How many carbon atoms are contained in one square centimeter of graphene? (See the Chemistry in Action essay "Buckyball, Anyone?" in Section 10.8 for a description of graphene.) What would be the mass of a \(1-\mathrm{cm}^{2}\) section of graphene?

Imagine a layer of interlocking hexagons, much like a honeycomb made from pencil-lead material. That is precisely the hexagonal structure of graphene. Graphene is a remarkable substance composed of a single layer of carbon atoms arranged in a hexagonal lattice, where each atom is bonded to three neighboring atoms at angles of 120 degrees.

The significance of the hexagonal structure lies in its efficiency and strength. The arrangement allows for a compact yet extraordinarily strong material while being extremely lightweight because it's only one atom thick. In the context of the exercise, understanding this hexagonal pattern is essential to calculate the number of carbon atoms in a given area of graphene. This structure is not just a design, but the key to graphene's superior electrical conductivity, flexibility, and thermal conductance.

Consider Avogadro's number as a bridge connecting the microscopic world of atoms to the macroscopic world we interact with daily. Named after Amedeo Avogadro, this constant represents the number of units in one mole of any substance, providing a common scale for chemists to use. Specifically, Avogadro's number is approximately \(6.022 \times 10^{23}\).

Why does this matter? This massive number allows us to discuss substances on an atomic level in understandable, tangible terms of grams and moles. For instance, when we say that one mole of carbon atoms weighs 12 grams, we signify that this is the mass of Avogadro's number of carbon atoms. This concept is a fundamental cornerstone in chemistry that enables the quantification of atoms in mass calculations, a key step in solving the exercise at hand.

At the core of many chemistry calculations is determining the mass of a single atom or molecule, known as atomic or molecular mass. Atomic mass can be thought of as the 'weight' of an atom on an incredibly tiny scale, and it's listed on the periodic table for each element as an average because of the presence of isotopes.

To calculate the mass of a single atom of an element like carbon, we refer to the atomic mass from the periodic table and then use Avogadro's number for a conversion. As the exercise suggests, we take the molar mass of carbon (12 grams/mole) and divide it by Avogadro's number to determine the mass of a single carbon atom. This step is vital to find out the mass of a specific number of atoms, such as those in a square centimeter of graphene, and thus, is an essential tool for chemists and materials scientists alike.