Just a note on dimensional analysis.
In the latest Trig & Calc PoW students need to find the density of titanium metal. As a Materials Engineering student, I know that this is a difficult question to answer. There are many factors that go into and ways to find the density of a metal, each of these I will get into briefly. Max Ray and I worked together to figure out how to accept answers to this problem regarding the wide varieties in the density of titanium metal, so for now the PoW is as solid as titanium.
Originally, the Ninja Mission PoW relied on the density of pure elemental titanium, though even this is variable. Let’s first consider how a scientist would calculate the density of titanium metal. We know that titanium atoms arrange themselves in a lattice pattern known as “Hexagonal Close Packed” or HCP. This means that the individual atoms can be divided into subdivisions of hexagonal prisms which contain a number of partial atoms. (It’d be a lot easier if titanium was arranged in a cubic lattice, but we can’t have everything.) These geometric subdivisions are known as a unit cell. The atoms are arranged on to various points in unit cell; one in each corner of the face hexagons, one in the center of the hexagon face, and three “floating” within the prism in a triangular formation. But, the whole volume of each atom is not entirely in the unit cell, so some are considered fractional atoms. Luckily, it’s easy to remember how many atoms are in an HCP unit cell. There are six total atoms made up of 12 twelfth atoms in the corners, two half atoms in the faces and three whole atoms in the volume.
I even drew you a picture!
So how do we get from the number of partial atoms in a unit cell to the bulk density of a metal? Well, what do we know?! We know how much an atom of titanium mass based on a weighted average of probabilistic distribution of titanium isotopes that exist on Earth. Essentially, it’s not a direct mass of the atom but, a product of the mass of protons, neutrons and electrons, and the number of each in one atom. The periodic table has this number; 47.867 grams per mole (1 mole = 6.02 x 10²³ atoms). The dimensions of the unit cell are 0.295 x 10^-9 meters for the length of a leg and the height is 0.466 x 10^-9 meters. With this information we can formulate the density of titanium metal.
Essentially, what I’m calculating here is how much mass there is in a 1 cubic centimeter hexagonal prism of titanium based on the weight of the total number of titanium atoms in such a volume. This calculates to 4.53 grams per cubic centimeter. This number represents the density of perfectly crystalline and pure titanium based on the standard distribution of isotopic titanium atoms on earth. What I do not account for are defects, dislocations, grain boundaries and impurities that occur from mechanical and thermodynamic forces. Metals (and most materials) do not crystallize perfectly on a macroscopic scale. What happens when a metal solidifies, lots of small crystals form in multiple crystallographic directions. Id est as metals solidify metal atoms arrange themselves into many separate volumes, or grains, with their own orientation. What ends up happening looks a lot like this.
All of those colored shapes are individual grains of crystalline titanium. If you looked at the atomic arrangement of a single grain, all of the atoms would align almost perfectly. Zooming in on a boundary between these grains would look something like this. (The image below is a boundary between a grain of Aluminum and Titanium.)
Though there are two different metals, you can see that the atoms align to form an angle. In the case above the boundary is fairly clean and doesn’t have many gaps, but in most cases, grain boundaries free up a lot of space in a material reducing its density. You can even see that within the two grains there are gaps and inconsistencies in the atomic patterns.
Beyond that, atoms don’t always stack perfectly. There is a sizable laundry list of multidimensional hiccups that can occur when a material solidifies, is heated up or is worked with. These are all classified under the umbrella term “Defects”. These can add or subtract from the density of a material depending on what they are. Single dimensional defects are generally missing, extra or displaced atoms, or other types of atoms crammed into a pure lattice (impurities). Two and three dimensional defects are often created when forces bend, slip and twist atoms out of crystalline alignments.
How does a materials scientist account for defects in terms of density? Short answer: carefully. Considering that in the context of the problem, the wise Sensei grafted titanium spikes to a titanium disks to produce a throwing star, –a process requiring a computer operated plasma cutter– the density of the metal is sure to be changed by defects created via the working and heating of the metal. Theoretically, it’s possible to calculate the density of a metal after all is said and done, though it’d be a massive mathematical undertaking. It’s possible to determine the average number of single dimensional defects using a simple thermodynamic equation based on experimental data. In plain terms, the formation energy of a single dimensional defect wherein an atom is added, removed or displaced from it’s crystalline pattern is the difference between the specific enthalpy of a material minus the product of the temperature and the specific entropy of the defect. Unfortunately, defects with more than two dimensions are only formed by mechanical forces, not thermodynamic, so to measure any changes in density from 2+ dimensional defects, we would need to run a computer simulation of the specific titanium material. All of these phenomenons affect the density of titanium by little more than 0.02 grams per cubic centimeter, but this is enough to affect the final answer by a whole degree.
(Science note: Enthalpy is a measure of energy within a substance based on it’s internal energy and the product of pressure and volume. Entropy is a measure of disorder within a system.)
Why does this matter in terms of our math problem? Well, it doesn’t so much in a practical sense, but mathematically the dimensions of the titanium disks are all functions of it’s density. We know that density is an incredibly variable thing. Though, it’s enough to confuse a student looking up the density of Titanium. It’s enough to know the total weight and volume of the titanium disk, and take the quotient to get the exact density of the particular chunk of metal. Though we don’t initially know the volume, so we have to rely on a carefully calculated density for a material with hundreds of forms and alloys (549 according to matweb). The problem states that the throwing star is made of pure titanium, though pure titanium is impossible to produce on Earth. There are theoretical results for the density of pure Titanium, but we know that it’s a finicky number. I chose annealed grade 1 titanium because it’s the purest Titanium we could possibly produce, and the density is experimental rather than theoretical, as in someone in a laboratory took a known volume of titanium, weighed it and then did the math. There’s lots of reasons why Max and I allowed for a range of answers for the Ninja Mission PoW. It isn’t because the math isn’t solid, but rather the solid isn’t!