factor of two of those calculated from the data of Farber et (21), which is certainly within the accuracy of both the experiment and the limits of the rigid-rotor, harmonic oscillator approximation at 2000 to 2500 K ( ). The physical and thermochemical data selected here are sumnarized in Table II and the ideal gas thermodynamic functions calculated to 1500 K from these data are summarized in Table III. 

These data are summarized in Table X, and the ideal gas thermodynamic functions, calculated from these data, are given in Table XI. 

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. 

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. 

K (66.46 e.u.) with the spectroscopic value calculated from experimental data (66.41 0.009 e.u.) (295, 289) indicates that the crystal is an ordered form at 0°K. Thermodynamic functions of thiazole were also determined by statistical thermodynamics from vibrational spectra (297, 298). 

Selected physical properties are given ia Table 4. The nmr data (97) and ir and Raman spectra (98) have also been determined. Thermodynamic functions have been calculated from spectral data (99). 

Once the values of thermodynamic functions, Aff, ASp. ate known at a given temperature the value for the function can be calculated at any other temperature by ... 

Boron Monoxide and Dioxide. High temperature vapor phases of BO, B2O3, and BO2 have been the subject of a number of spectroscopic and mass spectrometric studies aimed at developiag theories of bonding, electronic stmctures, and thermochemical data (1,34). Values for the principal thermodynamic functions have been calculated and compiled for these gases (35). 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

Once the partition function is known, thermodynamic functions such as the internal energy U and Helmholtz free energy A may be calculated according to... 

Macroscopic observables, such as pressme P or heat capacity at constant volume C v, may be calculated as derivatives of thermodynamic functions. 

These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A, and G. We should keep in mind that these equations apply to a reversible process involving pressure-volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of AZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The... 

Values for the thermodynamic functions as a function of temperature for condensed phases are usually obtained from Third Law measurements. Values for ideal gases are usually calculated from the molecular parameters using the statistical mechanics procedures to be described in Chapter 10. In either... 

Figure 4.16 demonstrates that thermodynamic functions are not very exciting. That is, no unusual changes or effects occur. But these functions are very useful in calculating the thermodynamic quantities ArV, Ar//, ArG,... for a chemical reaction. When the reaction is written as,f... 

Example 9.3 Calculate the pressure of atomic chlorine in Cl2(g) at a total pressure of 1.00 bar and a temperature of 2000 K. Do the calculation (a) using the thermodynamic functions in Table 4.3, and (b) using equation (9.58), which requires Cp m expressed as a function of T, obtained from Table 2.1, and compare the results. 

P10.5 The thermodynamic functions for solid, liquid, and gaseous carbonyl chloride (COCL) obtained from Third Law and statistical calculations... 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

First, we shall explore a conceptual relation between kinetics and thermodynamics that allows one to draw certain conclusions about the kinetics of the reverse reaction, even when it has itself not been studied. Second, we shall show how the thermodynamic state functions for the transition state can be defined from kinetic data. These are the previously mentioned activation parameters. If their values for the reaction in one direction have been determined, then the values in the other can be calculated from them as well as the standard thermodynamic functions. The implications of this calculation will be explored. Third, we shall consider a fundamental principle that requires that the... 

Vapor pressures and vapor compositions in equilibrium with a hypostoichiometric plutonium dioxide condensed phase have been calculated for the temperature range 1500 I H 4000 K. Thermodynamic functions for the condensed phase and for each of the gaseous species were combined with an oxygen-potential model, which we extended from the solid into the liquid region to obtain the partial pressures of O2, 0, Pu, PuO and Pu02 as functions of temperature and of condensed phase composition. The calculated oxygen pressures increase rapidly as stoichiometry is approached. At least part of this increase is a consequence of the exclusion of Pu +... 

The process we have followed Is Identical with the one we used previously for the uranium/oxygen (U/0) system (1-2) and Is summarized by the procedure that Is shown In Figure 1. Thermodynamic functions for the gas-phase molecules were obtained previously (3) from experimental spectroscopic data and estimates of molecular parameters. The functions for the condensed phase have been calculated from an assessment of the available data, Including the heat capacity as a function of temperature (4). The oxygen potential Is found from extension Into the liquid phase of a model that was derived for the solid phase. Thus, we have all the Information needed to apply the procedure outlined In Figure 1. 

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