No, most requirements can not be fully expressed in a class diagram.
A class diagram shows the static structure of how classes are related to each other, but many requirements also have a dynamic aspect to them. However, requirements do drive the structure of the class diagram.
For example, to be able to fulfil the requirement that a Student must have followed the Class "Spanish I" before being allowed to enroll in "Spanish II", the class diagram must show an optional relation of Class with itself to record the possible prerequisites for a class.