Kinetic theory, evaporation-condensation

It must also be realized that this thin surface region is in a very turbulent state. Since the liquid is in equilibrium with its vapor, then, clearly, there is a two-way and balanced traffic of molecules hitting and condensing on the surface from the vapor phase and of molecules evaporating from the surface into the vapor phase. From the gas kinetic theory, the number of moles striking 1 cm of surface per second is... 

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... 

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... 

Figure 16. Schematic illustration of envelopes of gas species i, in this case Mg, surrounding a volatilizing molten chondrule in space. The size of the gas envelope is a function of ambient background pressure P by virtue of the effect that pressure has on the gas molecule diffusivity D,. The diffusion coefficient can be calculated from the kinetic theory of gases, as shown. The level of isotopic fractionation associated with volatilization of the molten chondrule depends upon the balance between the evaporative flux J vap and the condensation flux Tom When the fluxes are equal (i.e., when = 0), there is no mass-dependent isotope fractionation associated with volatilization. This will be the case when the local partial pressure P, approaches the saturation partial pressure P,...

In marked contrast, the classical continuum theory by mullins describes the sim-ulational data (profile shapes and amplitude decay) above roughening for wires even with small geometries surprisingly well, both for surface diffusion and evaporation-condensation The agreement may be a little bit fortuituous, because of a compensation of the competing effects of the anisotropic surface tension and anisotropic mobility, whereas continuum theory assumes isotropic quantities. In any event, the predicted decay laws with w= 1/4 for surface diffusion and w= 1/2 for evaporation kinetics are readily reproduced in the simulations. 

The first theoretical equation relating the quantity of gas adsorbed to the equilibrium pressure of the gas was proposed by Langmuir [16]. Using the kinetic theory of gases, Langmuir balanced the rates of condensation and evaporation of the gas molecules at the solid surface, giving the volume adsorbed, V, as a fimetion of the gas pressure, P ... 

This is apparent if we consider the number of molecules that must evaporate each second from the surface to maintain the vapor pressure. At equilibrium, the number of liquid molecules that evaporate into the gas phase is equal to the number of gas molecules that condense at the liquid surface (which will take place in the intermediate phase). The number of molecules hitting the liquid surface is considered to condense irreversibly." From the kinetic theory of gases, this number can be estimated as follows ... 

Based on gas-kinetic theory, the net evaporation rate is taken to be the difference between the evaporation and condensation rates [33] and is expressed... 

Young-Ping Pao Application of Kinetic Theory to the Problem of Evaporation and condensation, P.Fluids, Vol.14, No.2,... 

Consider a semi-infinite expanse of an initially uniform gas bounded by its plane condensed phase. Depending on the conditions of the gas and the condensed phase, condensation or evaporation will take place on the condensed phase the disturbance induced by their interaction will propagate in the gas and after a long time a steady condensation or evaporation flow will be established. The senior author (Y. S.) considered the problem on the basis of kinetic theory in Ref. 1 when condensation takes place. In Ref. 1 the behavior of the gas is analyzed numerically by a finite difference method for a large number of initial situations, from which the transient behavior to a final steady state is classified and the steady behavior, especially the relation satisfied among the parameters at infinity and on the condensed phase in a condensation flow, is clarified. 

Various flow problems involving evaporation and condensation phenomena are quite common in ordinary circumstances and have aroused an interest of scientists not only in the field of fluid dynamics but also of kinetic theory. The reason for this is that the ordinary continuum-based fluid dynamics cannot describe qualitatively correctly the process of evaporation and condensation occurring at the interface even in ihe continuum limit because of the existence of a nonequilibrium region, the thickness of which is of the order of the molecular mean free path, in the close vicinity of the interface between the condensed phase and the gas phase. Such a nonequilibrium region is called the Knudsen layer, in which collisions between molecules are not so frequent that the momentum and energy exchanges between the molecules leaving the interface... 

Sone, Y. Onishi, Y. Kinetic theory of evaporation and condensation - Hydrodynamic equation and slip boundary condition -. J. Phys. Soc. Japan 44 (1978)... 

From the kinetic theory of gases, an expression for the net-mass flux at the interphase can be derived based on the works of Hertz [223] and Knudsen [224]. From a statistical consideration under the assumption of a Maxwell-Boltzmann distribution for the velocity of the gas molecules, the maximum condensation mass flux can be calculated. The evaporation mass flux has to equal the condensation mass flux at equilibrium. The resulting Hertz-Knudsen equation for calculating the area specific net-mass flux is given below ... 

If 6 is the fraction of the total catalyst surface covered by adsorbed molecules at any instant, then the fraction of bare catalyst surface available for adsorption is (1 — 6). According to kinetic theory, since the rate at which molecules strike a unit area of a surface is proportional to the pressure of the gas, the rate of condensation of molecules should be determined both by the partial pressure and the fraction of bare catalyst surface or i(l — 6)p, where ki is a proportionality constant. If 2 is the rate at which molecules evaporate from a unit surface when the surface is fully covered, then for a fraction 0 of a fully covered surface, the rate of evaporation will be k2d. For adsorption equilibrium, these rates must be equal. Therefore,... 

Dubrovskiy, G.V. (1991b) Microscopic Models of Heat and Mass IVansfer During Adsorption Film Growth , in Proc. of Int. School-Seminar on Kinetic Theory of Transport Processes in Evaporation and Condensation (Lykov Heat and Mass Transfer Inst., Minsk) pp. 38-50 (in Russistn)... 

Kinetic-molecular theory provides an explanation on a molecular level for this equilibrium. Evaporation from the liquid occurs as fast moving molecules on the surface escape from the liquid. In turn, molecules in the gas phase strike the liquid and condense, As the concentration (pressure) of gas molecules builds up in the gas phase, the rate of condensation increases. Eventually, a pressure is reached where the rate of condensation and rate of evaporation just balance, and equilibrium is achieved. The equilibrium pressure is denoted by p and is known as the vapor pressure. The magnitude ofp depends upon the substance, composition of the liquid, and any two of our thermodynamic variables such as temperature and total pressure. The criteria for equilibrium that we will now derive provide the thermodynamic relationships that will help... 

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