His eyes goggled with eagerness.

他的眼珠热切地转动着。

Their faces are blazing with eagerness.

他们的脸上充满了急切的期待。

Her hands trembled with eagerness as she opened the letter.

她拆信时，急得双手发抖。

To undertake ( a task, for example ) with eagerness and vigor.

冲劲十足地工作迫切而干劲十足地着手 ( 如一项任务 )

He looked and spoke with eagerness: his old impetuosity was rising.

他的神态和说话都很急切，他焦躁的老脾气又发作了。

His voice sounded louder and higher, as if he were afire with eagerness and rage.

他的声音又大又尖, 仿佛他已经怒不可遏,迫不及待了.

Anxiety has some kinship with eagerness but it will not do as a substitute in this context.

焦虑确与”热切“相关，但此类语境下不宜替换。