Hopefully a simple question.

[bbowman00]

In the sentence below the word "den" has been substituded for the word Der because it is in Accusative Case.
- Das Mädchen arbeitet für den Vorsitzenden.
But in the sentence below this statement you can see that "dem" has been substituded for the word Der because of I think the Dative rules.
- Der Professor isst mit dem Vorsitzenden.
Am I correct or not? And if so, isn't in both sentences "Vorseitzenden" the direct object?
How I got the direct object:
Sentence one: the girl works for what or whom? The chairman
Sentence two: the professor eats with what or whom? The chairman

[kliapatra]

"für" always takes the Accusative. "mit" always takes the Dative. It is worth looking up prepositions that always take one case or another. The difference in cases here is only due to the nature of the words in German-- "for whom" vs. "with whom".