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The Alchemist subclass of the artificer has a 5th level feature Alchemical Savant which states (E:RftLW, pg 58)

Whenever you cast a spell using your alchemist's supplies as the spellcasting focus, you gain a bonus to one roll of the spell. That roll must restore hit points or be a damage roll that deals acid, fire, necrotic, or poison damage, and the bonus equals your intelligence modifier

Elemental Weapon is an Artificer spell gives the option to grant an extra 1d4 of acid/fire damage when the enchanted weapon hits a target. Does this extra damage count as a roll of the spell and so can the roll benefit from the alchemist's intelligence modifier bonus for a single hit?

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No, this does not work.

First, let's look at Alchemical Savant:

At 5th level, you develop masterful command of magical chemicals, enhancing the healing and damage you create through them. Whenever you cast a spell using your alchemist’s supplies as the spellcasting focus, you gain a bonus to one roll of the spell. That roll must restore hit points or be a damage roll that deals acid, fire, necrotic, or poison damage, and the bonus equals your Intelligence modifier (minimum of +1).

This gives you a bonus to one damage roll for the spell's damage.

Let's look at elemental weapon:

For the duration, the weapon has a +1 bonus to attack rolls and deals an extra 1d4 damage of the chosen type when it hits.

The spell elemental weapon does not do damage. It changes a weapon so that the weapon deals an additional 1d4 damage. This damage is from the weapon when it hits, not from the spell.

Therefore, the 1d4 roll is not an eligible damage roll for Alchemical Savant, since it is weapon damage, not a damage roll from the spell.

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(I've looked over the spell a couple more times, and had to change my answer)

Elemental Weapon applies a persistent effect to a non-magical weapon for its duration. Emphasis mine

A nonmagical weapon you touch becomes a magic weapon. Choose one of the following damage types: acid, cold, fire, lightning, or thunder. For the duration, the weapon has a +1 bonus to attack rolls and deals an extra 1d4 damage of the chosen type when it hits.

The spell alone does not have you roll damage or healing, and so would not benefit from Alchemical Savant.

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