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We know that the colour we see of different objects around us is because they absorb all the radiation except a specific wavelength which gets reflected back to us. So, in an emission spectrum, when an element's electron is excited and when it goes back to its normal state, it radiates only specific wavelengths.

Does this mean than if I pass white light to the element, it should absorb all the other wavelengths except those specific wavelength and reflect them back?

Though it's a scenario of absorption spectrum and shows a negative photograph of an emission spectrum, but this confuses me — why would an element emit certain wavelength in one case while absorb in the other?

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The color of objects due to reflection from the surface has nothing to do with the atomic spectra (absorption or emission). Your question seems to have two parts and your attempt to connect them is not justified. The color of a solid object is due to reflection from the surface and indeed some wavelength ranges are reflected more than others in order to produce a specific color of the object when illuminated with white light. It is nothing like " they absorb all the radiation except a specific wavelength which gets reflected back to us". Most common solid objects reflect all the wavelengths in the white light, some more than others. Same for absorption. All the wavelengths are absorbed but the ones that are reflected more are absorbed less.

For atomic absorption and emission spectra, it is not very clear what are you asking. If you send light through a gas, some wavelengths will be absorbed, the atoms get excited and a little later they emit back the same wavelengths. The reason you see dark lines is just a matter of contrast. The emission that follows sends the absorbed wavelength in all directions so their intensity is reduced in the beam that goes straight from the source through the gas and reaches the spectrograph. They are still there, same as the stars are still up during the day but you don't see them. They are not turned off during the day.

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I will try to answer your question using the example of the atomic hydrogen spectrum.

Take a look at the below image:

 The hydrogen emission spectrum is the Balmer series.
ttsz / Getty Images

The continuous spectrum would be your white light.

If you shine this white light onto some hydrogen, the hydrogen atoms will absorb specific wavelengths of this incoming white light. The resulting spectrum is the hydrogen absorption spectrum. You see that there are regions with black lines such as at 656nm. These are the hydrogen absorption lines. In fact, what happens for the 656nm line is that an electron in hydrogen is excited from the n = 2 energy level to the n = 3 energy level. In doing so a photon of 656nm wavelength is absorbed by the atom.

Now turn off the white light source. What happens to the hydrogen? The electrons return from the excited energy states to their original energy states. In doing so the hydrogen emits specific wavelengths of light. So what does this look like?

Since the white light is turned off we expect a completely black spectrum if the hydrogen was not emitting. Since the hydrogen is emitting there will be emission lines on top of the black spectrum. This looks like the hydrogen emission spectrum above. You see that there is an emission line at 656nm this occurs when an electron is deexcited from the n = 3 energy level to the n = 2 energy level.

You can see that the absorption and emission lines correspond to one another between the absorption and emission spectra. That is, hydrogen absorbs and emits at the same wavelengths*.

*This isn't strictly true as I am simplifying here

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