(a) develop a general mathematical relationship for the number of surface atoms (N_s), number of volume atoms (N_v), and total atoms (N_t) for cubic particles in terms of “n,” where n = number of atoms along a side of cubic nanoparticle.

(b) How is Ns/Nt for cubic particles related to that for square particles in:

equation 1: N_s=4n-4
equation 2: N_t=n^2

? Explain.

(c) Plot the fraction of surface atoms (N_s/N_t) vs particle size (n*d) for cubic (3-D) particles, where d = atom diameter. Use a logarithmic particle size scale for n = 3-100.

(d) On the same graph, plot surface area-to-volume ratio (S_v) vs. particle size.