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The tubes glow their respective colors when the Tesla coil connects with them. The video shows the gases in the numerical order presented in the tables above.
The Tesla coil is an energy source. This energy excites the electrons in the gases to higher energy states. In order to return to the ground state, electrons release excess energy in the form of light. The light is differently colored in each tube because of the different wavelengths of light that must be released in each instance.
Different elements emit different wavelengths of light to return to their respective ground states, so the tubes' colors are varied. These colors can be used to produce atomic emmision spectra of the elements electrically excited. Using known values of emmision spectra, one can perform a similar discharge test on un unknown gas, gather an emmision spectrum from it, and determine which elements are in the unknown gas.