The Eucharist, or viaticum, given to the sick
“I desire therefore, first of all, that supplications, prayers, intercessions, and thanksgivings be made for all men: . . .”
Here in the Greek passage, thanksgiving is εὐχαριστία which is the same term used by early Church Fathers to designate the Lord’s Supper which consists of the bread and wine transmutated into the Body and Blood of Christ as Saint Just Martyr wrote. Though I would not say that such thanksgiving is excluded from private prayer, the liturgical act of thanksgiving or Eucharist should not be ignored in this passage among many others which mention the liturgical act of εὐχαριστία.
So, Saint Paul is exhorting Christians to offer all Eucharists for all. The Church teaches this also includes the faithful departed in Purgatory . We bear in mind what Judas Maccabeus did for his soldiers slain in battle when he “made atonement for the deed that they might be freed from sin” . We see the early Church did offer a Sacrifice for the faithful departed as Tertullian said, “We offer sacrifices for the dead on their birthday anniversaries” . Saint Cyril of Jerusalem commented on this saying that “we believe that it will be of very great benefit to the souls of those for whom the petition is carried up, while this holy and most solemn sacrifice is laid out” .
We pray for the faithful departed, that they may enjoy the beatific vision.
 Catechism of the Catholic Church, nn. 1371, 1689
 2 Maccabees 12:46
 Tertullian. The Crown 3:3.
 Saint Cyril of Jerusalem. Catechetical Lectures 23:5:9.