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Each element has its own unique emission spectrum. Each time the electrons in an atom absorb or emit light, it has to gain or lose an exact amount of energy to be able to jump up or down, and each jump has a specific wavelength of light. The emissions spectrum is made up of all the different jumps that the electrons in an atom can have, so each one is distinct for each element, as each element has a different number of protons and electrons. This causes unique light or wavelengths to be emitted, unique to the element.
The spectra of light emitted is not continuous; atoms can only absorb and emit specific energy levels. When an atom emits light, it doesn't emit a smooth band of light, but many specific lines of light. These specific lines of light emitted are unique to only one element, which is how spectroscopy is used to identify elements. Element's emission spectra shows a line pattern unique to the energy levels associated with a specific element which allows an element to be identified.
Going along with what was said above, light is also quantized, so it exists in discrete units. This is why you see jumps in energy levels and lines on an emission spectra—each element has a unique number of electrons with different sets of energy levels. These unique energy levels cause different elements' electrons to have different gaps with subshells.
To elaborate, the specific lines that form as a result of an atom emitting light can be compared to a fingerprint that is unique to a certain atom. Therefore, each atom has a different identity when it comes to the lines that define them. I hope this makes sense
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