The death penalty fails as a rational form of punishment on several counts.
First, it does not act as a deterrent. If it did then nations like the USA would have much lower murder rates. What is interesting is that in general nations without the death penalty tend to have lower murder rates.
Murders (per capita) statistics - countries compared - NationMaster
You might note that nations with high murder rates often tend to have serious social and economic problems that the death penalty does nothing to curb.
Second, in nations offering fair trials, the cost of execution is about six time what it costs to incarcerate a criminal for life.
Third, the death penalty tends to be applied more severely against the poorer members of society - especially racial minorities.
Fourth, a mistaken conviction cannot be reversed.
Fifth, criminals facing the death penalty have little to lose by murdering their victims or violently resisting arrest - they can only be executed once.
Sixth, applying the death penalty more widely will simply lead to more violence on the part of the criminal for the same reason stated in point five. For example, if rape is punishable by death, then the criminal has little to lose by murdering his victim.
Seventh, studies have shown that juries faced with convicting someone who is likely to be executed will often allow a criminal to go free rather than senctence him to death
Eighth, there is as yet no truly humane method of execution.