At 1st level, you learn three Alchemical Formula options:Alchemical Fire, Alchemical Acid, and one other option of your choice.
You learn an additional formula of your choice at 3rd, 9th, 14th, and 17th levels. To use any of these options, your Alchemist’s Satchel must be within reach. If an Alchemical Formula option requires a saving throw, the DC is 8 + your proficiency bonus + your Intelligence modifier.
Alchemical Fire. As an action, you can reach into your Alchemist’s Satchel, pull out a vial of volatile liquid, and hurl the vial at a creature, object, or surface within 30 feet of you (the vial and its contents disappear if you don’t hurl the vial by the end of the current turn). On impact, the vial detonates in a 5-foot radius. Any creature in that area must succeed on a Dexterity saving throw or take 1d6 fire damage. This formula’s damage increases by 1d6 when you reach certain levels in this class: 4th level (2d6), 7th level (3d6), 10th level (4d6), 13th level (5d6), 16th level (6d6), 19th level (7d6).
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the alchemist fire created by the Artificer do the fire damage at the start of each of its turns like the alchemist fire you can buy?
If the class doesn't specify then it is the same as the fire you can buy. You are essentially just creating it so I'd assume it works the same.
Is the damage also the same?
Alchemical Formula
At 1st level, you learn three Alchemical Formula options:Alchemical Fire, Alchemical Acid, and one other option of your choice.
You learn an additional formula of your choice at 3rd, 9th, 14th, and 17th levels. To use any of these options, your Alchemist’s Satchel must be within reach. If an Alchemical Formula option requires a saving throw, the DC is 8 + your proficiency bonus + your Intelligence modifier.
Alchemical Fire. As an action, you can reach into your Alchemist’s Satchel, pull out a vial of volatile liquid, and hurl the vial at a creature, object, or surface within 30 feet of you (the vial and its contents disappear if you don’t hurl the vial by the end of the current turn). On impact, the vial detonates in a 5-foot radius. Any creature in that area must succeed on a Dexterity saving throw or take 1d6 fire damage. This formula’s damage increases by 1d6 when you reach certain levels in this class: 4th level (2d6), 7th level (3d6), 10th level (4d6), 13th level (5d6), 16th level (6d6), 19th level (7d6).