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If we know the molecular weight of a substance we implicitly know the atomic composition of that substance - provided that the molecular weight is known to an accuracy which is sufficient to distinguish between molecules having the same nominal mass.
To explain this, let's have a look at a very simple example: suppose we know that our substance has a molecular weight of 28. A quick calculation (considering only the nominal masses of the most abundant isotopes of carbon, hydrogen, oxygen, and nitrogen - 12, 1, 16, and 14, respectively) shows that there are three substances which all have the same molecular mass of 28: C2H4 (ethene), CO (carbon monoxide) and N2 (nitrogen gas).
Now, the important point is not to forget that the nominal masses are just for convenience. The actual masses of the four elements are:
1H = 1.007825037
16O = 15.99491464
14N = 14.003074
From this you can easily calculate that the actual molecular masses of the three substance having a nominal mass of 28 are:
The smallest difference between the three masses is between CO and N2, showing a difference of 0.0112 mass units. In other words, if we know the molecular mass to an accuracy of 0.0056 masses (half the minimum difference) we can easily distinguish between the three molecules having a nominal mass of 28.
This simple example shows the principle of finding the atomic composition by means of mass spectrometry. You simply have to determine the mass with an accuracy high enough to reduce the number of possible candidates to a single molecule. Of course, in practice the molecular masses of interest are much higher and the number of possible candidates increases exponentially if we do not take countermeasures, such as a chemical plausibility check.