Graphene is sheet of carbon atoms arranged in hexagonal rings (like benzene, see

Question

Graphene is sheet of carbon atoms arranged in hexagonal rings (like benzene, see illustration below). It is currently being studied for its interesting electronic and optical properties. The HOMO-LUMO energy gap is an important property for these systems. In this problem, we will use the particle-in-a- box model to study the behavior of the ?-electrons associated with each carbon atom. For example, if we consider the length (length corresponds to the horizontal direction in the illustration below) of the ribbon to be very large (relative to atomic dimensions), we can adjust the energy gap by varying only the ribbon width (width = vertical direction). So the 1 dimension is the width.

Suppose the desired energy gap is 2.5 eV (electron volts). What is the corresponding photon wavelength?

Now, consider the zig-zag ribbon illustrated below. Use 1D particle-in-a-box theory to determine the width (labeled b in the diagram below) in nm necessary to achieve HOMO- LUMO energy gap of 2.5 eV. Assume the average C-C bond length to be 140 pm. Do not assume the width to correspond to that in the illustration below. Hint: To determine the HOMO, you will need to determine the number of electrons which depends on the number of carbons, an unknown. Because the number of carbons also determines the width, you will need to set up an equation that includes the relationship between the number of carbons and the width. To do this, follow the steps below.

Write the basic equation for the energy difference between the adjacent HOMO/LUMO levels.

Write an equation for how the HOMO level depends on the number of carbons.

Write an equation for how the length depends on the number of carbons.

Substitute the relationships from (ii) and (iii) into the equation from (i) and solve.

Explanation / Answer

Graphene is pure carbon in the form of a very thin, nearly transparent sheet, one atom thick. It is remarkably strong for its very low weight (100 times stronger than steel) and it conducts heat and electricity with great efficiency. While scientists had theorized about graphene for decades, it was first produced in the lab in 2004.Because it is virtually two-dimensional, it interacts oddly with light and with other materials. Researchers have identified the bipolar transistor effect, ballistic transport of charges and large quantum oscillations.
Graphene, unlike carbon

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