A.T. Robertson Commentary Hebrews 9:13

"For if the blood of goats and bulls, and the ashes of a heifer sprinkling them that have been defiled, sanctify unto the cleanness of the flesh:" — Hebrews 9:13 (ASV)

Ashes (σποδος). Old word, in N. T. only here, Matthew 11:21; Luke 10:13. Common in LXX.

Of a heifer (δαμαλεως). Old word (δαμαλις), a red heifer whose ashes mingled with water (μετα υδατος, verse 19) were sprinkled (ραντιζουσα, present active participle of ραντιζω, in LXX, though ραινω more common) on the contaminated or defiled ones (Numbers 19) as the blood of bulls and goats was offered for sins (Leviticus 16).

Sanctify (αγιαζε). First-class condition, assumed as true. This ceremonial ritual does serve "for the cleansing (καθαροτητα, old word here only in N.T.) of the flesh," but not for the conscience (verse 9). The cow was αμωμον, the individual καθαρος.