Por que "in" e não "on" ?

No trecho do texto: "My mother is a nurse at a big hospital. My brothers are very smart and work hard in school."

Por que não posso colocar "work hard AT school" e "IN a big hospital" ?

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The thing with in school vs at school is that the former refers to being a student and the type of grades you get while the latter is more about physically being at a school building but not necessarily playing the role of a student, so it could refer to an employee or really just being there but not in a classroom.

As for hospital one it's just that "at" is the preferable preposition over "in" in this specific context when referring to being at specific establishments. So you'll more often hear "at the hospital" "at the library" at the supermarket" etc.
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Nuances, nuances...
At -
“At” is used when you are at the top, bottom or end of something; at a specific address; at a general location; and at a point.
Source: YourDictionary.

“In” is used in a space, small vehicle, water, neighborhood, city and country.
Source: ditto.

I would elaborate more about the "in-at" definition, "AT" would mean "being physically present at someplace" (generally where there a "crowd", restaurants, schools, etc. And "in" meaning "inside" the premises.
That is, as Marcio (a Forum participant) explained elsewhere (EMB forum):
At the office = nas imediações do escritório, no escritório no sentido geral. Também pode significar "no trabalho".
In the office = Exatamente dentro do escritório, do ponto de vista físico

At home = Em casa ; no lar.
In the house = inside the house = na casa ; dentro de casa, do ponto de vista físico.

Same goes to both, in our case here, hospital or school. You could use both prepositions, depending on what you are thinking at the moment. Plus, the writer, may (or may not) have avoided to keep repeating a word (that turned out to be a preposition).