Pressure, kinetic theory

McLeod gages, we propose to use the reactivity of zirconium as a means to estimate the true pressures within the tube. We call this method the measurement of the apparent pressure. Kinetic theory allows one to formulate an expression for the weight gain of a sample of zirconium of given surface, as a function of the temperature, pressure, molecular weight of the reacting gases, and the efficiency of the surface reaction. 

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. 

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. ... 

As a consequence of these simple deductions, Graham s experiments c effusion through an orifice came to be regarded as one of the earliest direct experimental checks on the kinetic theory of gases. However, a closer examination of his experimental conditions reveals that this view is mistaken. As mentioned earlier, his orifice diameters ranged upwards from 1/500 in., while the upstream pressure was never very much less thai atmospheric. Under these circumstances the molecular mean free path len ... 

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... 

During the nineteenth century the growth of thermodynamics and the development of the kinetic theory marked the beginning of an era in which the physical sciences were given a quantitative foundation. In the laboratory, extensive researches were carried out to determine the effects of pressure and temperature on the rates of chemical reactions and to measure the physical properties of matter. Work on the critical properties of carbon dioxide and on the continuity of state by van der Waals provided the stimulus for accurate measurements on the compressibiUty of gases and Hquids at what, in 1885, was a surprisingly high pressure of 300 MPa (- 3,000 atmor 43,500 psi). This pressure was not exceeded until about 1912. 

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. 

Electron spectroscopic techniques require vacuums of the order of 10 Pa for their operation. This requirement arises from the extreme surface-specificity of these techniques, mentioned above. With sampling depths of only a few atomic layers, and elemental sensitivities down to 10 atom layers (i. e., one atom of a particular element in 10 other atoms in an atomic layer), the techniques are clearly very sensitive to surface contamination, most of which comes from the residual gases in the vacuum system. According to gas kinetic theory, to have enough time to make a surface-analytical measurement on a surface that has just been prepared or exposed, before contamination from the gas phase interferes, the base pressure should be 10 Pa or lower, that is, in the region of ultrahigh vacuum (UHV). 

As we have implied, diffusion is a rather complex process so far as molecular motion is concerned. Effusion, the flow of gas molecules at low pressures through tiny pores or pinholes, is easier to analyze using kinetic theory. 

Pressure is an important quantity in a discussion of gas behavior. The applicability of the kinetic theory to an understanding of gas pressure is, then, an important success (see Section 2-1.1). We shall investigate this success in more detail, but first we should investigate how pressure is measured. 

The pressure behavior shown in Figure 4-3 is readily explained in terms of the kinetic theory of gases. There is so much space between the molecules that each behaves independently, contributing its share to the total pressure through its occasional collisions with the container walls. The water molecules in the third bulb are seldom close to each other or to molecules provided by the air. Consequently, they contribute to the pressure exactly the same amount they do in the second bulb—the pressure they would exert if the air were not present. The 0.0011 mole of water vapor contributes 20 mm of pressure whether the air is there or not. The 0.0050 mole of air contributes 93 mm of pressure whether the water vapor is there or not. Together, the two partial pressures, 20 mm and 93 mm, determine the measured total pressure. 

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. 

There are great advantages to an absolute temperature scale that has its zero point at — 273°C. Whereas the zero of temperature in the Centigrade scale is based upon an arbitrary temperature, selected because it is easily measured, the zero point of the absolute scale has inherent significance in the kinetic theory. If we express temperatures on an absolute temperature scale, we find that the volume of a fixed amount of gas (at constant pressure) varies directly with temperature Also, the pressure of a fixed amount of (at constant volume) varies directly with temperature. And, according to the kinetic theory, the kinetic energy of the molecules varies directly with the absolute temperature. For these reasons, in dealing with gas relations, we shall usually express temperature on an absolute temperature scale. 

There is a reasonable explanation for this type of deviation. The kinetic theory, which explains the pressure-volume behavior, is based upon the assumption that the particles exert no force on each other. But real molecules do exert force on each other The condensation of every gas on cooling shows that there are always attractive forces. These forces are not very important when the molecules are far apart (that is, at low pressures) but they become noticeable at higher pressures. With this explanation, we see that the kinetic theory is based on an idealized gas—one for which the molecules exert no force on each other whatsoever. Every gas approaches such ideal behavior if the pressure is low enough. Then ihe molecules are, on the average, so far apart that then-attractive forces are negligible. A gas that behaves as though the molecules exert no force on each other is called an ideal gas or a perfect gas. 

Avogadro s Hypothesis is consistent with the kinetic theory. Therefore a perfect gas follows Avogadro s Hypothesis. At one atmosphere pressure and 0°C, one mole (6.02 X 10 molecules) of a perfect gas occupies 22.414 liters. How closely real gases approximate a perfect gas at one atmosphere pressure and 0°C is shown by measur-... 

We have explored the meaning of temperature. According to the kinetic theory, when two gases are at the same temperature, the molecules of the two gases have the same average kinetic energies. Changing the temperature of a sample of gas at constant pressure reveals that the volume is di-... 

Why does the pressure build up in a tire on a hot day Answer in terms of the kinetic theory. 

Kinetic theory A theory of matter based on the mathematical description of the relationship between pressures, volumes, and temperatures of gases (PVT phenomena). This relationship is summarized in the laws of Boyle s law, Charle s law, and Avogadro s law. 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

It has been assumed in the deduction of (1) that the solute is an ideal gas, or at least a volatile substance. The extension of the result to solutions of substances like sugar, or metallic salts, must therefore be regarded as depending on the supposition that the distinction between volatile and non-volatile substances is one of degree rather than of kind, because a finite (possibly exceedingly small) vapour pressure may be attributed to every substance at any temperature above absolute zero. This assumption is justified by the known continuity of pleasure in measurable regions, and by the kinetic theory of gases. 

The kinetic theory of gases shows that the pressure p exerted by a gas is given by ... 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

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. 

The ordinary multicomponent diffusion coefficients D j and the viscosity and thermal conductivity are computed from appropriate kinetic theory expressions. First, pure species properties are computed from the standard kinetic theory expressions. For example, the binary diffusion coefficients are given in terms of pressure and temperature as... 

Conventional bulk measurements of adsorption are performed by determining the amount of gas adsorbed at equilibrium as a function of pressure, at a constant temperature [23-25], These bulk adsorption isotherms are commonly analyzed using a kinetic theory for multilayer adsorption developed in 1938 by Brunauer, Emmett and Teller (the BET Theory) [23]. BET adsorption isotherms are a common material science technique for surface area analysis of porous solids, and also permit calculation of adsorption energy and fractional surface coverage. While more advanced analysis methods, such as Density Functional Theory, have been developed in recent years, BET remains a mainstay of material science, and is the recommended method for the experimental measurement of pore surface area. This is largely due to the clear physical meaning of its principal assumptions, and its ability to handle the primary effects of adsorbate-adsorbate and adsorbate-substrate interactions. 

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