I'm interested in solving the diffusion equation for gas in vacuum. I have a general question and a more specific questions.
What I know:
The Diffusion Equation: For density function $\phi(\vec{\mathbf{r}},t)$ the diffusion equation is: $$\frac{\partial}{\partial t} \phi(\vec{\mathbf{r}},t) = D \nabla^{2} \phi(\vec{\mathbf{r}},t)$$ where D is the diffusion coefficient.
For a gas of two constituent atom types, Chapman-Enskog theory predicts that the Diffusion Coefficient – at 1 atm and 300 K – will be: $$ D = \dfrac{9.65445\cdot\sqrt{1/M_{1} + 1/M_{2}}}{\sigma_{12}^{2}\Omega} $$ where $\sigma_{12}^{2}$ is the average of the collision diameters for the two gasses ($\sigma_{1}$ and $\sigma_2$) in Angstroms and $\Omega$ is a temperature-dependent collision integral (apparently, usually on order 1).
What I don't know
How would one calculate the diffusion coefficient if instead of a gas of two atoms, we have a gas of just one atom type diffusing?
General Question
Let's say we know that at time t=0 we have a gas of one type of atom confined to a volume $V_c$ in the corner of a box of size $V_b$ (with $V_c$ = $V_b / 1000$), we let it diffuse out of the corner, and we know the time at which the gas diffuses to uniformly fill the box of size $V_b$ (t=T). How could we calculate the proportion of the room that is filled by gas at T/2 seconds?
Specific question
Let's say we're dealing with Argon Gas at a density of 2 g/cm^3 (g/ml) or 0.002g/m^3 initially in a 100 ml container. The diffusion time to to fill a 100 cubic meter box at vacuum is 56 seconds. I know that the initial and final density could be defined as follows, with $\phi(\vec{\mathbf{r},t})$ expressed in cartesian coordinates with the corner the gas stars in defined as the origin, and the box defined as the positive quadrant from $x,y,z=0…10m$: $$ \phi(\vec{\mathbf{r}},0) = 0.002 \ [g/m^3] \ \text{for} \ x,y,z < 0.01 \ [m] $$ and $$ \phi_{f}(\vec{\mathbf{r}},56) = 0.02 [g/m^3] \ \text{for} \ x,y,z < 10 \ [m] $$
How could I analytically/numerically solve for the density as a function of time, and are there any free software packages I could use to solve for and plot this?
Best Answer
The expansion of a gas into the vacuum is not a diffusive process. Depending on the initial conditions (density and density gradient) the expansion is either described by the Euler/Navier-Stokes equation of fluid dynamics, or the Boltzmann equation of kinetic theory.
For the parameters that you mention the gas is sufficiently dense that most of it is in the regime of short mean free paths, and the expansion is described by the Euler equation (or the Navier-Stokes equation, if you would like to keep dissipative corrections). The Euler description breaks down near the dilute edge (and this is difficult to treat correctly), but this is not where most of the particles are.
If the initial density has a sharp edge, then the expansion takes the form of shock wave (in 1d, this is the Riemann problem). This is discussed in standard text books on fluid dynamics. If the initial density is smooth (say a Gaussian), then the expansion is approximately described by self-similar scaling solutions of the Euler equation.
Order of magnitude estimates can be obtained just from energy conservation. The initial energy is $E=\frac{3}{2}NT_0$ (for an ideal gas). The expansion cools the gas and converts this into kinetic energy, $$ E=\int d^3x \frac{1}{2}\rho u^2. $$ This means that after some initial acceleration phase the gas moves with mean velocity $u_{rms}=\sqrt{3T_0/m}$. This is not surprising, it says that the gas moves (approximately) with the speed of sound in the initial cloud. However, it is clearly different from a diffusive front, which slows down with time (the mean position goes as $\langle x^2\rangle^{1/2}\sim \sqrt{t}$).