Kinetic theory calculations

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 Lennard-Jones (6-12) potential has served very well as an inter-molecular potential and has been widely used for statistical mechanics and kinetic-theory calculations. It suffers, however, from having only two adjustable constants, and there is no reason why it should not gradually be replaced by more flexible and more realistic functions. Recently a number of applications have been made of the Buckingham (6-exp) potential [Eq. (82)], which has three adjustable parameters. For this potential the first approximation to the coefficient of diffusion is written by Mason (M3) in the form... 

Before applying the vacuum microbalance or any similar method to the study of the rate of a particular surface reaction it is essential to understand as much as possible concerning the chemistry of the main reaction and the possible side reactions which may occur in a given system. This requires detailed thermochemical calculations to be made for all conceivable reactions to determine the specifications for the vacuum system and furnace tubes, the preparation of the specimens, the experimental procedures, and the interpretation of the data obtained. Kinetic theory calculations should be applied to aid in interpretation of the rates of certain vacuum and low-pressure reactions. 

II. Thermochemical and Kinetic Theory Calculations 1. Type Reactions... 

Different types known as gaseous, condensed, and expanded films are distinguished. In the gaseous films there appears to be free motion of the molecules over the surface, and in certain examples an equation FA = kT has been verified. This is the analogue of Boyle s law, and k itself has approximately the value to be expected from a variant of the kinetic theory calculation of gas pressure. The equation holds only for large values of. 4. As the pressure increases, FA passes through a minimum, just as pV does with an imperfect gas. At certain pressures condensation occurs to a film which is very resistant to further compression. 

It has been recognized for some time that a fully microscopic treatment of the dynamics of fluids is feasible at low densities. For dilute gases methods based on density expansion prove to be useful and a Boltzmann-like equation can be used to calculate time correlation functions. " The notion that microscopic or kinetic theory calculations could be formulated for dense fluids is much more recent. The difficulty with dense systems is that there is no small parameter in the problem. Consequently a direct density expansion where one computes the first few corrections to the low-density result is not likely to give useful results. At first sight things are worse than this because the first few density corrections are not only mathematically exhausting, but the... 

When you think about the kinetic-theory calculations of molecular speed, you might ask why diffusion is not even much faster than it is. Why does it take minutes for the gas to diffuse throughout a room when the molecules are moving at perhaps a thousand miles per hour This was, in fact, one of the first criticisms of kinetic theory. The answer is simply that a molecule nevo- travels very far in one direction (at ordinary pressures) before it collides with another molecule and moves off in anodiCT direction. If you could trace out the path of an individual molecule, it would be a zigzagging trail. For a molecule to cross a room, it has to travel many times the straight-line distance. 

KAPRAL - My comment concerns the effect on the rate coefficient of including spatial dependence in the diffusion or friction coefficient. We investigated this problem some time ago (M. Schell, R. Kapral and R.I. Cukier, Chem. Phys. Lett.) for a model bimolecular reaction using a space-dependent friction coefficient obtained from a kinetic theory calculation (R.I. Cukier, 3.R. Mehaffey and R. Kapral, J. Chem. Phys.). The effects are rather small, amounting to no more than about forty per cent corrections to the rate coefficient. The small size of the effect arises from averaging over a range of spatial configurations. 

One Experimental Data Point An ion gauge built into a life-test a dewar shows a pressure of about 5 x 10 torr while the dewar is cold, and we observe an apparent water condensation rate of about 1 pm/day (from the responsivity loss with time). This agrees fairly well with the kinetic theory calculation above. 

In the second part of hla memoir Reynolds gave a theoretical account of thermal transpiration, based on the kinetic theory of gases, and was able CO account for Che above "laws", Chough he was not able to calculate Che actual value of the pressure difference required Co prevent flow over Che whole range of densities. ... 

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. 

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 ... 

For a short pipe of circular cross section, the conductance as calculated for an orifice from Eq. (6-82) is multiplied by a correction factor iC which may be approximated as (Kennard, Kinetic Theory of Gases, McGraw-Hill, New York, 1938, pp. 306-308)... 

The simplest system in which useful products are obtained by thermochemical processing involves the evaporahon of an element or elements in vacuum in order to produce thin hlms on a selected substrate. This process is usually limited to the production of thin hlms because of the low rates of evaporation of the elements into a vacuum under conditions which can be controlled. These rates can be calculated by the application of the kinetic theory of ideal gases. 

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

Z, the collision frequency, which gives the number of molecular collisions occurring in unit time at unit concentrations of reactants. This quantity can be calculated quite accurately from the kinetic theory of gases, but we will not describe that calculation. 

When mass transfer rates are very high, limitations may be placed on the rate at which a component may be transferred, by virtue of the limited frequency with which the molecules collide with the surface. For a gas, the collision rate can be calculated from the kinetic theory and allowance must then be made for the fact that only a fraction of these molecules may be absorbed, with the rest being reflected. Thus, when even a pure gas is brought suddenly into contact with a fresh solvent, the initial mass transfer rate may be controlled by the rate at which gas molecules can reach the surface, although the resistance to transfer rapidly builds up in the liquid phase to a level where this effect can be neglected. The point is well illustrated in Example 10.4. 

Study, the students are taught the basic concepts of chemistry such as the kinetic theory of matter, atomic stmcture, chemical bonding, stoichiometry and chemical calculations, kinetics, energetics, oxidation-reduction, electrochemistry, as well as introductory inorgarric and organic chemistry. They also acquire basic laboratory skills as they carry out simple experiments on rates of reaction and heat of reaction, as well as volrrmetric analysis and qualitative analysis in their laboratory sessions. 

Figure 5 Stopping power for protons on He calculated with the standard basis (basis A), with two consistent bases (B and C), and in the Bethe approximation using the kinetic theory [17, 18],...

Specific heat of each species is assumed to be the function of temperature by using JANAF [7]. Transport coefficients for the mixture gas such as viscosity, thermal conductivity, and diffusion coefficient are calculated by using the approximation formula based on the kinetic theory of gas [8]. As for the initial condition, a mixture is quiescent and its temperature and pressure are 300 K and 0.1 MPa, respectively. 

Ab initio methods allow the nature of active sites to be elucidated and the influence of supports or solvents on the catalytic kinetics to be predicted. Neurock and coworkers have successfully coupled theory with atomic-scale simulations and have tracked the molecular transformations that occur over different surfaces to assess their catalytic activity and selectivity [95-98]. Relevant examples are the Pt-catalyzed NO decomposition and methanol oxidation. In case of NO decomposition, density functional theory calculations and kinetic Monte Carlo simulations substantially helped to optimize the composition of the nanocatalyst by alloying Pt with Au and creating a specific structure of the PtgAu7 particles. In catalytic methanol decomposition the elementary pathways were identified... 

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