To work or to visit?

[DreamFounder]

This is the factory where ( in which/at which) they want to work.

This is the factory where ( in which/ at which )they want to visit.

question:

Why is it the first is correct, but not the second ?

Thanks

[Casiopea]

The verb 'work' takes the prepositions 'at' and 'in'.

1. This is the factory where they want to work.
at: This is the factory they want to work at.
in: This is the factory they want to work in.

The prepositions 'at' and 'in' coupled with the noun they modify (i.e., factory) refer to a location:

Where do they want to work?
They want to work in this factory.
They want to work at this factory.

When prepositional phrases, like 'at this factory' and 'in this factory', refer to a location, they function as adverbs; they answer the question, Where?, and that's why 'where' works here as a relative adverb:

This is the factory where they want to work.
This is the factory at which they work.
This is the factory in which they work.

As for our second example sentence, below, the verb 'visit' doesn't require a prepositional phrase. These phrases are ungrammatical: *visit to + <location>, *visit at + <location>, so 'where' isn't needed here:

2. *This is the factory where they want to visit.
at: *This is the factory they want to visit at. <ungrammatical>
in: *This is the factory they want to visit in. <ungrammatical>

The verb 'visit' doesn't take 'at' or 'in', or even 'to', but the verb 'go' does:

They want to go to the factory.
This is the factory they want to go to.
This is the factory where they want to go.

Hope that helps.