From kinetic theory

It is instructive to consider just how mobile the surface atoms of a solid might be expected to be. Following the approach in Section III-2, one may first consider the evaporation-condensation equilibrium. The number of molecules hitting a 1-cm surface per second is from kinetic theory... 

A knowledge of the viscous and thermal properties of non-Newtonian fluids is essential before the results of the analyses can be used for practical design purposes. Because of the nonlinear nature, the prediction of these properties from kinetic theories is as of this writing in its infancy. Eor the purpose of design and performance calculations, physical properties of non-Newtonian fluids must be measured. 

The chance of a collision will obviously depend upon the number of gas molecules per unit volume or, alternatively, upon the pressure. The chance of a collision will also depend upon the size of the gas molecules. For example, the chance of two basketballs thrown toward one another undergoing a collision is much greater than the chance of having a similar collision between two golf balls. An expression for the mean free path in terms of pressure and molecular diameter may be derived from kinetic theory. We give only the result, which may be expressed as... 

Figure 15 A schematic of the model for evaluating sublimation rate and evaporation rate from kinetic theory.

Contamination from the gas phase will be proportional to the number of collisions a surface undergoes. Hence, from kinetic theory, the number of collisions per unit time and unit area, Zc, is given by ... 

Increasing the pressure shifts the position of the above equilibrium to the right. The system has responded to counteract the effect of the change (an increase in pressure). From kinetic theory, two moles of gas molecules exert less pressure on the walls of a vessel than four moles. As predicted, the position of equilibrium has altered in such a way as to reduce the total pressure. 

This equation has a different temperature coefficient from that calculated from kinetic theory, and shows a shortcoming in the theory which is due to the assumption of non-interaction between the molecules. 

From kinetic theory the average energy of a hydrogen molecule at 300K, calculated as... 

This represents an upper limit for the dimensions of the nucleus. Compared with the estimates for the size of the atom, obtained from kinetic theory calculations on gases, which are typically 4x10 9 m. we can see that the nucleus is very small indeed compared to the atom as a whole - a radius ratio of 10-5, or a volume ratio of 10 15, which supports Rutherford s observation that most of an atom consists of empty space. We can also conclude that the density of the nucleus must be extremely high - 1015 times that encountered in ordinary matter, consistent with density estimates in astronomical objects called pulsars or neutron stars. 

The macroscopic properties of a material are related intimately to the interactions between its constituent particles, be they atoms, ions, molecules, or colloids suspended in a solvent. Such relationships are fairly well understood for cases where the particles are present in low concentration and interparticle interactions occur primarily in isolated clusters (pairs, triplets, etc.). For example, the pressure of a low-density vapor can be accurately described by the virial expansion,1 whereas its transport coefficients can be estimated from kinetic theory.2,3 On the other hand, using microscopic information to predict the properties, and in particular the dynamics, of condensed phases such as liquids and solids remains a far more challenging task. In these states... 

The parameters D and Dk > whether for macro (denoted by subscript m) or for micro (denoted by subscript ju) regions, are normal bulk and Knudsen diffusion coefficients, respectively, and can be estimated from kinetic theory, provided the mean radii of the diffusion channels are known. Mean radii, of course, are obtainable from pore volume and surface area measurements, as pointed out in Sect. 3.1. For a bidisperse system, two peaks (corresponding to macro and micro) would be expected in a differential pore size distribution curve and this therefore provides the necessary information. Macro and micro voidages can also be determined experimentally. 

We will be estimating the velocity of a molecule from kinetic theory, not the x-component of this velocity. Thus, from the Pythagoras theorem, the velocity of a molecule, v, is given by... 

The mean velocity of a molecule can be similarly estimated from kinetic theory (Kauzmann, 1966) ... 

