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The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State

The theory of diffusion in gases based on Boltzmann’s equation[edit]

f_i(x,c,t)
C_i(x,t)=\frac{1}{n_i}\int_c c f(x,c,t) \, dc
C_i(x,t)

In Boltzmann’s kinetics of the mixture of gases, each gas has its own distribution function, {\displaystyle f_{i}(x,c,t)}, where t is the time moment, x is position and c is velocity of molecule of the ith component of the mixture. Each component has its mean velocity {\displaystyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}. If the velocities {\displaystyle C_{i}(x,t)} do not coincide then there exists diffusion.

In the Chapman–Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:[8]

  • individual concentrations of particles, {\displaystyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc} (particles per volume),
  • density of momentum {\displaystyle \sum _{i}m_{i}n_{i}C_{i}(x,t)} (mi is the ith particle mass),
  • density of kinetic energy

{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}

{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}

The kinetic temperature T and pressure P are defined in 3D space as{\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}

{\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}
n=\sum_i n_i

where {\displaystyle n=\sum _{i}n_{i}} is the total density.

C_1-C_2

For two gases, the difference between velocities, {\displaystyle C_{1}-C_{2}} is given by the expression:[8]{\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}

{\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}
F_{i}
k_{T}

where {\displaystyle F_{i}} is the force applied to the molecules of the ith component and {\displaystyle k_{T}} is the thermodiffusion ratio.

The coefficient D12 is positive. This is the diffusion coefficient. Four terms in the formula for C1C2 describe four main effects in the diffusion of gases:

  1. {\displaystyle \nabla \,\left({\frac {n_{1}}{n}}\right)} describes the flux of the first component from the areas with the high ratio n1/n to the areas with lower values of this ratio (and, analogously the flux of the second component from high n2/n to low n2/n because n2/n = 1 – n1/n);