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Looked and looked, can't find the answer anywhere (maybe I'm blind):

How long does Alchemist's Fire burn for?

In the item description within the Adventuring Gear section of the rules, it says:

On a hit, the target takes 1d4 fire damage at the start of each of its turns. A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames.

Yet no mention of the burn rate. One round after a hit? 2? A minute? Continuous until extinguished?

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It burns until extinguished

Alchemist’s Fire.
... On a hit, the target takes 1d4 fire damage at the start of each of its turns. A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames. (Basic Rules, p. 47; PHB, p. 151)

If no effort is made to extinguish the fire, it will keep burning.

The attack can miss, however, so it's not a guaranteed problem for the target.

Make a ranged attack against a creature or object, treating the alchemist’s fire as an improvised weapon

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    \$\begingroup\$ Does this mean that Alchemist's Fire can sit on something non-flammable, and will continue to burn forever without intervention? Because that sounds like a poor man's Continual Flame. \$\endgroup\$ – Daniel Zastoupil Jul 31 '18 at 20:14
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    \$\begingroup\$ @DanielZastoupil I suggest that you ask that as a separate question. If you look at the text I cited, if you don't take fire damage in the first place (not being able to burn, immune to fire) then you can't take more fire damage the next turn ... but I really thing you should ask that as a separate question. Might attract some very interesting answers. \$\endgroup\$ – KorvinStarmast Jul 31 '18 at 20:23
  • \$\begingroup\$ You can add further support that there are things that DO have time limit for the burning (like the oil flask). The absence of the limitation is telling here because there are examples where limits are given. No limit given is no limit. \$\endgroup\$ – NautArch Jun 27 at 15:46
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The rules don't say. Make a ruling.

The description of Alchemist's Fire covers only one use case: throwing the flask at a creature that you want to set on fire. The only described way for the fire to go out is for the creature to extinguish it. (Note that extinguishing it doesn't require magic or special tools, just dexterity, implying that it is a mundane fire that needs air.)

Other than that, the rules say nothing about how long the fire keeps burning.

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