How does Christ’s atonement achieve salvation for the elect?

According to the doctrine of particular redemption, Christ's atonement is both efficacious and definitively applied to the elect. This means that His sacrifice on the cross effectively paid for their sins, and as a result, those whom He substituted for will undoubtedly be saved. Romans 3:24-26 affirms this idea by highlighting that God is both just and the justifier of those who have faith in Jesus. If Christ died for all without application, many would still face eternal punishment, contradicting the idea of effective atonement. Thus, the efficacy of Christ’s sacrifice comes from its intention and application to the elect, guaranteeing their salvation.