Concentration of Proteins

Question

You release a billion protein molecules at position $x = 0$ in the middle of a narrow capillary test tube. The molecules’ diffusion constant is $10^{−6} \ cm^2 s^{−1}$. An electric field pulls the molecules to the right (larger x) with a drift velocity of $1\ μms^{−1}$. Nevertheless, after $80 \ s$ you see that a few protein molecules are actually to the left of where you released them. How could this happen? What is the ending concentration exactly at $x = 0$?

[Note: This is a one-dimensional problem, so you should express your answer in terms of the concentration integrated over the cross-sectional area of the tube, a quantity with dimensions $L ^{−1}.$]

In this problem I tried to use diffusion equation $$ \frac{\partial C}{\partial t} = D \frac{\partial ^2 C}{\partial x^2} + v\frac{\partial C}{\partial x} $$ where $v$ is drift velocity.

Firstly, does this equation help me, or am I in the wrong woods? Secondly, I have trouble finding the initial conditions, since we have two parameters.

Answer 1

You will consider green's equation $$c(x, t) = (N / √4πDt) e^{−x2/4Dt}$$

And $N$ is small at $x=0$ and divide the equation by $N$ and using drift velocity $v=xt$ find the $x$

plug in all given values $D$,$t$,$x$ and found to be $2600$ particles per metre or $0.0026$ particles per micrometer