# Presents the Math Original article © by A. Daniel Klarmann, 2007

## How much Oxide makes a colorand how much Electricity does it take?

To start with, I'll assume you've read my pages on anodizing titanium and the basic physics of the color. Now, you want to know if you have the power.

Basic Numbers Needed to Find the Answer
Titanium dioxide numbers
TiO2 specific gravity = 4.2 g/cc
TiO2 refractive index = 2.7
TiO2 mole-weight = 47.87 + (2 x 16) = 79.87g/mole
Relevant Physical Constants
Avogadro's Number = 6.02 x 1023
Electrons per Coulomb = 6.24 x 1018
=> 1 Coulomb = 1 amp-second
Approximate Wavelength of titanium colors
- bronze = 3 x 10-5 cm
- blue = 4 x 10-5 cm
- yellow = 6 x 10-5 cm
- purple = 7 x 10-5 cm
- green (second order) = 10 to 12 x 10-5 cm

Assumptions for this approximate number:

• optically smooth metal surface
• incident and reflected light both normal to surface
• color produced purely by reflective/refractive reinforcement
• oxide thickness is uniformly half of wavelength in oxide at refractive index
• Pure titanium and pure titanium dioxide, no inclusions
• Benign electrolyte, nothing to bond directly to titanium (like nitrogen, chlorine, etc)
• No metal migration to cathode
• No sharp corners or burs to bleed current
• 1 oxygen atom/ion (O--) migrates per 2 electrons injected (100% efficient = each electron strips off one hydrogen)

Steps to calculate the volume of oxide under these ideal conditions and the amount of electricity needed:

• oxideVolume = area * (wavelength / 2) / refractiveIndex
The refractive index is needed because light is slower in solids than in a vacuum
Divide by 2 because the distance traveled inand out is twice the thickness therefore the wavelength is that much shorter at the same photon energy.
• oxideMass = oxideVolume * specGrav
• oxideMolecules = oxideMass / moleWt * Avagadro
Units: g / (g/mole) = moles, * molecules/mole = molecules
• total amp-seconds = 4 * oxideMolecules / ElectronsPerCoulomb
Times 4 because have 2 oxygen atoms per molecule and 2 free electrons per oxygen atom.

Example:

Let's turn a 1 square centimeter flat piece of titanium yellow:
Ignoring edges, that gives us 2 cm2
oxideVolume = area * (wavelength / 2) / refractiveIndex
= 2 * (6 x 10-5 / 2) / 2.7
= 2.22 x 10-5 = 0.0000222 cc

oxideMass = oxideVolume * specGrav
= 2.22 x 10-5 cc * (4.2 g/cc)
= 9.32 x 10-5 = 0.0000932 g

oxideMolecules = oxideMass / moleWt * Avagadro
= 9.32 x 10-5 g) / (79.87 g/mole) * 6.02 x 1023
= 7.0 x 1017 = 700,000,000,000,000,000 molecules

amp-seconds = 4 * oxideMolecules / ElectronsPerCoulomb
= 4 * (4.5 x 1021) / (6.24 x 1018)
= .45 amp-seconds or 100 ma for 4.5 seconds

BUT in the real world, you probably need several times that much for reasons alluded to in the Assumptions section above.
Also, the current starts high, and drops exponentially fast as the oxide builds up.

Check out my anodizing page for more detailed instructions on coloring titanium with an anodizer.
Check out my physics page for how the color actually works.
Read my titanium details page for information about the history, chemistry, and so forth of element 22

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