how surface area decreases with decreasing size of nano particles.

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To be perfectly accurate, we cannot provide an exact answer to this question. This is because nanoparticles do not have a perfect shape, i.e. we cannot say whether the nanoparticles are spherical, cuboidal or some other regular shape. The nanoparticles are synthesized and their size is measured by optical or other means. The size that is presented is (most likely) the diameter of an 'equivalent' sphere. In other words, we measure the volume of nanoparticles and convert this to diameter of a sphere that will have the same volume as that of the nanoparticle. Also, the nanoparticles are so small (around 1 nanometer in size) that properties such as diameter are averaged over a large sample. 

If we are to assume size as the diameter, d (nm) and shape to be spherical, the surface area, A, will be given as:

`A = pi d^2`

i.e., the surface area varies as square of particle diameter. So if the size decreases by half, surface area will decrease by a factor of 4.

Hope this helps

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