# Graphene

Alignments to Content Standards: F-LE.A.4

Graphite is a mineral with many technological uses and it is perhaps most familiar for its use in writing instruments. At the atomic level, it is made of many layers of carbon atoms, each layer arranged in the familiar pattern of hexagonal tiles:

The pattern continues on in all directions and there is a single carbon atom at each vertex.

Graphene is a 1 atom thick layer of graphite with many interesting properties and uses. Suppose the thickness of graphene is 200 picometers: one picometer is one trillionth of a meter. About how many times would you have to split a 1 mm thick sample of graphite in half in order to get a single layer of graphene? Explain.

## IM Commentary

This task provides a real world context for examining the incredible power of exponential growth/decay. Moving from the visible world around us to the atomic level takes only a few handfuls of successive halving! Students can actually experiment, at the macro level, using scotch tape to pull a chip of graphite apart into two pieces: of course the split is unlikely to be exactly into halves but one of the pieces will always be less than or equal to half the thickness of the original sample. More sophisticated techniques are of course necessary to reach the atomic level but these simple instruments will be sufficient to see how quickly the thickness of the graphite decreases.

While only the third solution fits the F-LE.4 standard, more elementary solutions have been included for comparison and because they are likely to arise in student work. These solutions use middle school mathematics but the level of the context and the numbers involved make this task more suitable for high school students.

Much useful information about graphene, including its history, production, and many uses can be found here:

http://en.wikipedia.org/wiki/Graphene

The thickness of graphene chosen for this problem is hypothetical and approximate. As a point of reference, the approximate distance between carbon atoms within a sheet of graphene is about 150 picometers. More about the picometer can be found here:

http://en.wikipedia.org/wiki/Picometre

This task aligns well at many levels with MP6, Attend to Precision. First, students have to use units carefully, particularly with picometers which they are unlikely to have seen. They also have to realize that the answer to the question has to be a whole number. In practice, at any given point in the splitting process, the ''halving'' is only approximate if there are an odd number of sheets of graphene present. While it seems like one layer more or less in each half could not make much of a difference, keep in mind that if this happens at the first step, this tiny difference is magnified in each of the successive 21 cuts.

## Solutions

Solution: 1 Table

We begin by making a table showing the thickness of the graphite after being cut in half a different number of times. The table does not show all values and after the fifth line in the table, the values are rounded to the nearest picometer. In making the table, it is important to understand that going, for example, from 4 times halved to 8 times halved means that you need to cut in half four additional times: so the thickness needs to be multiplied by $\frac{1}{2^4} = \frac{1}{16}$. In the table, pm stands for picometer.

Number of times halved Thickness of graphite
0 1,000,000,000 pm
1 500,000,000 pm
2 250,000,000 pm
4 62,500,000 pm
8 3,906,250 pm
16 $\approx$ 15,259 pm
24 $\approx$ 60 pm

With 24 halvings we have gone too far since the thickness of a single layer of graphene is given as about 200 picometers. We can see, however, that if we slice the graphite in half 22 times, this should be in the right range. We can then check that 22 slices will give a thickness of about 238 picometers. So it should take 22 cuts to get to a single layer of graphene.

Solution: 2 Using the tenth power of two

Here we use the fact that $2^{10} = 1024 \approx 1000$. This means that if we cut the graphite chip in half 10 times, we will have cut the thickness by approximately 1000 times. The original piece of graphite has a thickness of 1 mm or 1,000,000,000 picometers. So after being halved 10 times we have a thickness of about 1,000,000 picometers. After halving 10 more times we will be down to a thickness of about 1,000 picometers. Two more cuts gets us to 250 picometers which is about the thickness of graphene. So it should take 22 total cuts to go from the original piece of graphite to a single sheet of graphene.

Solution: 3 Solving an exponential equation (F.LE.4)

We start with a thickness of 1 mm and wish to get to a thickness of 200 picometers by successively cutting in half. First we need to find common units. One millimeter is $\frac{1}{1000}$ of a meter. Since there are $1,000,000,000,000 = 1 \times 10^{12}$ picometers in a meter, this means that there are $$\frac{1}{1000} \times 1 \times 10^{12} = 1 \times 10^9$$ picometers in one millimeter. Cutting the graphite in half $x$ times will give a thickness of $\frac{1}{2^x}$ millimeters or $\frac{10^9}{2^x}$ picometers. We therefore need to solve the equation $$\frac{10^9}{2^x} \approx 200$$ or, equivalently, $2^x \approx \frac{10^9}{200}$, so

\begin{align} x &\approx \log_2\left(\frac{10^9}{200}\right) \\ &\approx 22.3. \end{align}

So it should take 22 cuts to reach a single layer of graphene: 23 cuts will produce a thickness just over half of the given value. The fact that the number is not exact has to do with the fact that the number of layers of graphene in the original sample of graphite is not a power of 2 and therefore the ''cuts'' cannot all be exactly in half.