Application to Gases

If we take a gas that consists of identical point particles that exert no mutual forces and rebound elastically at the walls of the container, then the potential energy t/ = 0 (except at the walls where C/ = oo) and we can calculate Z directly. The integration over the position coordinates for each molecule simply gives the value K, the volume of the vessel, and since there are N particles, we obtain V. The rest of the integral is 

We see that in general, when the potential function U depends only on the position coordinates and not on the velocities, the integration of Z over the momenta can be obtained, and so in general we may write 

If we consider a gas made of hard sphere molecules of diameter r, but exerting no forces, we can evaluate Qc approximately as follows Let us consider the first molecule. As its coordinates vary over the container, U is constant except at the walls or when its center is at a distance cr from the center of another molecule, where U becomes infinite. The result of the integration over the coordinates of this first molecule is then V (the volume of the container) minus A — 1 times %Tr r (the volume excluded by the other molecules). Thus 

If now the gas is dilute so that V SNv, that is, the total volume is much greater than 8 times the volume of all molecules, Ave can take the logarithm of both sides and expand all the logarithmic terms in a series. We find 

If we substitute this in the formula for Z and differentiate to obtain the thermodynamic pressure [Eq. (IX.2.6)], we have, since d In Z/dF = d In QddV, 

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Diffusion The mixing of substances by molecular motion to equalize a concentration gradient. Applicable to gases, fine aerosols and vapors. (See Brownian diffusion.)... 

Corrosion Product metal reaction product resulting from a corrosion reaction although the term is normally applied to solid compounds it is equally applicable to gases and ions resulting from a corrosion reaction. 

If the kinetic theory is applicable to gases, we should expect pressure to be affected by other factors than the number of moles per unit volume. For example, the mass of the molecules and their velocities should be important, as well. After all, a baseball exerts more push on a catcher s mitt than would a ping-pong ball thrown with the same velocity. Also, a baseball exerts more push on the mitt if a fast ball is thrown rather than a slow ball. To see how the mass of the molecules and their velocities are dealt with in the kinetic theory, we must consider temperature. 

Paterson, An Equation of State Applicable to Gases at Densities Near That of The Solid and Temperatures Far Above The Critical , PrRoy-Soc A-213,214-25 (1952) CA48,10735... 

A number of other laser spectroscopic techniques are of interest but space does not permit their discussion. A few specialized methods of detecting laser absorption worthy of mention include multiphoton ionization/mass spectrometry (28), which is extremely sensitive as well as mass selective for gas-phase systems optically detected magnetic resonance (29) laser intracavity absorption, which can be extremely sensitive and is applicable to gases or solutions (30) thermal blooming, which is also applicable to very weak absorbances in gases or liquids (31) and... 

A rapid volumetric method for the determination of sulphur is as follows 2 The sulphur is dissolved in a known volume of hot standard sodium hydroxide solution and after cooling is oxidised to sulphate by the addition of hydrogen peroxide the excess of alkali is then titrated with standard acid.3 The method is applicable to gases containing any common sulphur compound except thiophen (e.g. coke oven gas).4... 

CottrelUPoterson Equation of State, An equation of state, applicable to gases at densities near that of the solids and to temps far above the critical, is derived by Cottrell Paterson (Ref 1). It is shown that this equation is likely to hold in the range of density temperature characteristics of the detonation wave in condensed expls. The hydrodynamic equations of deton are developed on the basis of the equation of state. They were applied to PETN and the theory predictions were shown to agree with observations. Murgai (Ref 2) extended the application of the equations to oxygen-deficient expls, specifically TNT... 

W. Kauzmann, Thermodynamics and Statistics With Applications to Gases, Benjamin, New York, 1967. 

Immediately it is clear that a plot of Pm versus Tutilizing measurements of k as a function of temperature will yield both oq and F- This technique is readily applicable to gases. In principle it is applicable to liquids and solutions also, but it is seldom convenient owing largely to the small temperature range accessible between the melting point and boiling point. 

Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is = k for small IVp,. Thus in the solu-... 

Residual properties have validity for both gases and hquids. However, the advantage of Eqs. (6.49) and (6.50) in application to gases is that and the temis wliicli contain all the complex calculations, are residuals that generally are quite small. They liave the nature of corrections to the major temis, and S. For hquids, this advantage is largely lost, because... 

It will be observed that Table XXV contains values for the free energy function of graphite this has not been obtained from equation (33.43), which is applicable to gases only. The method for the calculation of — (F — HD/T for solids is based on the use of heat capacities. For a pure solid, since So is zero, by the third law of thermodynamics, equation (23.1) becomes... 

In equation (127), U0 denotes the heat evolution of the reaction at the absolute zero, and E the energy content of each separate gas the summation is to be taken as described on page 126. The expression in formula (128) is, however, by no means zero for T — o, i.e. our Heat Theorem in this case again does not hold, in its direct application to gases. This is, of course, not surprising because we have so far assumed the applicability of the gas laws down to the lowest temperatures, which, according to equation (126) precludes the applicability of our Heat Theorem. 

The above discussion presumes the availability of a volume-explicit equation of state. For applications to gases at moderate to high pressures or densities or to vapors and liquids, realistic equations of state are not volume explicit but are instead pressure explicit. That is, Z is expressed as a function of T. v. and x or. equivalently, of T, p (molar density e o->), and Jt ... 

Ferrari [96] also solved the von Mises form of the energy equation, but with the velocity distributions in the inner and outer portions of the boundary layer represented by polynomials of a velocity potential function. This analysis is applicable to gases in that it accounts for compressibility, but the solutions may be in error for high Prandtl numbers because of the rather approximate fit to the law of the wall. 

Basic laws, using the Caratheodory approach. Applications to gases, mixtures and solutions, chemical and phase equilibria,... 

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