Not sure if this is allowed here, but I have a very basic question, I can't quite wrap my head around.
One glass has $1$kg of powder with particle size of $4$ microns. A second one is double, $8$ microns.
How much more surface area has the smaller grained powder and how does the relationship between particle size and surface are look? I know its not linear, but not sure if its close to exponential?
Meaning the $4$ micron powder doesn't have double the surface are, but many times more.
How do I calculate that and write down in simple terms?
Assuming that the particles are smooth surfaces, generally speaking the surface area will scale proportional to the square of the linear measurements. So if the diameter of the particles doubles, the surface area will quadruple; if the diameter increases by a factor of ten, the surface area will increase by a factor of one hundred, and so on.
You can prove this directly for simple shapes - for example, if we assume the particles are spherical with radius $R$ then their surface area is $4 \pi R^2$; if they're cubes with side length $R$ then their surface area is $6R^2$.
On the other hand, if the particles are better approximated as being of a fractal dimension $D$, which could be the case if they are very coarse and have lots of detail at a microscopic level, then the surface are will scale relative to $R^D$. $D$ will be a value between 2 and 3, so the scaling will be somewhere between the square and the cube of the particle size.