Pressure from partition function

As seen earlier ( 12k), Aflo for a reaction may be evaluated from thermal measurements, including heat capacities at several temperatures. However, instead of using experimental heat capacity data to derive AHq from A/f values, the results may be obtained indirectly from partition functions (cf. 16c). The energy content of an ideal gas is independent of the pressure, at a given temperature hence, E — Eo in equation (16.8) may be replaced by E — JSo, so that... 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

Other thermodynamic functions may be derived from the partition function Q, or from the expression for the osmotic pressure. The chemical potential of the solvent in the solution (not to be confused with the excess chemical potential (mi —within a region of uniform segment expectancy, or density) is given, of course, by ... 

Fig. 20.1. Correlation between the air/water partition coefficient, Kaw, determined from measurements of the surface pressure as a function of drug concentration (Gibbs adsorption isotherm) in buffer solution (50 mM Tris/HCI, containing 114 mM NaCI) at pH 8.0 and the inverse of the Michaelis Menten constant, Km obtained from phosphate release...

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

If we have a mixture of AT molecules of one gas, N2 of another, and so on, the general phase space will first contain a group of coordinates and momenta for the molecules of the first gas, then a group for the second, and so on. The partition function will then be a product of terms like Eq. (3.5), one for each type of gas. The entropy will be a sum of terms like Eq. (1.14), with n in place of n, and Pt, the partial pressure, in place of P. But this is just the same expression for entropy in a mixture of gases which we have assumed thermodynamically in Eq. (2.7). Thus the results of Sec. 2 regarding the thermodynamic functions of a mixture of gases follow also from statistical mechanics. 

At that date, palladium hydride was regarded as a special case. Lacher s approach was subsequently developed by the author (1946) (I) and by Rees (1954) (34) into attempts to frame a general theory of the nature and existence of solid compounds. The one model starts with the idea of the crystal of a binary compound, of perfect stoichiometric composition, but with intrinsic lattice disorder —e.g., of Frenkel type. As the stoichiometry adjusts itself to higher or lower partial pressures of one or other component, by incorporating cation vacancies or interstitial cations, the relevant feature is the interaction of point defects located on adjacent sites. These interactions contribute to the partition function of the crystal and set a maximum attainable concentration of each type of defect. Conjugate with the maximum concentration of, for example, cation vacancies, Nh 9 and fixed by the intrinsic lattice disorder, is a minimum concentration of interstitials, N. The difference, Nh — Ni, measures the nonstoichiometry at the nonmetal-rich phase limit. The metal-rich limit is similarly determined by the maximum attainable concentration of interstitials. With the maximum concentrations of defects, so defined, may be compared the intrinsic disorder in the stoichiometric crystals, and from the several energies concerned there can be specified the conditions under which the stoichiometric crystal lies outside the stability limits. 

Entropy of activation (continued) sign of, 256 Entropy unit, 242 Enzyme catalysis, 102 Enzyme-substrate complex, 102 Equilibrium, 60, 97, 99, 105, 125, 136 condition for, 205 displacement from, 62, 78 in transition state theory, 201, 205 Equilibrium assumption, 96 Equilibrium constant, 61. 138 complexation, 152 dissociation, 402 ionization, 402 kinetic determination of, 279 partition functions in, 204 pressure dependence of, 144 temperature dependence of, 143, 257 transition state, 207 Equivalence, kinetic, 123 Error analysis, 40 Error propagation, 40 Ester hydrolysis, 4 Euler s method, 106 Excess acidity method, 451 Exchange... 

The first stage in the calculations is to select a transition complex so that the rate coej0 cient in the high pressure limit is equal to that predicted by transition state theory. This requires that g+, the total partition function per cm for the complex with the contribution from motion along the reaction coordinate removed, is given by... 

Through the vehicle of partition functions expressions can be derived for macroscopic quantities from molecular parameters. Examples of such qucintities cire the surface pressure, and, for Gibbs monolayers, the adsorbed amount. Fluctuations in extensive quantities, like the number density or the interfacial excess energy, may also be obtained (sec. 1.3.7). 

The quantity — (F — Eo)/T or — (F — Ho)/T, that is, the left-hand side of equation (33.42) or (33.43), is known as the free energy function of the substance its value for any gaseous substance, at a given temperature, can be readily derived from the partition function for that temperature, at 1 atm. pressure. The data for a number of substances, for temperatures up to 1600 K, have been determined in this manner and tabulated (Table XXV). ... 

The critical micelle concentrations (erne s) were determined with a slightly modified methodology of Floriano et al. (1999). Figure 7 illustrates the concept. The reduced osmotic pressure is obtained from the logarithm of the grand partition function, a quantity which can in turn be determined from the simulations to within an additive constant (Floriano et al., 1999). For the system H2T4 (12), L = 10, we obtain the two curves shown in Fig. 7 at the two temperatures indicated there. There is a clear break, indicating the... 

From a series of transformations of Equation 1 we obtain a new partition function (T) whose independent variables are temperature, pressure, solvent mole number, and the chemical potentials of the solutes (components 2 and 3). These transformations consist of first creating a partition function with pressure rather than volume as an independent variable, and then using this result to create yet another partition function in which we have switched independent variables from solute mole numbers to solute chemical potentials. These operations are analogous to the Legendre transforms commonly employed in thermodynamics. 

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