It is true, the word "purgatory" does not appear in Scripture, but purgatory itself is shown to exist in Sacred Scripture.
“Justice is uprightness (rectitudo)-of-will kept for its own sake.” [St. Anselm, On Truth, 12]. Continuing he said, “Justice is not rightness of knowledge or rightness of action but is rightness of will.” [St. Anselm, On Truth, 12]. The will is contained in the intellect of man, thus ‘to will’ something is an act or by the definition of some, “work”. Adam stood before God as a just man created with a soul that was ‘perfectly’ joined to the intellect and perfectly united with the will of God, overflowing with the knowledge of truth; the intellect functioned in the light of God's will disciplining the lower appetites through reason alone. Adam was just, ‘right with God’, “righteous”. However, because of Adam’s unjust act we now bear the guilt and punishment for the sin of this one man, the deprivation of justice.
Adam's original justice was a prevailing moral quality or habit that perfectly joined Adam’s will to an enlightened understanding of the will of God. The state of being just inexplicably joined other cardinal virtues giving the rights to honorable prudence, temperance, and fortitude in moral acts. Prior to his original act of rebellion, Adam 'abided' in God. This abiding is much like our invitation to abide in Christ when partaking in the Eucharist. [Cf. John 6:57]. Christ renews us. The punishment of original sin Adam’s progeny is inherited deprivation of original justice. All men inherit this one man’s sin, both guilt and punishment, being a part of the family of homo sapiens which is completely forgiven along with our own actual sins in Baptism. Justification then is the lifelong movement from being unjust to perfection of justice. As our desires remain disordered we sometimes continue in sin. While Adam and Eve did repent of their sin the punishment remains, death both physical and spiritual. Adam was deprived of being in a just state, the punishment of spiritual death. Through His sacrifice Jesus Christ brings to us and the rite of Baptism, a state going from unjust to perfect in justice.
We observe then that the forgiveness of sin is one aspect of atonement and that punishment is another aspect. Although we are absolved of our sins there is still the matter of punishment. One imperfection is unbelief or a weak faith. This is patently obvious when Moses and Aaron were incredulous that God would lead them to the Promised Land, Moses was left behind [Cf. Numbers 20:12] as punishment. Likewise, King David sinned yet confessed, while forgiven there remained a temporal punishment, the death of his young son. Consequently, though we might atone for sins, that confession may (sometimes not) have punishment associated with the transgression.
Accordingly, we hear Christ say, “Amen I say to thee, thou shalt not go out from thence till thou repay the last farthing." [Matthew 5:26]. And, “be perfect, even as your heavenly Father is perfect.” [Matthew 5:48]. While hampered with the human condition of disordered desire this would seem to be a herculean act. With purification of purgatory what isn’t completed on earth can be made perfect by Christ. We are handed over to prison [purgatory] until every last mite is paid [Cf. Matthew 5:26; 18:34; Luke 12:58]
In Luke we hear Christ describe His human vision of purgatory (one understood by mortals) where between the beggar and the rich man is a great chasm of chaos. The beggar in the bosom of Abraham receiving comfort from his hard life while the rich man in purgatory thirsting in purgatory for his uncharitable life. And then again, on the cross we hear the repentant thief given passage to paradeisos. Properly translated as ‘paradise’ but a paradise that was a different location other than heaven. Paradise considered a realm of the righteous dead, sheol. According to Thayer's Greek Lexicon this paradise is “that part of Hades which was thought by the later Jews to be the abode of the souls of the pious until the resurrection: Luke 23:43, cf. 16:23f;” otherwise known today as "purgatory".
We find then that purgatory can be a state of punishment or a pleasant place waiting for the resurrection of those who abide in Christ.
JoeT