Kinetic equation of gases

Problem 7 From kinetic equation of gases, how will you calculate the root mean square velocity of gas molecules under different conditions  

Problem 8 What are real and ideal gases In what respect does a real gas differ from an ideal gas (Meerut 2004) 

A gas which strictly obeys Boyle s and Charles laws at all temperatures and pressures is known as an ideal or perfect gas. The characteristics of an ideal gas are as follows  

Problem 9 (a)What are the limitations of the equation PV = RT Show in what respects vander Waals equation is an improvement over the simple gas equation. Derive vander Waals equation and discuss it or write the drawbacks of this equation. 

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Problem 2 Write the postulates of kinetic theory of gases. Derive the kinetic equation of gases. 

From the above postulates, the kinetic equation of gases can be derived easily as follows ... 

This equation is known as kinetic equation of gases. Equation (3) gives the pressure exerted by an ideal gas. 

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

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 fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. 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. 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

For example, the measurements of solution osmotic pressure made with membranes by Traube and Pfeffer were used by van t Hoff in 1887 to develop his limit law, which explains the behavior of ideal dilute solutions. This work led direcdy to the van t Hoff equation. At about the same time, the concept of a perfectly selective semipermeable membrane was used by MaxweU and others in developing the kinetic theory of gases. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

By analogy with the tlrermal conduction equation for a gas, given by the kinetic theoty of gases, we can write for tire thermal conductivity, K, of a solid... 

White s equation is widely used mainly because it is easy to use and because it gives values which are in reasonable agreement with the experimental ones. However, because this model is based on the kinetic theory of gases, it should be used for small particles only. This model (as many others) assumes that particle charge can be described with a continuous function. Especially in the case of small particles, only the lowest charge numbers (0, 1, 2) are possible, and therefore the model—which does not take into account the discrete charge numbers—is somewhat questionable. 

Example 1.1 Use the kinetic theory of gases to rationalize the functional form of Equation (1.8). 

It has been established by Dong and Bloom [29] and Courtney and Armstrong [30] that this equation can be translated into one based upon the kinetic theory of gases using... 

Parallel with the phenomenological development, an alternative point of view has developed toward thermodynamics, a statistical-mechanical approach. Its philosophy is more axiomatic and deductive than phenomenological. The kinetic theory of gases naturally led to attempts to derive equations describing the behavior of matter in bulk from the laws of mechanics (first classic, then quanmm) applied to molecular particles. As the number of molecules is so great, a detailed treatment of the mechanical problem presents insurmountable mathematical difficulties, and statistical methods are used to derive average properties of the assembly of molecules and of the system as a whole. 

The Freundlich equation proved to be applicable to the adsorption of liquids with only limited ranges of concentration. It was replaced by the Langmuir equation (see later on) and others which had a theoretical basis in the kinetic theory of gases. It is clear that neither the Freundlich nor the Langmuir equation can describe isotherms of the shape shown in Figure 10.5. 

The kinetic theory of gases has therefore given us a fairly simple equation for the diffusion coefficient of a molecule. All that remains to be determined is the mean velocity of the molecule and the mean free path of the molecule. 

Although the kinetic theory of gases is not generally used, as is, to estimate the diffusion coefficients of gases, it does provide a framework for the characterization of data through predictive equations. This simple kinetic theory has shown us the following ... 

The bi are derived from the collision radii of the molecular species at high temperature and, as in the kinetic theory of gases at moderate pressure, are equal to four times the molecular volume multiplied by Avogadro s number. Despite the use of diminished covolumes in the equation and despite the apparent theoretical basis of the model, the equation is oversimplified and the results of detonation calculations quite clearly show it to be inaccurate. 

Distribution functions are usually first met in physical chemistiy when the crude treatment of molecular velocities in the kinetic theory of gases (all the molecules taken as having the same mean speed) is replaced by Maxwell s seminal equation showing that the number of molecules having velocities between narrow limits depends very much on what velocities are chosen. This is shown in Fig. 9.1. Thus, this first and basic distribution law of Maxwell, the distribution of velocities, gives an unexpected result (the nonsymmetrical nature of the distribution), which still causes us to think, more than a century after its publication. 

Finally, we note that all the statistical equations of this chapter could have been borrowed directly from the kinetic theory of gases by simply changing the variables. We illustrate this now by going in the opposite direction. For example, if we replace the quantity 3/n 2 by m/ kBT and replace L by v in Equation (69), we obtain the Boltzmann distribution of molecular velocities in three dimensions. If we make the same substitutions in Equation (73), we obtain an important result from kinetic molecular theory ... 

To bridge the gap between molecular processes and empirical coefficients and between laboratory determinations of input data and an engineering approach to predictions, we want to develop the above fundamental equations in terms of the kinetic theory of gases and reaction rate theory. There are three principal candidates for the rate-controlling... 

The situation is similar to the one encountered in the kinetic theory of dilute plasmas. 510 To lowest order in the density+) the one-particle distribution function of the electrons obeys the Vlasov equation. The next order approximation consists of two coupled equations for the one-particle and two-particle distribution functions. On the other hand, in the kinetic theory of gases Bogolyubovft) has proposed an... 

The previous equations which describe the behavior of an ideal gas now will be verified using the kinetic theory of gases. This will illustrate the reasons for the previously given three conditions imposed on the molecules of an ideal gas. Also, you will gain an understanding of the meanings of pressure and temperature. 

Boyle s Equation —Charles Equation—Avogadro s Law — The Equation of State for an Ideal Gas — Density of an Ideal Gas — Kinetic Theory of Gases Mixtures of Ideal Gases 100... 

As Equations 2.19 and 2.20 illustrate, the diffusion curve is determined by time (time elapsed from beginning to end of separation) and the value of D (diffusion coefficient which is different for each gas). One may obtain the value of D from the kinetic theory of gases. 

This is of the same form as Equation 30, but involves the mixed diffusion coefficient, Jci9, instead of the thermal conductivity of the mixture. However, as seen from the kinetic theory of gases, the thermal conductivity is proportional to the diffusion coefficient. Therefore agreement of experimental results with either Equation 30 or 53a is not an adequate criterion for distinguishing between first explosion limits in branching chain reactions and purely thermal limits. It has been reported (52), that, empirically,... 

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