The way I understand it one of the reasons we no longer have the ashes of the para aduma to purify people is because they were used up. I'd like to know if it would have been theoretically possible for them to have dumped the ashes of the para aduma into a lake or some other large body of water so that it could have lasted a long time. Would that have been halachically valid?
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The water that the ashes of the parah adumah are placed in must be drawn by a person from a body of water using a vessel. (Rambam Hilchos Para Adumah 6:1-8).
The Chazon Ish (Parah siman 12) proves that the ashes do not actually have to mix in with the water, they just have to be stirred a bit (see Temurah 12b for details of this mixing). Therefore, there does not need to be any proportion of ashes to water, and a body of water of any size would hypothetically be possible. This water would still need to be in a vessel (Mishna Parah 5:5), but as large of a vessel as you could procure would be OK.