Fullerenes are a group of well-known novel nanomaterials with hollow spherical structures; these nan — Analytical Chemistry Chemistry Question
Structures of nanomaterials
Fullerenes are a group of well-known novel nanomaterials with hollow spherical structures; these nanomaterials are novel allotropes of carbon. Fullerenes with n carbon atoms have 12 pentagons and (n/2-10) hexagons, where n is an even number and 20 or more.
Answer the following questions by assuming that the length of each carbon-carbon bond in fullerene is 0.14 nm and that the carbon atoms are point masses.
Calculate the surface area of fullerenes with n carbon atoms in terms of nm2 (1 nm2 ≡ 10-18 m2).
Model Answer
The areas of the hexagon and pentagon (S6 and S5) in the fullerenes are
respectively. Therefore, the total area of a fullerene with n carbon atoms is
.
Calculate the radius of fullerenes (in nm) as a function of n by considering the fullerene molecule as a perfect sphere.
Model Answer
Since the total area of a perfect sphere is 2 sphere = 4S π r, and this area is equal to the area of the fullerene, the radius of the fullerene would be .
Figure 1 shows a large fullerene with C1500. One of hypothetical applications of these large fullerenes is as a “molecular balloon” that can float in air. At 300 K and 101325 Pa, the density of these hollow spherical molecules can be smaller than that of air (80 % N2 and 20 % O2). Calculate the minimum number of carbon atoms and the minimum radius of the fullerene (in nm) required to satisfy this condition. Here, the fullerene molecule is rigid enough to retain its structure under air pressure and is considered to be a perfect hollow sphere.
Model Answer
The mass of the fullerene with n carbon atoms is Since the volume of a sphere with radius r is which is also the volume of the fullerene, the density would be The density of air under standard conditions is This air density is larger than that of the fullerene with n carbon atoms, and n should be large enough so that In this case, the minimum radius of the “molecular balloon” is .