Deriving expression for steady state flux and concentration: questions relating to diffusion Can anyone tell me where to begin?
How do I find the expression for steady state flux and steady state concentration for example?
What assumed knowledge is implicit in the question?
What common mathematical facts are relevant?

Question.

(4.) Consider a substance diffusing in one dimension with diffusivity $D$ from $x=0$ where the concentration is maintained at $c(0,t)=c_0$ to $x=L$ where the concentration is maintained at $c(L,t) = 0$ (i.e., the substance is removed as soon as it gets to $x = L$). 
(a) Find an expression for the steady state flux and the steady state concentration.
(b) Find an expression for the total amount of substance $m$ in the region $(0,L)$ in steady state.
(c) The average transit time $\tau$ from $x=0$ to $x=L$ can be estimated as the time for the total amount $m$ to leave the region given the flux $q$ for the amount that leaves per unit time, i.e., $\tau = m/q$. Show how this estimate of $\tau$ relates to the mean square displacement.
(d) A typical neurotransmitter has a diffusivity $\approx 10^{-6} \mathrm{cm}^2 \mathrm{s}^{-1}$. How long does it take the neurotransmitter to diffuse across a synaptic junction that is about $0.02$ micron. How does this synaptic time delay compare with the typical speed of a neutron pulse ($\approx 27 \mathrm{m}\mathrm{s}^{-1}$).
(e) NEW UNANSWERED QUESTION: The concise edition of the Encyclopedia Britannica http://concise.britannica.com/ebc/article-9030421/diffusion [sic] defines diffusion as the "process resulting from random motion of molecules by which there is a net flow of matter from a region of high concentration to a region of low concentration.  A familiar example is the perfume of a molecule that quickly permeates the still air of a room.
    A typical perfume molecule has a diffusivity of $\approx 10^{-5} m^2 s^{-1}$.  How long would it take a typical perfume molecule to diffuse across the still air of a room that is $\approx 10m$ across?

To be helpful, please explain the solution thoroughly in a way that a beginner can follow.
 A: This is an exercise in Fick's law, that postulates the following formula for the diffusion flux: 
$$
q = - D \frac{\partial c}{\partial x}, \tag{$\ast$}
$$ 
where 


*

*$q$ is the diffusion flux, i.e., the amount of substance that flows through the cross-section at a given position. To prevent accumulation or draining of material at any $x \in (0, L)$, we assume that this flux is the same, say $q$, at all points. 

*$D$ is the diffusivity (again given in the question).

*$c(x)$ is the concentration at time $t$. I am showing only the dependence on $x$, not on $t$. 
It just remains to solve $(\ast)$. In steady state, $c$ depends on $x$, not on $t$. Therefore,
$$
\frac{dc}{dx} = -\frac{q}{D},
$$
which gives $c = C - \frac{q}{D} x$ for some constant $C$. We are yet to use the two boundary conditions:


*

*Setting $c=c_0$ at $x=0$ gives $C = c_0$, which implies that $c(x) = c_0 - \frac{qx}{D}$. 

*Setting $c = 0$ at $x=L$ gives $c_0 = \frac{qL}{D}$. You are supposed to solve for the flux $q$ in terms of the other constants $c_0$, $L$ and $D$.  


Plugging this back in $(\ast)$, we get $$c = c_0 \left( 1 - \frac{x}{L} \right) .$$

(a) The steady state flux is $q = \cdots$, and the state state concentration is $c(x) = \cdots$.
(b) The total amount of substance at steady state is
$$
m = \int_0^L c(x) ~dx = \int_{0}^L c_0 \left( 1 - \frac{x}{L} \right) ~dx = \cdots.
$$
(c) You can find the average transit time $\tau$ from $\tau = m / q = \cdots$. 
Finally, I believe the mean square displacement is given by
$$
\frac{\int_{0}^{L} c(x) x^2 ~ dx}{\int_0^L c(x) ~ dx} = \cdots.
$$
