Statistical Thermodynamics and Rate Theories/Diffusion

Contents

Diffusion

Diffusion is the movement of chemicals from an area of high concentration to an area of lower concentration due to random movement of particles. Molecules diffuse so that they are spread out evenly in they space they are occupying, which can effect where a chemical reaction takes place.

Diffusion is pressure,temperature, and mass dependent. Diffusion is slower at high pressure and for heavier particles, and diffusion is faster at high temperatures.

Fick's Law

Fick's law relates the concentration gradient to the rate of diffusion. In other words, the rate of flow of a molecule is related to the concentration gradient of that species and its diffusion coefficient. The equation for the rate of diffusion in one dimension is

$J=-D\operatorname {d} \!\rho _{N}/\operatorname {d} \!x$ where D is the diffusion coefficient, and ρN is the number density of the particle.

For particular boundary conditions, Fick's law can be integrated to give the position-dependent number density of the diffusing gases at a function of time.

$\rho _{N}(x,t)=\rho _{N,0}{\text{erfc}}({\sqrt {x/2\surd (Dt)}}$ Diffusion of two Gases

If there are two gases in a container with different masses and velocities, the diffusion coefficients of the gases will be

$D_{N}=1/3l_{N}(v_{N})$ where l is the mean free path length of particle N, and vN is the average velocity of particle N.

The mean free path length of two component forms is complex due to the fact that more than one type of collision can occur, and the collision cross section of each species is involved,

$l_{1}=4/(4\pi ^{2}\sigma _{1}^{2}\rho _{1,N}+\pi (\sigma _{1}+\sigma _{2})^{2}\rho _{2,N})$ where σ is the collision cross section.

Combining the diffusion coefficient equation and the mean free path length equation, the diffusion coefficient for one particle becomes,

$D_{1}=1/3(4/(4\pi ^{2}\sigma _{1}^{2}\rho _{1,N}+\pi (\sigma _{1}+\sigma _{2})^{2}\rho _{2,N}))(v_{N})$ or,

$D_{1}=1/3(8k_{B}T/\pi m_{2})^{1/2}(4/(4\pi ^{2}\sigma _{1}^{2}\rho _{1,N}+\pi (\sigma _{1}+\sigma _{2})^{2}\rho _{2,N}))$ Self Diffusion

The self diffusion of a gas occurs when there is only one component in the system. Self diffusion occurs constantly as a gas molecule moves around in it's container. It can be observed by isotopic labeling. Since the gas molecules are the same the equation for the diffusion constant simplifies to

$D_{simple}={\frac {1}{3}}\left({\frac {2}{\pi }}\right)^{3/2}\left({\frac {k_{B}T}{m}}\right)^{1/2}{\frac {1}{\rho _{N}\sigma ^{2}}}$ or,

$D_{collision}={\frac {3}{8\pi ^{1/2}}}\left({\frac {k_{B}T}{m}}\right)^{1/2}{\frac {1}{\rho _{N}\sigma ^{2}}}$ for the diffusion coefficient in terms of collisions.