Objection 1. It would seem that Christ's Passion did not bring about our salvation by way of atonement. For it seems that to make the atonement devolves on him who commits the sin; as is clear in the other parts of penance, because he who has done the wrong must grieve over it and confess it. But Christ never sinned, according to 1 Peter 2:22: "Who did no sin." Therefore He made no atonement by His personal suffering.
Objection 2. Further, no atonement is made to another by committing a graver offense. But in Christ's Passion the gravest of all offenses was perpetrated, because those who slew Him sinned most grievously, as stated above (Question 47, Article 6). Consequently it seems that atonement could not be made to God by Christ's Passion.
Objection 3. Further, atonement implies equality with the trespass, since it is an act of justice. But Christ's Passion does not appear equal to all the sins of the human race, because Christ did not suffer in His Godhead, but in His flesh, according to 1 Peter 4:1: "Christ therefore having suffered in the flesh." Now the soul, which is the subject of sin, is of greater account than the flesh. Therefore Christ did not atone for our sins by His Passion.
On the contrary, It is written (Psalm 68:5) in Christ's person: "Then did I pay that which I took not away." But he has not paid who has not fully atoned. Therefore it appears that Christ by His suffering has fully atoned for our sins.
I answer that, He properly atones for an offense who offers something which the offended one loves equally, or even more than he detested the offense. But by suffering out of love and obedience, Christ gave more to God than was required to compensate for the offense of the whole human race. First of all, because of the exceeding charity from which He suffered; secondly, on account of the dignity of His life which He laid down in atonement, for it was the life of one who was God and man; thirdly, on account of the extent of the Passion, and the greatness of the grief endured, as stated above (Question 46, Article 6). And therefore Christ's Passion was not only a sufficient but a superabundant atonement for the sins of the human race; according to 1 John 2:2: "He is the propitiation for our sins: and not for ours only, but also for those of the whole world."
Reply to Objection 1. The head and members are as one mystic person; and therefore Christ's satisfaction belongs to all the faithful as being His members. Also, in so far as any two men are one in charity, the one can atone for the other as shall be shown later (XP, 13, 2). But the same reason does not hold good of confession and contrition, because atonement consists in an outward action, for which helps may be used, among which friends are to be computed.
Reply to Objection 2. Christ's love was greater than His slayers' malice: and therefore the value of His Passion in atoning surpassed the murderous guilt of those who crucified Him: so much so that Christ's suffering was sufficient and superabundant atonement for His murderer's crime.
Reply to Objection 3. The dignity of Christ's flesh is not to be estimated solely from the nature of flesh, but also from the Person assuming it--namely, inasmuch as it was God's flesh, the result of which was that it was of infinite worth.
Source: Summa Theologica (New Advent)