There are lots of instruments chemists have at their disposal for identifying compounds, and you generally use quite a few of them, each which provides a little piece of the puzzle.
Mass spectrometry, gas chromatographs, nuclear magnetic resonance, these would give you a decent start, plus any number of little chemical tests for different functional groups.
Chemist here. There are a lot of ways to identify compounds and what they are made of. In terms of composition, you’d use a mass spectrometer. This is a device that measures the identity and masses of various fragments of a molecule you want to analyze. Since each element has a unique atomic mass, the fragments of the molecule can be identified and quantified. This is then typically put into a data base to match literature values to identify the molecule. There are many organic molecules that have the exact same mass and makeup but in different steuctures.
As for structure, there are a few ways. One is NMR (Nuclear magnetic ressonce) spectroscopy. This is essentially looking at the vibration enrgies of various atoms (usually Hydrogen 1 or Carbon 13) in a structure. This method gives you a graph that can be matched to literature values or worked out by hand. NMR is usually paired with Infrared spectroscopy in order to identify a compound’s structure. Infrared spectroscopy makes use of the stretching and bending properties of various chemical bonds. Since chemical bonds between various atoms have known values, a graph can be made of the Infrared data which can then be used to identify what kind of chemical bonds are in the structures. With NMR and IR paired up, it’s easy to figure a structure. Some honorable mentions are Gas chromatography and high performance liquid chromatography. Both of which are used to separate out differing chemicals In a mixture.
Note that this is a very broad topic and I only really covered the organic (carbon bearing) molecules.
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