Statistical thermodynamics example calculation

The entropy of formation is calculated from 5° values obtained from Third Law measurements (Chapter 4) or calculated from statistical thermodynamics (Chapter 10). The combination of AfS with Af// gives AfG. For example, for the reaction at 298.15 K... 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

The following example shows that the influence of statistical thermodynamical calculations on qualitative assertions is often insignificant. The reactions (3) <5) describe three of the first propagation steps of the cationic copolymerization of ethene and isobutene. 

The examples cited above are only two of the many possible cases of H-bond isomerization. Because of the low kinetic barriers separating these species, equilibration of H-bonded isomer populations to limiting thermodynamic values is generally expected to be much faster than for covalent isomers. Methods of quantum statistical thermodynamics can be used to calculate partition functions and equilibrium population distributions for H-bonded isomers,41 just as in the parallel case for covalent isomers and conformers. 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

As expected, the total interaction energies depend strongly on the van der Waals radii (of both sorbate and sorbent atoms) and the surface densities. This is true for both HK type models (Saito and Foley, 1991 Cheng and Yang, 1994) and more detailed statistical thermodynamics (or molecular simulation) approaches (such as Monte Carlo and density functional theory). Knowing the interaction potential, molecular simulation techniques enable the calculation of adsorption isotherms (see, for example, Razmus and Hall, (1991) and Cracknell etal. (1995)). 

Based upon experimentally observed spectroscopic data, statistical thermodynamic calculations provide thermodynamic data which would not be obtained readily from direct experimental measurements for the species and temperature of interest to rocket propulsion. If the results of the calculations are summarized in terms of specific heat as a function of temperature, the other required properties for a particular specie, for example, enthalpy, entropy, the Gibb s function, and equilibrium constant may be obtained in relation to an arbitrary reference state, usually a pressure of one atmosphere and a temperature of 298.15°K. Or alternately these quantities may be calculated directly. Significant inaccuracies in the thermochemical data are not associated generaUy with the results of such calculations for a particular species, but arise in establishing a valid basis for comparison of different species. 

Transport Limitation For the estimation of the mass transport limitation, Equation (20) has an important drawback. In many cases neither the rate constant k nor the reaction order n is known. However, the Weisz-Prater criterion, cf. Equation (21), which is derived from the Thiele modulus [4, 8], can be calculated with experimentally easily accessible values, taking < < 1 for any reaction without mass transfer limitations. However, it is not necessary to know all variable exactly, even for the Weisz-Prater criterion n can be unknown. Reasonable assumptions can be made, for example, n - 1, 2, 3, or 4 and / is the particle diameter instead of the characteristic length. For the gas phase, De can be calculated with statistical thermodynamics or estimated common values are within the range of 10-5 to 10 7 m2/s. In the liquid phase, the estimation becomes more complicated. A common value of qc for solid catalysts is 1,300 kg/m3, but if the catalyst is diluted with an inert material, this... 

Thermodynamics is complementary to kinetic theory and statistical thermodynamics. Thermodynamics provides relationships between physical properties of any system once certain measurements are made. Kinetic theory and statistical thermodynamics enable one to calculate the magnitudes of these properties for those systems whose energy states can be determined. There are three principal laws of thermodynamics. Each law leads to the definition of thermodynamic properties which help us to understand and predict the operation of a physical system. Here you can find some simple examples of these laws... 

With this example we also address the issue that quasi-chemical approaches sometimes offer flexibility in designing an inner shell, and differently designed approaches permit us to learn different features from the molecular statistical thermodynamic calculations. 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

A completely separate branch of science is statistical mechanics which is concerned with microscopic properties. This subject makes use of what quantum mechanics tells us about the energy levels of molecules, and allows us to calculate macroscopic properties on the basis of this information. The area of overlap between statistical mechanics and thermodynamics is known as statistical thermodynamics which allows us, for example, to calculate equilibrium constants for chemical reactions using the molecular properties obtained from quantum mechanics. 

We have explored some of the simpler aspects of statistical thermodynamics, a very powerful theoretical tool. If the energy levels of the molecules composing the system can be obtained by solution of the Schrodinger equation, the partition function can be calculated then any thermodynamic property can be evaluated. One of the great virtues of statistical thermodynamics is its ability to reveal general laws. For example, we reached the conclusion that all monatomic solids should have the same heat capacity at high temperatures. Restrictions on the laws are made apparent f or example, the heat capacity of a monatomic solid at low temperatures depends on what is assumed about the frequencies in the solid. 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

A numerical calculation needs knowledge of the solvent activity of die corresponding homopolymer solution at the same equilibrium concentration (here characterized by the value of the Flory-Huggins %-function) and the assumption of a deformation model that provides values of the factors A and B. There is an extensive literature for statistical thermodynamic models which provide, for example, Flory A = 1 and B = 0.5 Hermans A = 1 and B = 1 James and Guttf or Edwards and Freed A = 0.5 and B = 0. A detailed explanation was given recently by Heinrich et al. ... 

In order to describe the three-phase (Hydrate - Liquid Water - Vapor) equihbria (H-Lw-V) the theory developed by van der Waals-Platteeuw [9, 10] is traditionally used. The theory is based on Statistical Thermodynamics and according to Sloan and Koh [1] it is probably one of the best examples of using Statistical Thermodynamics to solve successfully a real engineering problem. An excellent description of the theory for the three-phase equilibrium calculations is provided in a number of publications [1,9, 10] and will not be repeated here. In addition, extensive details on the methodology for the calculation of two-phase equilibrium (H-L ) conditions can be foimd in the review papers by Holder et al. [12], and Tsimpanogiannis et al., [11]. 

For certain reasons the lattice model will be chosen here to give some examples. Following Koningsveld et al. [71], we extend the random mixing assumption and the surface contact statistics to a system consisting of two random polydisperse copolymer, each built up by two different units, B(a, P) + C(y5). Applying the principles of continuous thermodynamics in calculating the contact probabilities in such mixtures [34,40], we obtain for [66]... 

A microscopic theory may be developed by using a calculational scheme based on following the trajectories (position and velocity) of each molecule in the system. At each molecule-molecule or molecule-wall collision, new trajectories would have to be computed. Such calculations can be performed for limited number of molecules and short periods of time. Such calculations yield the probability distribution of particle velocities or kinetic energies. For example, the temperature of a monoatomic gas could then be computed from the average kinetic energy. Therefore, statistical thermodynamics determine probability distributions and average values of properties when considering all possible states of the molecules consistent with the constraints on the overall system. 

Figure 8.2 summarizes the methods discussed here. The organization of this chapter is as follows. First, methods for calculating the rate constant of an elementary step are described. Then DFT is briefly introduced for estimation of adsorption properties and barriers, followed by an outline of selected statistical Ihermodynantics. Examples of the thermochemistry on Ni(l 11) and Pt(lll) are presented to address thermodynamic consistency of the DFT-predicted adsorption properties. S iempirical methods for predicting adsorbate thermodynamic properties and kinetic parameters are also presented. With this input, microkinetic models can be solved. Finally, analytical tools are described to develop and analyze a nticrokinetic model, with the water-gas shift (WGS) reaction on Pt-based catalysts taken as an example. 

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