To the first sentence, see what this site says:At, on and in are not usually used before expressions of time with next, this, that, last, one, any, each, every, some or all
As for the second:In is used with a part of the day or with longer periods of time.
Example: "It's even more difficult in the winter
, when it's still dark."http://www.spotlight-online.de/language ... ns-of-time
used to specify, used when you talk about a particular thing. In the case of the second sentence "THE" semester is defined by "the following", indicating that is not any semester or "a semester" in general, it´s the following.
Hence, "in the" following semester
. As per the above rules to the usage of preposition IN, combined with the rule to the definite article "THE".