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I'm looking for a science-fiction short story, probably written by either Isaac Asimov or Poul Anderson. (I was binging on short stories collections from these two authors around the same time, 5 to 10 years ago, and I'm pretty sure the story I'm looking for came from one of these books.)

I recall from this story that an alien species has met with humanity, but they originated from a very different kind of world. On their planet, life evolved in another direction, and is based on nitrogen and/or arsenic (I think, it might be something else), whereas Earth-like life is based on carbon and oxygen.

To this alien species, these elements are poison, pretty much like arsenic and nitrogen are poison to humans, and this information plays a role in the story: there is an attempted murder on an alien or a suspicion of murder attempt, maybe at a dinner party. Before that, the relationship between the two species seemed relatively okay.

The story you are looking for might be "Hostess" by Isaac Asimov.

The Hawkinsites are from a planet that has hydrogen cyanide in the atmosphere. The Hawkinsites need it. Part of their matabolism depends on the cyanide. While on Earth, they have a kind of "oxygen mask" that trickles a small amount of hydrogen cyanide into their respiration system through a tube that runs into the corner of the mouth.

There is a scene in the story where a Hawkinsite has supper with the human couple he is staying with on Earth. The human male of the pair is in some high position in the Earth government's security forces. He ends up taking away the Hawkinsite's cyanide cylinder to force him to explain the real reason behind the Hawkinsite's visit to the Earth. The security guy then kills the Hawkinsite after he reveals that he is there for the reason the security guy suspected.

Here's why the security guy killed the Hawkinsite:

The Hawkinsite suspected that the Earth was the source of an infection that had begun killing Hawkinsites and other intelligent beings, but not Earth humans. His suspicion was correct, and the Earth security guy knew it. Earth was not attacking the other planets. There was some intelligent parasite that lived on Earth (in humans) so humans were adapted and lived normal lives. The parasite was, however, deadly to the other races. The parasite needed humans to live and reproduce, but liked to inhabit the other races for the novelty. The security guy killed the Hawkinsite because the infection was known to the Earth government, but there was no cure. The Hawkinsite was the first alien to figure it out. The security guy killed him to keep the secret from spreading.

The alien species might the Ymirites, brainchildren of Poul Anderson. They are a race of hydrogen breathers, who normally don't get into contact with oxygen breathers that much. In the short story Hunters of the Sky Cave, the agent of the Terran Empire Dominic Flandry is investigating an attack on the planet Vixen. The evidence seems to point to the Ymirites als culprits, but are they really?

This short story has also been expanded into a novel first published as We Claim These Stars!.

Hostess by Asimov?

Starring alien needs low concentrations of cyanide in the air he breathes, so he carries a small cylinder of it with him, and takes a breath from time to time. However oxygen is not poisonous. He is is shot and killed at the end of the story, at the end of a dinner he had been invited to.

I think is C-Chute by Asimov.
Plot (Wikipedia):

During Earth's first interstellar war, a civilian transport traveling to Earth is captured by the Kloros, a chlorine-breathing race of intelligent beings. The ship is commandeered by two Kloros along with six human civilians as prisoners of war. The humans fall into argument and dispute, some coming to blows, with contradictory feelings on what should be done. Opinions range from a violent counteroffensive to a passive acceptance of their situation. Stuart, who has previously spent time as a guest of the Kloros, where he was provided with prosthetic hands when his own were damaged in an accident, posits that the Kloros are masters of chemistry (thus easily able to maintain an atmosphere and provide food for the captives) but less proficient at engineering, hence prefer to steal human ships to use in the war.

Only Mullen, a shy, mild-mannered, short bookkeeper, is willing to make an attempt to take back control of the ship, which he does by exiting via the C-Chute (short for "casualty chute", normally used for launching corpses for burial in space) and entering the control room via the navigational steam-tubes. He successfully kills the two Kloros by spraying them with oxygen.

As an unlikely hero, Mullen admits that he was not motivated by bravery, anger, or fear, but by homesickness for Earth (specifically his hometown, Richmond, Virginia), where he has not returned for 17 years, and that he could not face the prospect of waiting out the war in captivity when on the cusp of returning home.

Probably not the story you're looking for, but I think it's still worthy as an answer since it matches some of the elements.

The 2001 movie Evolution is a sci-fi comedy about a meteor that crashes into Earth that brings an alien lifeform that begins to evolve on Earth itself. Their molecules are nitrogen-based, and the logic is that, since carbon lifeforms are vulnerable to arsenic, then nitrogen lifeforms are vulnerable to selenium. Then they proceed to kill the aliens with Head & Shoulders because it has selenium sulfide as an ingredient. (that's one weird sponsor, but I guess it works for the plot).