Learn Hardcore German: Die Verantwortlichen im Büro bleiben gelassen, obwohl der Tag chaotisch ist. - The people in charge at the office remain calm, although the day is chaotic.

Verantwortlichen is capitalized because it is a noun made from an adjective (a nominalized adjective). German turns adjectives into nouns very often.

The base adjective is verantwortlich = responsible.
die Verantwortlichen literally means the responsible ones → the people in charge / the people responsible.

Whenever an adjective is used as a noun like this, German:

capitalizes it, and
adds normal adjective endings (here: -en).

Why is it die Verantwortlichen and not die Verantwortliche?

Because die Verantwortlichen is plural, referring to several responsible people.

die Verantwortliche (singular, feminine) = the responsible (woman)
der Verantwortliche (singular, masculine) = the responsible (man)
die Verantwortlichen (plural, any mix of genders) = the responsible ones / the people in charge

In the sentence, die Verantwortlichen is the subject of the verb bleiben, so it’s in the nominative plural:

Nominative plural ending for nominalized adjectives is -en → Verantwortlichen.

What case is im Büro, and what does im stand for?

im Büro is dative case, and im is a contraction:

im = in dem (in + the)
Büro is neuter: das Büro → dem Büro in the dative.
The preposition in with a static location (no movement) uses the dative:
→ They stay in the office (no movement into it).

So:

in dem Büro → contracted in normal speech/writing to im Büro.

Why is the verb bleiben used with gelassen? Could I say sind gelassen instead?

bleiben + adjective expresses remaining in a state: to stay / remain X.

bleiben gelassen = to remain calm / to stay composed (despite something).
sind gelassen = are calm (describing their state, without the idea of staying that way).

You can say Die Verantwortlichen im Büro sind gelassen, but you lose the nuance of maintaining calmness in a chaotic situation. Bleiben fits nicely with the contrast introduced by obwohl (although).

Is gelassen here an adjective or a past participle from lassen?

Historically, gelassen is the past participle of lassen (to let, to leave), but in modern usage here it functions clearly as an adjective:

gelassen = calm, unruffled, composed (a character trait or current state)
As a predicative adjective (after sein, bleiben, werden), it does not take an ending:
- Sie bleiben gelassen. (They stay calm.)
- Sie sind gelassen. (They are calm.)

So in this sentence, treat gelassen as an adjective meaning calm.

Why isn’t there an ending on gelassen (for example gelassene)?

German adjectives only take endings when they come before a noun (attributive position):

ein gelassener Chef (a calm boss) → adjective has ending.
eine gelassene Reaktion (a calm reaction).

When the adjective is after the verb (predicative position) – after sein, bleiben, werden, etc. – it has no ending:

Der Chef bleibt gelassen.
Die Verantwortlichen bleiben gelassen.

So gelassen here is correctly used without an ending.

Why is there a comma before obwohl?

Because obwohl der Tag chaotisch ist is a subordinate clause, and German always separates a subordinate clause from the main clause with a comma.

Main clause: Die Verantwortlichen im Büro bleiben gelassen
Subordinate clause: obwohl der Tag chaotisch ist

Rule: main clause , obwohl + subordinate clause.
Hence the comma before obwohl is obligatory.

Why is the verb at the end in obwohl der Tag chaotisch ist?

obwohl is a subordinating conjunction. In German, subordinating conjunctions (like obwohl, weil, dass, wenn) send the conjugated verb to the end of the clause:

Main clause word order: subject – verb – rest
- Der Tag ist chaotisch.
Subordinate clause with obwohl:
- obwohl der Tag chaotisch ist

So ist goes to the end because obwohl introduces a subordinate clause.

Why is it chaotisch and not chaotischer or chaotische?

Here chaotisch is used as a predicate adjective describing the subject der Tag:

Der Tag ist chaotisch. → The day is chaotic.

Predicate adjectives in German do not take endings and are not inflected for case, gender, or number:

Der Plan ist gut.
Die Pläne sind gut.
Der Tag ist chaotisch.
Die Tage sind chaotisch.

Forms like chaotische or chaotischer would be used before a noun or in comparisons:

ein chaotischer Tag (a chaotic day)
ein noch chaotischerer Tag (an even more chaotic day).

Can I put the obwohl-clause at the beginning of the sentence?

Yes. You can say:

Obwohl der Tag chaotisch ist, bleiben die Verantwortlichen im Büro gelassen.

This is very natural German.
Two points to note:

The subordinate clause still has the verb at the end: obwohl der Tag chaotisch ist.
Because the obwohl-clause comes first (it occupies the first position), the finite verb of the main clause (bleiben) still stays in second position:
- … ist, bleiben die Verantwortlichen im Büro gelassen.

What is the difference between using obwohl and using trotzdem in a similar sentence?

Both express contrast, but they work differently in the sentence:

obwohl introduces a subordinate clause:
- Die Verantwortlichen im Büro bleiben gelassen, obwohl der Tag chaotisch ist.
- Structure: main clause + comma + obwohl-clause (verb at the end).
trotzdem is an adverb in the main clause:
- Der Tag ist chaotisch. Trotzdem bleiben die Verantwortlichen im Büro gelassen.
- Two main clauses; trotzdem usually comes early, often in position 1 or 3.

Meaning is very close: Although the day is chaotic, they stay calm vs. The day is chaotic. Nevertheless, they stay calm.
The obwohl version is a bit more tightly connected in one sentence.

Is the word order Die Verantwortlichen im Büro bleiben gelassen fixed, or can I move im Büro?

You can move im Büro within the main clause; German word order is quite flexible as long as the conjugated verb stays in second position:

All of these are possible (with different emphasis):

Die Verantwortlichen im Büro bleiben gelassen, obwohl der Tag chaotisch ist.
(default; neutral)
Im Büro bleiben die Verantwortlichen gelassen, obwohl der Tag chaotisch ist.
(emphasis on in the office.)
Die Verantwortlichen bleiben im Büro gelassen, obwohl der Tag chaotisch ist.
(slight emphasis on remaining calm in the office.)

What you cannot do is move bleiben away from the second position in the main clause:

❌ Die Verantwortlichen gelassen im Büro bleiben … (ungrammatical in standard German)