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The way the particles diffuse depends on the dimensions of the system. In a one-dimensional system there is only one direction and with it less dilution which facilitates diffusion. In a two-dimensional system and more in three-dimensional system there is the possibility of lateral displacements and therefore slower diffusion.

ID:(1022, 0)

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ID:(15302, 0)

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Equation

$c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$

c(r)=(c_1ln(r_2/r)+c_2ln(r/r_1))/ln(r_2/r_1)

$c(r)=\displaystyle\frac{r_1c_1(r_2-r)+r_2c_2(r-r_1)}{r(r_2-r_1)}$

c(r)=(r_1c_1(r_2-r)+r_2c_2(r-r_1))/(r(r_2-r_1))

$c(x)=c_1+\displaystyle\frac{c_2-c_1}{x_2-x_1}(x-x_1)$

c(x)=c_1+(c_2-c_1)(x-x_1)/(x_2-x_1)

$c(x,t)=\displaystyle\frac{1}{2}c_0\left(\textrm{erfc}\displaystyle\frac{h-x}{2\sqrt{Dt}}+\textrm{erfc}\displaystyle\frac{h+x}{2\sqrt{Dt}}\right)$

c(x,t)=c_0/2 (erfc(h-x/2 sqrt(Dt))+erfc(h+x/2 sqrt(Dt)))

$c(x,t)=\displaystyle\frac{1}{2}c_0\textrm{erfc}\displaystyle\frac{x}{2\sqrt{Dt}}$

c(x,t)=c_0/2 erfc(x/2 sqrt(Dt))

$c(x,t)=\displaystyle\frac{M}{\sqrt{\pi Dt}}e^{-x^2/4Dt}$

c(x,t)=M/sqrt(pi Dt) e^(-x^2/4Dt)

$\displaystyle\frac{\partial c}{\partial t}=\displaystyle\frac{D}{r}\displaystyle\frac{\partial}{\partial r}\left( r\displaystyle\frac{\partial c}{\partial r} \right)$

dc/dt=(D/r)d(rdc/dr)/dr

$\displaystyle\frac{\partial c}{\partial t}=D\left(\displaystyle\frac{\partial^2c}{\partial r^2}+\displaystyle\frac{2}{r}\displaystyle\frac{\partial c}{\partial r} \right)$

dc/dt=D(d^2c/dr^2+(2/r)dc/dr)

$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$

dc/dt=Dd^2c/dx^2

$j=-D\displaystyle\frac{c_2-c_1}{x_2-x_1}$

j=-D(c_2-c_1)/(x_2-x_1)

$J=\displaystyle\frac{2\pi D(c_2-c_1)}{\ln(r_2/r_1)}$

J=2pi D(c_2-c_1)/ln(r_2/r_1)

$J=4\pi D\displaystyle\frac{r_1r_2}{r_2-r_1}(c_2-c_1)$

J=4pi Dr_1r_2(c_2-c_1)/(r_2-r_1)

ID:(15360, 0)

Diffusion Equation, 1D

Equation

The temporal and spatial evolution of concentration $c$ in one dimension is governed by the equation:

$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$

where $D$ is the diffusion constant.

ID:(8381, 0)

Solution, 1D, stationary

Equation

The solution to the equation

$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$

for the steady-state case with concentration $c_1$ at position $x_1$ and concentration $c_2$ at position $x_2$ yields the following distribution:

$c(x)=c_1+\displaystyle\frac{c_2-c_1}{x_2-x_1}(x-x_1)$

ID:(8388, 0)

Solution, 1D, stationary, flow

Equation

With the solution

$c(x)=c_1+\displaystyle\frac{c_2-c_1}{x_2-x_1}(x-x_1)$

and the Fick's law equation

$j =- D \displaystyle\frac{ dc_n }{ dz }$

the flux is calculated as follows:

$j=-D\displaystyle\frac{c_2-c_1}{x_2-x_1}$

ID:(8389, 0)

Solution, 1D, point

Equation

The solution to the equation

$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$

for the case of a point concentration $c$ (Dirac delta) with a total volume of $M$ is as follows:

$c(x,t)=\displaystyle\frac{M}{\sqrt{\pi Dt}}e^{-x^2/4Dt}$

ID:(8383, 0)

Solution, 1D, non-point zone

Equation

The solution to the equation

$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$

for the case of a concentration $c$ in a semi-infinite system with a fixed concentration at the origin $c_0$ is as follows:

$c(x,t)=\displaystyle\frac{1}{2}c_0\left(\textrm{erfc}\displaystyle\frac{h-x}{2\sqrt{Dt}}+\textrm{erfc}\displaystyle\frac{h+x}{2\sqrt{Dt}}\right)$

ID:(8385, 0)

Solution, 1D, semi-infinite

Equation

The solution to the equation

$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$

for the case of a concentration $c$ in a semi-infinite system with a fixed concentration at the origin $c_0$ is as follows:

$c(x,t)=\displaystyle\frac{1}{2}c_0\textrm{erfc}\displaystyle\frac{x}{2\sqrt{Dt}}$

ID:(8384, 0)

Diffusion Equation, 2D

Equation

The temporal and spatial evolution of concentration $c$ in two dimensions with rotational symmetry is governed by the equation:

$\displaystyle\frac{\partial c}{\partial t}=\displaystyle\frac{D}{r}\displaystyle\frac{\partial}{\partial r}\left( r\displaystyle\frac{\partial c}{\partial r} \right)$

where $D$ is the diffusion constant.

ID:(8382, 0)

Solution, 2D, stationary

Equation

The solution to the equation

$\displaystyle\frac{\partial c}{\partial t}=\displaystyle\frac{D}{r}\displaystyle\frac{\partial}{\partial r}\left( r\displaystyle\frac{\partial c}{\partial r} \right)$

for the steady-state case with concentration $c_1$ at radius $r_1$ and concentration $c_2$ at radius $r_2$ yields a flux as follows:

$c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$

ID:(8386, 0)

Solution, 2D, stationary, flow

Equation

For the solution

$c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$

the flux is calculated as follows:

$J=\displaystyle\frac{2\pi D(c_2-c_1)}{\ln(r_2/r_1)}$

ID:(8387, 0)

Diffusion Equation, 3D

Equation

The temporal and spatial evolution of concentration $c$ in two dimensions with rotational symmetry is governed by the equation:

$\displaystyle\frac{\partial c}{\partial t}=D\left(\displaystyle\frac{\partial^2c}{\partial r^2}+\displaystyle\frac{2}{r}\displaystyle\frac{\partial c}{\partial r} \right)$

where $D$ is the diffusion constant.

ID:(8390, 0)

Solution, 3D, stationary

Equation

The solution to the equation

$\displaystyle\frac{\partial c}{\partial t}=D\left(\displaystyle\frac{\partial^2c}{\partial r^2}+\displaystyle\frac{2}{r}\displaystyle\frac{\partial c}{\partial r} \right)$

for the steady-state case with concentration $c_1$ at radius $r_1$ and concentration $c_2$ at radius $r_2$ yields a flux as follows:

$c(r)=\displaystyle\frac{r_1c_1(r_2-r)+r_2c_2(r-r_1)}{r(r_2-r_1)}$

ID:(8391, 0)

Solution, 3D, stationary, flow

Equation

For the solution

$c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$

the flux is calculated as follows:

$J=4\pi D\displaystyle\frac{r_1r_2}{r_2-r_1}(c_2-c_1)$

ID:(8392, 0)