# Surface area to volume ratio in nanoparticles

Surface area to volume ratio in nanoparticles have a significant effect on the nanoparticles properties. Firstly, nanoparticles have a relative larger surface area when compared to the same volume of the material. For example, let us consider a sphere of radius r:

The surface area of the sphere will be 4πr^{2}

The volume of the sphere = 4/3(πr^{3})

Therefore the surface area to the volume ratio will be 4πr^{2}/{4/3(πr^{3})} = 3/r

It means that the surface area to volume ration increases with the decrease in radius of the sphere and vice versa. It can also be conclude here that when given volume is divided into smaller piece, the surface area increases. Therefore as particle size decreases, a greater portion of the atoms are found at the surface compared to those inside. For example, a particle of size 3 nm has 50% of its atoms on its surface, at 10 nm 20% of its atoms and at 30 nm has 5% of its atoms on its surface. Therefore nanoparticles have a much greater surface area per unit volume compared with the larger particles. It leads to nanoparticles more chemically reactive. As growth and catalytic chemical reaction occurs at surfaces, therefore a given mass of nanomaterial will be much more reactive than the same mass of material made up of large particles. It is also found that materials which are inert in their bulk form are reactive when produced in their nanoscale form. It can improve their properties.