Separation is the most straightforwardly technical of the operations, and the one whose laboratory sense survives most obviously into modern practice.
What the word describes
Separatio means dividing a mixture into its components: decanting a liquid off a residue, filtering, skimming an oily layer, drawing off what rises from what settles. Every one of those is still done, under other names, in any working laboratory.
In the alchemical texts the operation carries the further sense of dividing the pure from the impure, and the two senses are not separable in the sources themselves. An operator who filtered a liquor and an operator who spoke of separating the subtle from the gross were often the same person doing the same thing and describing it in two registers.
Where the number seven comes from
A fixed run of seven numbered operations mapped onto the seven planets is a modern tidying. The historical sources give varying counts: Ripley's fifteenth-century Compound of Alchymy has twelve gates, and other treatises list other operations in other orders. The seven-step version now in general circulation is a synthesis rather than a transmission, and the atlas records it as the common modern scheme while saying so.
That distinction matters when reading a primary text, because a reader expecting seven ordered stages will not find them.