The Statistical Thermodynamics of a Dilute Gas

This section is meant only as a compilation of some of the important results of statistical thermodynamics. No proof will be given.The fundamental quantity which relates the mechanical properties of molecules to the thermodynamic properties of a dilute gas is the molecular partition function q T)... 

In the previous chapter we obtained the probability distribution for molecular states in a dilute gas and obtained a formula for the thermodynamic energy of a dilute gas in terms of the partition function. Statistical mechanics will not be very useful until we obtain formulas for the other thermodynamic functions, which is the topic of this chapter. 

The thermodynamic functions of a dilute gas can be calculated from the molecular partition function of the gas. The necessary formulas are based on the postulates of statistical mechanics and on the definition of the statistical entropy... 

Ben-Naim and Marcus [1] discussed a process termed solvation that applies to a particle of a (non-ionic ) substance transferring from its isolated state in the gas phase into a liquid irrespective of the concentration. The particle would then be surrounded by solvent molecules only in an ideally dilute solution (infinite dilution), or by solvent molecules as well as by molecules of its own kind at any arbitrary mole fraction with regard to the solvent, and by molecules identical with itself only on condensation into its own liquid. The interactions involved and their thermodynamics are aU covered by the same concept of solvation. The solvation process of a solute S is defined [1] as the transfer of a particle of S Ifom a fixed position in the (ideal) gas phase (superscript G) to a fixed position in a liquid (superscript L) at a given temperature T and pressure P. Statistical mechanics specifies the chemical potential of S in the ideal gas phase as ... 

Adsorption isotherms represent a relationship between the adsorbed amount at an interface and the equilibrium activity of an adsorbed particle (also the concentration of a dissolved substance or partial gas pressure) at a constant temperature. The analysis of adsorption isotherms can yield thermodynamic data for the given adsorption system. Theoretical adsorption isotherms derived from statistical and kinetic data, and using the described assumptions (see 3.1), are known only for the gas-solid interface or for dilute solutions of surfactants (Gibbs). Those for the system gas-solid are of a few basic types that can be thermodynamically predicted81. From temperature relations it is possible to calculate adsorption and activation energies or rate constants for individual isotherms. Since there are no theoretically founded equations of adsorption isotherms for dissolved surfactants on solids, the adsorption of gases on solides can be used as a starting point for an interpretation. 

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... 

First, consider a very dilute gas of molecules [16]. The conventional theory of the static dielectric susceptibility % of such a gas invokes the notion of polarizable molecules with permanent dipole moments that are partially aligned by the external electric field . Standard techniques of statistical thermodynamics produce the Langevin-Debye formula for x Per molecule that reads... 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the formalism of classical thermodynamics. Earlier sections in this article have shown how these experimental laws lead to simple thermod5mamic equations, but these results are added to thermod5mamics they are not part of the formalism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

There are statistical, and perhaps systematic, uncertainties associated with the primary measured quantities used to define the thermodynamic state these should be considered in the reporting of experimental uncertainties of the measured quantity and accounted for in the regressions used to determine a correlation. All the uncertainties are likely to have state point dependence, so that the assessment in the dilute gas regime will differ markedly firom that in the critical region or in the compressed liquid (Perkins et al. 1991a). The uncertainty reported for a correlation should be a function of the fluid state point. The coverage factor (based on, perhaps, two standard deviations in a normal distribution) should be applied to the appropriate distribution associated with the combined standard uncertainty of the correlation. 

Cv can thus be calculated, at least for an ideal gas, and the whole construction can be put to a stringent scientific test by comparing it with measured heat capacities. No need to say, it turns out that statistical thermodynamic calculations provide extremely accurate evaluations of the heat capacities of diluted gases. In fact, these calculations are so reliable that for small molecules the CvS of gases found in thermodynamic repertories are usually calculated from experimental vibration frequencies, rather than measured. On the other hand, heat capacities for condensed phases cannot be calculated, but are much more easily measured than for gases. In this case, calculation and experiment match and complement each other perfectly. 

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