Kinetic Theory: An Introduction 

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The Chapman-Enskog kinetic theory starts with the Boltzman Equation, then, given a model for molecular interaction, proceeds through various derivations to arrive at the equations of continuity, motion, and energy: equations of particular importance in fluid transport. 

 

The Boltzman Equation: For a low density, monatomic gas 

 

> Diff(f[alpha],r)=-(Diff(r[alpha]^dot*f[alpha],r))-(Diff(g[alpha]*f[alpha],r[alpha]^dot))+J[alpha];
 

(Typesetting:-mprintslash)([Diff(f[alpha], r) = -(Diff(r[alpha]^dot*f[alpha], r))-(Diff(g[alpha]*f[alpha], r[alpha]^dot))+J[alpha]], [Diff(f[alpha], r) = -(Diff(r[alpha]^dot*f[alpha], r))-(Diff(g[alph... 

 

Here:  f[alpha] is the probable number of molecules in a differential sphere, `∂`(r), with a differential volume,  g[alpha] contains all external forces acting on f[alpha], and J[alpha] is a complicated term involving the intermolecular potential energy (e.g. Lennard-Jones Potential). 

    

 

When these quantities are multiplied by a molecular property, Ψ_α, and integrated over all velocities.  When Ψ_α is a property that is conserved during a collision, such as momentum, then the J[alpha] term disappears.  When we integrate over mass, momentum, and energy, we arrive at the equations of change for mass, energy, and momentum, which lead directly to the equations of continuity, motion, and energy.  During that procedure, the three primary Transport Properties are found. 

 

Transport Properties as functions of the "Collision Integrals" Omega[D], Omega[u], Omega[k].       

 

D - Self-diffusivity 

> d:= 3/8*sqrt(Pi*m*kappa*T)/(Pi*sigma^2*Omega[D]*rho);
 

(Typesetting:-mprintslash)([d := 3/8*(Pi*m*kappa*T)^(1/2)/(Pi*sigma^2*Omega[D]*rho)], [3/8*(Pi*m*kappa*T)^(1/2)/(Pi*sigma^2*Omega[D]*rho)]) 

μ - Viscosity 

> mu := 5/16*sqrt(Pi*m*kappa*T)/(Pi*sigma^2*Omega[u]);
 

(Typesetting:-mprintslash)([mu := 5/16*(Pi*m*kappa*T)^(1/2)/(Pi*sigma^2*Omega[u])], [5/16*(Pi*m*kappa*T)^(1/2)/(Pi*sigma^2*Omega[u])]) 

k - Thermal Conductivity 

> K:=(25)/32*(sqrt(Pi*m*kappa*T))/(Pi*sigma^2*Omega[k])*(C[V]);
 

(Typesetting:-mprintslash)([K := 25/32*(Pi*m*kappa*T)^(1/2)*C[V]/(Pi*sigma^2*Omega[k])], [25/32*(Pi*m*kappa*T)^(1/2)*C[V]/(Pi*sigma^2*Omega[k])]) 

We are focusing on calculating the viscosity of gases, and are concerned primarily with Ωu.  We are looking to modify the Lennard-Jones potential to incorporate polar interactions. 

 

Applicability 

 

This approach is valid for low density, monatomic, non-polar molecules, but accounting for each of these assumptions can be made either by introducing correction factors or with sligh changes in the equations' forms.