In the same way that many people confuse hazard and risk when it comes to nanotechnologies, many in the financial sector tend to think that any old curve is an exponential (maybe its the bit about tending towards infinity that attracts them). Of course everyone learns about curve fitting in high school mathematics, especially of you have a nice graphical calculator to play with, but most of the information is promptly forgotten unless you head off to do physics, maths or engineering at University.
We came across a nice example recently of an explanation of solubility (something we almost tripped up on ourselves). According to Josh Wolfe Without making you think back to your high school physics, I’d note that as you make something smaller with the same volume, the surface area increases exponentially.
To be fair, many comments about nanotechnologies are often distorted through the lens of non technical journalism, but its worth a closer look at this particular myth.
Issues of density and nit picking over the relationship between size and volume aside, we do remember our high school physics, where the surface area of a sphere does not increase exponentially.
The formula we remember used to be 4 times pi times the square of the radius. This curve is very different from an exponential, as you can see from the graph below where we compare squares (blue), cubes (green) and exponentials (red). So different is the result that five squared is 25, whereas the exponential of five is 148.4131591 a significant difference. Similarly one hundred squared is, of course, 10,000, one hundred cubed is 1,000,000 whereas the exponential of 100 is 26880000000000000000000000000000000000000000!!!
The trick with nanoparticles is that the ratio of surface area (a square function) to volume (a cube) increases as you reduce the size (diameter or radius) making the particles pretty much all surface, and therefore making all of the atoms in the particle sit at the surface where they are available to take part in reactions, rather than being buried deep in the core.
Perhaps this mathematical confusion explains some of those outlandish market projections?