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I am trying to wrap my head around the flame test, and for the most part, it is making sense. My question is their a way to figure out the colour of a flame of an unknown element given the emission spectrum. In other words, how does the eye combine the mess of an emission spectrum into a single colour? I have been researching for a while now, so I am prepared for some pretty heavy maths. Some links would also be helpful for further reading.

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    $\begingroup$ Your brain receives information from your eye’s rods and cones and you see colors. What you see with an element’s flame atomic emission is just whatever your brain makes it. The element thallium gets its name from a Greek word that means “green twig” because it has a dominant strong emission at 535 nm. The story goes that it was identified spectroscopically, but the discoverer was color blind, so they called over their assistant to see it. Then the assistant apparently tried to lay claim to the discovery, being first to see the green emission. They lost! $\endgroup$
    – Ed V
    Nov 21 at 23:23
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    $\begingroup$ I am afraid this is not really a question about chemistry, but about the physiology of vision (probably better suited for Biology StackExchange). Anyhow, take a look at various colour spaces that connect spectrum to the perceived colour, such as CIE 1931. $\endgroup$
    – Domen
    Nov 21 at 23:24
  • $\begingroup$ Lots of information at physics stack exchange. Example: physics.stackexchange.com/a/488695/313612. Notice at the linked answer that the answerer says the purest yellow is 571 nm and that is shown in the figure. But sodium’s famous yellow D lines are at 589.0 and 589.6 nm and and no other sodium lines are nearly as intense. They look pure yellow to me. $\endgroup$
    – Ed V
    Nov 21 at 23:26
  • $\begingroup$ Sorry, I just want to turn an atomic emission spectrum into a wavelength. What spectral line does a flame choose, and why? $\endgroup$
    – Yak
    Nov 22 at 0:07
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    $\begingroup$ It does not pick a wavelength and is not a single color! What you perceive depends on your eyes and your brain. Look at some questions and answers such as this one: physics.stackexchange.com/q/564362/313612. $\endgroup$
    – Ed V
    Nov 22 at 0:12
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It is not simple -- for example, to many people, red + blue spectral lines appear purple/violet, i.e., shorter wavelength than pure blue! Though you could assign a color name to a pure spectral line, as you state, the appearance of mixed spectral lines becomes much more difficult to describe.

In this color wheel, names are assigned to mixed colors.

An interesting case is didymium glass, used in photography and by glass-workers to block the bright sodium yellow lines. Under incandescent illumination, the glass has a distinctly pink appearance, and under skylight, it's decidedly mauve.

So take description of spectral appearance with a grain of salt... oops, that's definitely yellow!

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    $\begingroup$ (+1) Nice answer! Glad you see the sodium D lines emission as yellow and my Nd:glass laser rod behaves exactly as you describe in terms of colors. Also, if I remember correctly, Isaac Newton saw only 5 colors in the rainbow, so even our terms and names for colors are not so intrinsic as might be hoped. $\endgroup$
    – Ed V
    Nov 21 at 23:53
  • $\begingroup$ I would have thought we could predict the colour of a flame by the emission spectrum, is there any way to get an approximate colour. The colour between people with normal vision doesn't vary dramatically for example I don't think anyone sees a sodium flame as green, so can we pinpoint the colour with RGB or something. I am probably misunderstanding this so if you could clarify is this is impossible that would be benificial. $\endgroup$
    – Yak
    Nov 22 at 0:03
  • $\begingroup$ @EdV Hmm, I have never perceived sodium lines as pure yellow, always seeing there some warmer orangish/pinkish/salmonish tints. $\endgroup$
    – Poutnik
    Nov 22 at 7:14
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    $\begingroup$ There are also impossible colors and so on. But getting back to chemistry, a digital photograph of a green-colored flame can have its intensity of green quantified by intensities of RGB pixels, yet this says nothing about what actually caused the green flame color, i.e., boron, barium, copper, thallium, etc. $\endgroup$
    – Ed V
    Nov 22 at 12:52
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    $\begingroup$ @EdV, thanks for pointing out "impossible colors." The crossed-eye color illusions are interesting! BTW, seven is considered a "magical" number, so colors such as indigo were added to make the spectrum more "complete". There was no color orange in English until the fruit (naranj, نارنج‎, and then misnamed) was imported, $\endgroup$ Nov 22 at 19:53
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As I understand your issue, you need to convert spectra into RGB triplets. Quantitative colorimetry theory addressing that issue is quite thoroughly stated in this wikipedia article on CIE 1931 color space

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In other words, how does the eye combine the mess of an emission spectrum into a single colour?

Although your query is related to vision and others have addressed it, let me clarify a couple of things that might be causing confusion. First, you are talking about simple flame tests for alkali metals, alkaline earth metals, and a few elements like boron and copper. Bunsen flame spectra are not a forest of emission lines but a very simple spectrum (indeed not a mess at all). You may have been misled by Google images search. Most of them are just cartoons and fake. Bunsen flame is a low-temperature flame. Not many lines are expected.

If you have a pocket spectroscope or even a homemade CD spectroscope, you will see that sodium flame is just one orange-yellow line around 589 nm. Similarly, potassium flame spectrum is just one or two lines. If you are lucky with sensitive eyes, you might the deep red and violet lines.

See this excellent video of the true flame colors Flame emission spectra. The video shows the simple spectra. You will also note that the yellow sodium line is usually present in most spectra...Na is everywhere.

You can try making a CD spectroscope to try it yourself. Internet is full of such easy constructions, but the spectra on Google Images is disappointly cartoonish (or rather software generated) from at a first glance.

Another misconception to be noticed is that many people think that the Bunsen flame colors are always due to atomic emission. This point is not valid. Ca, Ba, Sr, Cu, B, flame colors are due to small molecular compounds of those elements, not their atoms. Only the alkali metal emission is atomic emission, and the rest are molecular emission in a Bunsen burner.

For real analytical work, Bunsen flame color is useless. We have to use a high-temperature from a spark or electric discharge or a plasma around 5000 K to see true atomic emission of most of the elements in the periodic table. Then, indeed, the emission spectrum is a mess, mainly in the ultraviolet, and you certainly need a device to separate wavelengths. The concept of color then vanishes because we cannot see ultraviolet.

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