Several ways, but it all comes down to keeping track of your ins and outs. We know what each molecule weighs, and we know what each atom weighs. Since you can’t have fractions of an atom (we’re not doing nuclear here), there’s only certain combinations that can give you the right molecular weight.
For something like water you can combine known quantities of hydrogen and oxygen into water (by burning them) and measuring how much oxygen or hydrogen is left. Whatever got used must be in the water.
More complex molecules take more work to create but if you just keep track of what you started with and what you have now you can track where all the atoms must be.
Scientists use a technique called spectroscopy to determine the composition of molecules. When a molecule is exposed to light, it absorbs specific wavelengths related to the type and arrangement of its atoms. By analyzing the absorbed light, scientists can identify the elements in the molecule and their quantities. For example, a glucose molecule absorbs light in a pattern that indicates it has 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. Similarly, a water molecule’s absorption pattern reveals it has 2 hydrogen atoms and 1 oxygen atom.
Today we usually do it by working out what the molecule is based on its properties then looking it up from a resource that has spent more effort working it out.
A bottle of water, bottle of methanol and bottle of dichloromethane will *look* identical but have very different physical properties like boiling point and different chemical properties like pH. If you can measure the boiling point you can look it up in a table of data to see what clear colourless liquid boils at 39.6 degrees.
But if you’re starting from scratch then you need to use more complicated techniques to work out what the molecule is. Water was originally determined to contain a ratio of two hydrogen atoms to one oxygen atom by trial and error. When someone found a way to produce ‘inflammable air’ that we now call hydrogen and also ‘dephlogisticated air’ that we now call oxygen, they found these two gases would burn together and leave condensation on the inside of the container. Trial and error showed that a ratio of 2:1 gave the biggest flash and most energy output and also the most condensation. Therefore, they concluded, water must be composed of two parts hydrogen and one part oxygen.
For really really complicated molecules you need to spend months analysing it using various complicated techniques like NMR Spectroscopy, Mass Spectroscopy and X-Ray Crystallography. If you determine the correct shape of a really important molecule it could lead to the development of new medicines and might earn you a PhD today or a Nobel Prize in the past (Watson and Crick getting the Nobel Prize for the structure of DNA). It’s a complicated task that hasn’t got one clear path to the answer, you need to analyse the molecule with different tools and techniques and study the results, sometimes for months or years.
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