Alchemist's Fire is a kind of Adventuring Gear which you can throw to deal 1D4 points of Damage to a creature and set it on-fire, dealing 1D4 damage at the start of each of its turns. From Roll20 (and the SRD):
Make a ranged Attack against a creature or object, treating the alchemist's fire as an improvised weapon. 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.
Using a Crafting Skill, what would happen if a Character took a regular arrow and attached a vial of Alchemist's Fire to it, say with sinew, then a Rogue fired it from a bow?
Could the Rogue then use this to Assassinate and Sneak Attack a Target? Would a Critical from the Assassinate effect (Surprised Target) also apply to the Fire Damage, or would that be exempt?
Best Answer
Yes, you get assassinate and sneak attack.
No, the critical would not apply to the fire damage.
Let us assume the DM allows you to craft this contraption, possibly giving it a -2 to hit or something of the sorts due to its unwieldy nature.
PHB, page 96, Sneak Attack:
Sneak attack is applied on any attack roll with advantage as long as it is a ranged or finesse weapon. A bow is a ranged weapon.
PHB, page 97, Assassinate:
We have our advantage and critical against a surprised creature that hasn't taken their turn yet.
PHB, page 196, Critical Hits:
The key phrases here is when you score a critical hit. You would roll double the damage dice for the hit on the turn you hit the creature (including double the sneak attack damage), but at start of each of that creature's turns you would roll the 1d4 fire damage separately without doubling it.