Objection 1. It would seem that the effect of Christ's priesthood is not the expiation of sins. For it belongs to God alone to blot out sins, according to Isaiah 43:25: "I am He that blot out thy iniquities for My own sake." But Christ is priest, not as God, but as man. Therefore the priesthood of Christ does not expiate sins.
Objection 2. Further, the Apostle says (Hebrews 10:1-3) that the victims of the Old Testament could not "make" (the comers thereunto) "perfect: for then they would have ceased to be offered; because the worshipers once cleansed should have no conscience of sin any longer; but in them there is made a commemoration of sins every year." But in like manner under the priesthood of Christ a commemoration of sins is made in the words: "Forgive us our trespasses" (Matthew 6:12). Moreover, the Sacrifice is offered continuously in the Church; wherefore again we say: "Give us this day our daily bread." Therefore sins are not expiated by the priesthood of Christ.
Objection 3. Further, in the sin-offerings of the Old Law, a he-goat was mostly offered for the sin of a prince, a she-goat for the sin of some private individual, a calf for the sin of a priest, as we gather from Leviticus 4:3-28. But Christ is compared to none of these, but to the lamb, according to Jeremiah 11:19: "I was as a meek lamb, that is carried to be a victim." Therefore it seems that His priesthood does not expiate sins.
On the contrary, The Apostle says (Hebrews 9:14): "The blood of Christ, Who by the Holy Ghost offered Himself unspotted unto God, shall cleanse our conscience from dead works, to serve the living God." But dead works denote sins. Therefore the priesthood of Christ has the power to cleanse from sins.
I answer that, Two things are required for the perfect cleansing from sins, corresponding to the two things comprised in sin--namely, the stain of sin and the debt of punishment. The stain of sin is, indeed, blotted out by grace, by which the sinner's heart is turned to God: whereas the debt of punishment is entirely removed by the satisfaction that man offers to God. Now the priesthood of Christ produces both these effects. For by its virtue grace is given to us, by which our hearts are turned to God, according to Romans 3:24-25: "Being justified freely by His grace, through the redemption that is in Christ Jesus, Whom God hath proposed to be a propitiation, through faith in His blood." Moreover, He satisfied for us fully, inasmuch as "He hath borne our infirmities and carried our sorrows" (Isaiah 53:4). Wherefore it is clear that the priesthood of Christ has full power to expiate sins.
Reply to Objection 1. Although Christ was a priest, not as God, but as man, yet one and the same was both priest and God. Wherefore in the Council of Ephesus [Part III, ch. i, anath. 10] we read: "If anyone say that the very Word of God did not become our High-Priest and Apostle, when He became flesh and a man like us, but altogether another one, the man born of a woman, let him be anathema." Hence in so far as His human nature operated by virtue of the Divine, that sacrifice was most efficacious for the blotting out of sins. For this reason Augustine says (De Trin. iv, 14): "So that, since four things are to be observed in every sacrifice--to whom it is offered, by whom it is offered, what is offered, for whom it is offered; the same one true Mediator reconciling us to God by the sacrifice of peace, was one with Him to Whom it was offered, united in Himself those for whom He offered it, at the same time offered it Himself, and was Himself that which He offered."
Reply to Objection 2. Sins are commemorated in the New Law, not on account of the inefficacy of the priesthood of Christ, as though sins were not sufficiently expiated by Him: but in regard to those who either are not willing to be participators in His sacrifice, such as unbelievers, for whose sins we pray that they be converted; or who, after taking part in this sacrifice, fall away from it by whatsoever kind of sin. The Sacrifice which is offered every day in the Church is not distinct from that which Christ Himself offered, but is a commemoration thereof. Wherefore Augustine says (De Civ. De. x, 20): "Christ Himself both is the priest who offers it and the victim: the sacred token of which He wished to be the daily Sacrifice of the Church."
Reply to Objection 3. As Origen says (Sup. Joan. i, 29), though various animals were offered up under the Old Law, yet the daily sacrifice, which was offered up morning and evening, was a lamb, as appears from Numbers 28:3-4. By which it was signified that the offering up of the true lamb, i.e. Christ, was the culminating sacrifice of all. Hence (John 1:29) it is said: "Behold the Lamb of God, behold Him Who taketh away the sins [Vulgate: 'sin'] of the world."
Source: Summa Theologica (New Advent)