help

[Walter]

can someone explain what the error function has to do with fick's second law I dont get what the relationship between the two is

[Mando Pandango]

Fick's second law predicts how diffusion causes the concentration to change with time:

    \frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2}\,\!

Where

    \,\phi is the concentration in dimensions of [(amount of substance) lengthâˆ'3], example (\tfrac\mathrm{mol}{m^3})
    \, t is time (s)
    \, D is the diffusion coefficient in dimensions of [length2 timeâˆ'1], example (\tfrac{m^2}{s})
    \, x is the position [length], example \,m

It can be derived from Fick's First law and the mass balance:

\frac{\partial \phi}{\partial t} =-\,\frac{\partial}{\partial x}\,J = \frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x}\phi\,\bigg)\,\!

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying by the constant:

    \frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x} \phi\,\bigg) = D\,\frac{\partial}{\partial x} \frac{\partial}{\partial x} \,\phi = D\,\frac{\partial^2\phi}{\partial x^2}

and, thus, receive the form of the Fick's equations as was stated above.

For the case of diffusion in two or more dimensions Fick's Second Law becomes

\frac{\partial \phi}{\partial t} = D\,\nabla^2\,\phi\,\!,

which is analogous to the heat equation.

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, Fick's Second Law yields

    \frac{\partial \phi}{\partial t} = \nabla \cdot (\,D\,\nabla\,\phi\,)\,\!

An important example is the case where \,\phi is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant \, D, the solution for the concentration will be a linear change of concentrations along \, x. In two or more dimensions we obtain

    \nabla^2\,\phi =0\!

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

DUH

[Walter]

oh ok but where does the error function come in and why is it used. my professor wants us to derive the second law and explain something about the error function but I dont know why or where the error function is used

wtf someone upstairs just hulk yelled and banged on something that was weird

[Mando Pandango]

idk i just copied that from wikipedia

[Walter]

idk i just copied that from wikipedia

OH YOU SLY DOG

[bluaki]

idk i just copied that from wikipedia

At least use boyah's tex tag around the latex equations

[tex]\frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2}\,\![/tex]
    [tex]\,\phi[/tex] is the concentration in dimensions of [(amount of substance) lengthâˆ'3]
    [tex]\, t[/tex] is time (s)
    [tex]\, D[/tex] is the diffusion coefficient in dimensions of [length2 timeâˆ'1]
    [tex]\, x[/tex] is the position [length], example [tex]\,m[/tex]

[tex]\frac{\partial \phi}{\partial t} =-\,\frac{\partial}{\partial x}\,J = \frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x}\phi\,\bigg)\,\![/tex]

[tex]\frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x} \phi\,\bigg) = D\,\frac{\partial}{\partial x} \frac{\partial}{\partial x} \,\phi = D\,\frac{\partial^2\phi}{\partial x^2}[/tex]

[tex]\frac{\partial \phi}{\partial t} = D\,\nabla^2\,\phi\,\!,[/tex]

[tex]\frac{\partial \phi}{\partial t} = \nabla \cdot (\,D\,\nabla\,\phi\,)\,\![/tex]

[tex]\nabla^2\,\phi =0\![/tex]

[applesauce]

well is it homework or a project/test?

If it's homework, don't do it.

[PLEASEHELP1991]

error function is just a statistical function and thermodynamics is statistically-derived
duh.

[FAMY2]

Are you guys talking dirty to each other?