Paul moves from emphasizing faith (or belief and trust in Jesus) (33 times in 1:1-4:25 to only 3 times in chapters 5-8) to life (24 times chapters 5-8), both eternal and now.
In Chapter 5, Paul deals with the apparent contradiction that we have been eternally declared to be in the right with God now through our belief in- and trust in- King Jesus to the long held Jewish belief that all people would be judged by God on "Judgment Day."
Paul believes in Judgment Day but also believes that those "in Christ" have already been judged through King Jesus. He states that we have this hope and that this hope will not disappoint us (v.5) for the Holy Spirit is poured out and this is a sign of salvation and the symbol of the messianic age.
Paul drags the problem back before the giving of the law to the entrance of sin and to the real remedy for the power of sin--the obedient life, death and resurrection of Jesus. Paul even anticipates an objection in v.13, "but there couldn't be sin before the law? to which Paul answers, "well, did people die before the law? Then there was sin before the law? Paul will go on to show at length that the law does not break the power of sin...only Christ.