Is communicating with a professor via his private email to benefit from his knowledge in his field of specialization considered a violation of an oath to refrain from entering "chat rooms" and invoking illness upon oneself if that occurs, especially given that the intention behind the oath was to avoid unbeneficial talk and unknown individuals?

The oath you swore is valid and must be fulfilled. It is not permissible to violate it because it aims to keep the soul away from forbidden things and places of temptation. Fulfilling an oath is obligatory when swearing to do something obligatory or to abstain from something forbidden. If a person swears about something using a general لفظ (wording) but intends to specify it, then the نيّة (intention) is what is considered and restricts the مطلق (unrestricted) wording. Therefore, if your intention for not entering chat rooms was to avoid useless conversations or conversations with those you do not know, then your intention specifies the oath. Consequently, there is no blame upon you for entering to benefit from the teacher, and you do not owe an expiation for breaking an oath, because the oath did not include every entry. You must uphold your oath to refrain from entering degrading and time-wasting chat rooms, as these are what you intended in your oath, and it should be fulfilled.