Using U $=3.$

$($a$)($i$)$ What is the surface area$-$to$-$volume ratio of a spherical gold nanoparticle $($NP$)$

of diameter $(2+U)nm?$

$5\%$

$($ii$)$ Gold NPs are often commercially sold in aqueous solutions with stabilising

agents to prevent aggregation of the NPs$.$ One common stabilising agent is

sodium citrate, which results in an effective negative charge on each NP$.$

Explain why sodium citrate helps to prevent NP aggregation. What will be the

effect on NP dispersion upon addition of a large amount of salt to an aqueous

solution of gold NPs$?$ Your answer can be completed in about $50$ words.

$[5\%]$

$($b$)($i$)$ What are the diameter, chiral angle, and electronic properties of a single

wall carbon nanotube $($CNT$)$ with chiral indices $(4+2U,3+U).$ The lattice

constant, $a,$ of a graphene unit cell is $0\mathrm{.}246nm.$

$15\%$

$($ii$)$ A regular square array of CNTs with nearest neighbour separation between

nanotubes of $(10+U)m$ is fabricated. What is the number of CNTs per $c{m}^2?$

$10\%$

$($iii$)$ If the maximum emission current from each nanotube is $(10+U)A,$ what is

the current density from the array? A current density of $4\frac{A}{c}{m}^2$ is measured,

what is the efficiency of the array? What design considerations might increase

the efficiency of array?

$[15\%]$