TL/DR: we exploit the differences in attraction between different molecules to separate them using a science called chromatography
The science behind the separation of different molecules is called chromatography. There are MANY, MANY different kinds of chromatography, but they all work on the exact same principle: exploit differences in intermolecular forces of attraction.
What are intermolecular forces? Well, they are the forces of attraction that occur between individual molecules! The type and strength of these forces are governed by the types of atoms that make up the molecule, and their structure/arrangement in 3-D space. Without getting too complicated, each element “wants” electrons to a different degree than other elements (we call this wanting of electrons electronegativity), and these differences in electronegativity in a molecule mean that the shared electrons spend more time near the highly electronegative part of the molecule and less time near the other parts. This gives a small negative charge to one side of the molecule, and a small positive share to another side (we call these small charges momentary dipoles because they only last very briefly). How strong these dipoles are is a trait we call polarity, the stronger the dipoles the more polar a compound is said to be. As an example of how dramatically this can affect the interactions of compounds, polarity is the reason why oil and water don’t mix.
Okay,so what does all that have to do with separating different compounds? Well, these small dipoles are different FOR EVERY SINGLE CHEMICAL based on their 3-D shape and structure. We can exploit this by dissolving multiple compounds into a solvent and running the solvent through a highly polar separation media. This causes the molecules to separate out based on their dipole properties, as the more polar molecules interact with the media more than the less polar compounds. Thus, the less polar compounds run through the media and out the bottom faster while the more polar compounds take longer to pass through the media.
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