Is it possible to solve the equation for the diffusion of gas through a sphere analytically by applying e.g. "combination of variables"?

In the case of diffusion through a thin film it is possible to combine the variables in the following way: 
$\sigma = \frac{x}{\sqrt{4Dt}}$ 


where 


* $x$=length
* $D$=diffusion coefficient
* $t$=time
* $\sigma$=a function of x and t


Thereby, x and t is substituted by the single variable sigma


Is there a similar expression for a sphere?


E. L. Cussler discribes in detail how to solve the equation for a thin film by applying "combination of variables".




Given the diffusion equation


$D(\frac{d^2y}{dr^2}+(\frac{2}{r})\frac{dy}{dr})=\frac{dy}{dt}$


You can put in


$y=z/r$


And then


$\frac{dy}{dr}=(\frac{1}{r})(\frac{dz}{dr}-\frac{z}{r})$


$\frac{d^2y}{dr^2}=(\frac{1}{r})(\frac{d^2z}{dr^2}-(\frac{2}{r})\frac{dz}{dr}+\frac{2z}{r^2})$


$\frac{dy}{dt}=(\frac{1}{r})\frac{dz}{dt}$


Plug these into your diffusion equation and a lot of terms cancel leaving the same equation you would have in rectangular coordinates:


$D\frac{d^2z}{dr^2}=\frac{dz}{dt}$


So you then use the same combination of variables as in the rectangular case except you put $z=ry$ as your dependent variable.


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Yes. Let's improve the wording here: that is not just an expression, it's the definition of a dimensionless variable that eases the understanding of the solution of the transient diffusion is a semi-infinite medium. Because there is no clear characteristic length as in a finite diffusion path problem, Buckingham π theorem will tell us the two dimensionless quantities we need are
$$\Pi\_1 = \frac{z}{\sqrt{D t}}$$ (Or any function of it, like squaring it or dividing it by 4, such as your $\sigma$)


And
$$\Pi\_2 = \frac{y-y\_{A\infty}}{y\_{As} - y\_{A\infty}}$$


With $s$ being the surface index and $\infty$ indicating a position "at infinity" (often we use $y\_{A\infty} = 0$).


Then we conclude $\Pi\_2 =\textrm{function}(\Pi\_1)$. This particular function can found by solving Fick's second law with the appropriate boundary and initial conditions. In the most common case, it yields a Gauss error function.


Had we be initially interested in a sphere instead of a planar surface, the only major difference would be the use of spherical coordinates instead of a Cartesian frame, which would result in a different functional solution. But the problem of defining the dimensionless variables would be the same, given we assume perfect angular symmetry. The $z$ position would just be substituted by a radial position $r$.


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