Kinetic theory, definition of temperature

Assuming that the kinetic theory definition of temperature and the thermodynamic counterpart are consistent, the entropy variable may be considered a function of p and T as obtained combining the first and second law of thermodynamics [32]. The entropy of the gas is then given by [12] (p. 41) ... 

Equation (3.56) is the dilute gas kinetic theory definition of temperature where k is Boltzmann s constant. From these relationships, the constants in Eq. (3.53) can be uniquely determined, leading to (Prob. 3.10)... 

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. 

Equation (1-1) is the defining equation for thermal conductivity. On the basis of this definition, experimental measurements may be made to determine the thermal conductivity of different materials. For gases at moderately low temperatures, analytical treatments in the kinetic theory of gases may be used to predict accurately the experimentally observed values. In some cases, theories are available for the prediction of thermal conductivities in liquids and solids, but in general, many open questions and concepts still need clarification where liquids and solids are concerned. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

This chapter covers the second fundamental concept used in chemical reaction engineering—chemical kinetics. The kinetic relationships used in the analysis and design of chemical reactors are derived and discussed. In Section 3.1, we discuss the various definitions of the species formation rates. In Section 3.2, we define the rates of chemical reactions and discuss how they relate to the formation (or depletion) rates of individual species. In Section 3.3, we discuss the rate expression that provides the relationship between the reaction rate, the temperature, and species concentrations. Without going into the theory of chemical kinetics, we review the common forms of the rate expressions for homogeneous and heterogeneous reactions. In the last section, we introduce and define a measure of die reaction rate—the characteristic reaction time. In Chapter 4 we use the characteristic reaction time to reduce the reactor design equations to dimensionless forms. 

The diffusivity, or diffusion coefficient, /) is a property of the system dependent upon temperature, pressure, and nature of the components. An advanced kinetic theory [12] predicts that in binary mixtures there will be only a small effect of composition. The dimensions of diffusivity can be established from its definition, Eq. (2.1), and are length /time. Most of the values for D reported in the literature are expressed as cm /s the SI dimensions are m /s. Conversion factors are listed in Table 1.5. A few typical data are listed in Table 2.1 a longer list is available in The Chemical Engineers Handbook [18]. For a complete review, see Ref. 17. 

A definite prediction of DLVO theory is that charge-stabilized colloids can only be kinetically, as opposed to thermodynamically, stable. The theory does not mean anything at all if we cannot identify the crystalline clay state (d 20 A) with the primary minimum and the clay gel state (d 100 to 1000 A) with the secondary minimum in a well-defined model experimental system. We were therefore amazed to discover a reversible phase transition of clear thermodynamic character in the n-butylammonium vermiculite system, both with respect to temperature T and pressure P. These results rock the foundations of colloid science to their roots and... 

According to the kinetic-molecular theory, a mole of solid particles has as much kinetic energy as a mole of liquid particles at the same temperature. By definition, the particles in a solid must be in constant motion. So why do solids have a definite shape and volume For a substance to be a solid rather than a liquid at a given temperature, there must be strong attractive forces acting between particles in the solid. These forces limit the motion of the particles to vibrations around fixed locations in the solid. Thus, there is more order in a solid than in a liquid. Because of this order, solids are much less fluid than liquids and gases. In fact, solids are not classified as fluids. 

In the absence of bulk flow one is led by arguments using a d)mamic version of the random phase approximation to an alternative expression to equation (4.4.11), in which it is the slow chains that dominate the kinetics (Brochard et al. 1983). This is sometimes referred to as the slow theory . There have been suggestions (Akcasu et al. 1992) that mutual diffusion should approach the slow-theory limit as the temperature is reduced towards the glass transition temperature of one of the components. A definitive solution to this problem awaits more work, both theoretical and experimental. 

The Zener theory of the incomplete transformation phenomenon sometimes attending the bainite reaction in steel, as described by the overall reactions kinetics definition, has been shown by [2001Aar] to be able to describe neither the upper temperature limit for bainite formation in C-Fe-Mo alloys nor the average carbon concentration in retained austenite at the beginning of incomplete transformation. 

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