The least resolved measurement is determination of the isothermal rate constant k(T), where T is the isothermal temperature. Although conceptually simple, such measurements are often exceedingly difficult to perform for activated process without experimental artifact (contamination) because they require high pressures to achieve isothermal conditions. For dissociative adsorption, k(T) = kcol (T) [S (Tg = TS = T)), where kcol(T) is simply the collision rate with the surface and is readily obtainable from kinetic theory and Tg and T, are the gas and surface temperatures, respectively [107]. (S ) refers to thermal averaging. A simple Arrhenius treatment gives the effective activation energy Ea for the kinetic rate as... 

From kinetic theory of gases (see Atkins 1994), the number of collisions per unit area per unit time Z between gas molecules and a wall equals... 

The obtainable current is limited by the maximum gas pressure that can be maintained without resulting in a discharge. Thus the rate-determining step is not the desorption of ions under the influence of the high field, but the diffusion of gas to the tip. The latter is an order of magnitude greater than would be expected from kinetic theory because gas near the tip is polarized and thus attracted. 

This vapor pressure can be deduced from kinetic theory and the rate of arrival of Cs atoms per square centimeter per second (3,4). This arrival rate. A, can be calculated from the measured value of the saturation positive ion current. Langmuir first showed that if the tungsten is hot enough every cesium atom that strikes the surface evaporates off as a positive ion of cesium. This saturated positive ion current, i.p, can be measured easily with a medium sensitive galvanometer when the collector is negative. It is related to A by the equation... 

Quantitative tests of the absolute modulus value from kinetic theory [Eq. (7.2)] are quite scarce independent determination of the network strand concentration is generally the difficulty. Schaefgen and Flory (278) prepared model... 

Besides the particle radius, the fundamental quantities indicated above are Dv, DP, KD, v, and a. v can be calculated from kinetic theory (5), Lai and Freiling have calculated Dv (6), and Norman has estimated DP and KP (9,12). Only the problem of estimating a remains. 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

In a low-density limit the binary diffusion coefficient between two gaseous species may be determined from kinetic theory as... 

Here the subscript oo represents a reference property at the inlet condition, which may be at the inlet manifold or the far-held value for semi-infinite situations. The simple power-law dependence follows from kinetic theory. Typically the temperature dependence for polyatomic gases is n 0.645 (Section 3.3). 

As a starting point in the study of equations of state of real gases, we will consider a hypothetical gas known as an ideal gas. We will develop the equation of state of an ideal gas in two ways, first from experimental evidence and then from kinetic theory. The form of the equation for ideal gases will be used as the basis of equations for real gases. 

Thus, the equation of state of an ideal gas can be derived from kinetic theory. All three properties of an ideal gas were used in the development of the equation. 

We have shown from kinetic theory that the ideal gas equation has the correct form. The behavior of most real gases does not deviate drastically from the behavior predicted by this equation. So the best way of writing an equation of state for a real gas is to insert a correction factor into the ideal gas equation.1 This results in... 

At 315°C. the rate constant ki has a value of 7.0 X 1016 molecules/sec.-cm.2-atm. From the definition of kh this represents the rate of adsorption of methylcyclohexane per cm.2 of bare platinum surface at a methylcyclohexane partial pressure of 1 atm. From kinetic theory and statistical mechanics, one can calculate the number of molecules striking a unit area of surface per unit time with activation energy Ea. This is given by... 

Uranium hexafluoride is probably the most interesting of the uranium fluorides. Under ordinary conditions, it is a dense, white solid with a vapor pressure of about 120 hull ai room temperature. It can readily be sublimed or distilled, and it is by far the most volatile uranium compound known. Despite its high molecular weight, gaseous UFg is almost a perfect gas, and many of the properties of the vapor can be predicted from kinetic theory. 

In both cases the zone width is many times greater than the mean free path. This is as it should be, since the velocity is small compared to the speed of sound as proof it is sufficient to substitute an expression from kinetic theory,... 

The friction on a tagged particle is expressed in terms of the time-dependent force-force autocorrelation function. Although the bare, short-time part of the friction that arises from binary collisions can be calculated from kinetic theory, the long-time part needs the knowledge of the solvent and the solute dynamics and the coupling between them. The solvent dynamic quantities... 

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