(i) In an experiment, a finite amount $M$ of marker gas of diffusivity $D$ is released at time $t=0$ into an infinite tube in the neighbourhood of the origin $x=0$. Starting from the one-dimensional diffusion equation for the concentration $C(x, t)$ of marker gas,

$$C_{t}=D C_{x x}$$

use dimensional analysis to show that

$$C=\frac{M}{(D t)^{1 / 2}} f(\xi)$$

for some dimensionless function $f$ of the similarity variable $\xi=x /(D t)^{1 / 2}$.

Write down the equation and boundary conditions satisfied by $f(\xi)$.

(ii) Consider the experiment of Part (i). Find $f(\xi)$ and sketch your answer in the form of a plot of $C$ against $x$ at a few different times $t$.

Calculate $C(x, t)$ for a second experiment in which the concentration of marker gas at $x=0$ is instead raised to the value $C_{0}$ at $t=0$ and maintained at that value thereafter. Show that the total amount of marker gas released in this case becomes greater than $M$ after a time

$$t=\frac{\pi}{16 D}\left(\frac{M}{C_{0}}\right)^{2} .$$

Show further that, at much larger times than this, the concentration in the first experiment still remains greater than that in the second experiment for positions $x$ with $|x|>$ ${ }_{4} C_{0} D t / M$.

[Hint: $\operatorname{erfc}(z) \equiv \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-u^{2}} d u \sim \frac{1}{\sqrt{\pi} z} e^{-z^{2}}$ as $\left.z \rightarrow \infty .\right]$