In chemistry, all properties of a molecule arise from how the electrons within the molecule interact with the environment around them. In the simplest case, this can lead to a molecule giving up, or taking on an electron to participate in an oxidation-reduction reaction; the basic process that allows for cellular metabolism. In the most complicated case, in organometallic based light-emitting diodes, an electron within the organic part of the molecule can violate quantum rules to flip their spin thanks to the metallic center to form a forbidden triplet state, and then release their energy through phosphorescence to create light.
However, chemists could not immediately understand these phenomena without creating a fundamental model of how an arbitrary particle behaved in an enclosed box, and this lead to the aptly named "particle in a box" model. In such a box, an observer can only determine the velocity or the position of the particle at a given moment, but not both simultaneously. Strangely, the particle in this box takes on a wave-like quality, and delocalizes throughout the box.
In the realm of organic semi-conducting materials "conjugated acenes" have very special properties thanks to electrons in these molecules essentially acting like a particle in a box. A conjugated acene consists of fused rings, all of which have roughly the same bond length due to their aromaticity. This means that the highest occupied molecular orbital (HOMO) will delocalize over the whole molecule. Below I give the HOMO of tetracene I computed with density functional theory (DFT).
The blue and red correspond to the bonding and anti-bonding orbitals of the HOMO respectively, but both have equal representation over the whole molecule.
While the behavior of the HOMO of tetracene has more complexity than this model, as shown by DFT, we can pretend that the positive part of this wavefunction corresponds to the blue blobs in the tetracene picture and the negative part corresponds to the red blobs.
Below I give the general form of the particle in a box function, which describes the behavior of an electron in the box at a given point.
\[\Psi(x) = \sqrt[]{\frac{2}{L}}\sin (\frac{n \pi\ x}{L})\]
\(L\) refers to box length, \(n\) refers to principle quantum number, and \(x\) refers to position in the box.
For tetracene, I assume a value of \(L = 2\) for a single orbital blob to simplify the math. Likewise, \(n\), has a value of 6 for Carbon. Since Carbon drives the electronic properties of tetracene, the principle quantum of Hydrogen does not apply here. Below I give the equation of a particle in a tetracene box with these parameters and its graph. I will note that the value of \(L\) for a blob lies closer to 1.4 Angstroms, but that causes the math for this problem to look quite nasty.
\[\Psi(x) = \sin(3 \pi\ x)\]
To model a blue blob, I limit my position values between \(x = 0\) and \(x = \frac{1}{3}\) to get a single peak on the wavefunction. I then revolve this peak around the y-axis, and we can pretend this has similar behavior to a blue blob in tetracene if we rotated the molecule about an axis in space.
So our surface of revolution will correspond to the behavior of an electron in a bonding orbital in tetracene if we rotate tetracene in space.
To find the volume of the solid if we rotate about the y-axis, we have to solve for x in terms of y!
Our formula thus evolves to what I give below:
\[x = \frac{\arcsin(y)}{3 \pi}\]
In general, if we assume \(f\) and \(g\) represent functions of y, we can find the volume enclosed by a region of space bounded by \(x = a\) and \(x = b\) with the formula below.
\[\pi\ \int_{a}^{b} (f^2 - g^2) dy\]
Here, \(g(y)\) is the mirror of f so. Also, since we have to solve in terms of y, we have to change out limits of integration. While x varies from 0 to \(\frac{1}{3}\), y varies from 0 to 1. Thus, our function will have the exact volume given by:
\[\frac{2}{3} \int_{0}^{1} \arcsin(y) ^2 dy\]
Interestingly, the solid of revolution takes on the form below:
The full antiderivative has the horrifying final form of
\[\frac{2}{3} (2\sqrt[]{1- x^2}\arcsin(x) - 2x + x \arcsin(x)^2)\]
This horrendous expression evaluates to only 0.31!
However, an easier way exists to approximate this area, because no-one wants to compute an arcsine by hand. Since we revolved around the y-axis we can approximate the area with washers!
In a washer, we specify \(k\) number of vertical slices. We have a fixed height of each slice which we will call \(\Delta y\). This term has a value we can derive by using out limits of integration, and k and therefore \(\Delta y = \frac{b - a}{k}\). Here we integrate over 0 and 1, and we will have 10 slices, therefore \(\Delta y = \frac{1}{10}\).
A hollow cylinder gives the volume of each washer, and such a cylinder has an inner radius I denote by r, and an outer radius I denote by R. R extends from the center to the edge of the cylinder, and since we bound our volume by \(x = 0\) and \(x = \frac{1}{3}\). given this, R simply has the value of \(\frac{1}{3}\). Below I give the formula, which we will then generalize to a sum of cylinders to give an approximation of our integral.
\[V = \pi h (R^2 - r^2)\]
And now, since we have \(k\) hollow cylinders and each has height \(\Delta y\) our formula takes on the form below.
\[V = \sum_{k=1}^{10}{\pi \Delta y (R_k^2 - r_k^2)}\]
So, we have a pretty decent approximation! We have an error of only about 0.01 cubic units!
Therefore, washers provide an easier to understand and fairly accurate portrayal of the volume enclosed by a solid of revolution.
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