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We see the use or mention of nuclear weapons in sci-fi world all the time. Examples can be Terminator, Star Trek, X-Men etc.
There are two related questions here:
Which Sci-Fi work first mentioned Nuclear Weapons?
Which Sci-Fi work first showed the use of Nuclear Weapons? Meaning, Nuclear Weapons are actually used.
H. G. Wells predicted the atom bomb in his 1914 book, The World Set Free.
His story not only mentioned nuclear weapons, but showed them in use with a fore-knowledge that seems scarily accurate. (Kind of like how he accurately predicted the Apollo missions to the moon.)
He predicted bombs based on radioactive elements that were far more destructive than any conventional weapons. His knowledge of atomic physics came from reading William Ramsay, Ernest Rutherford, and Frederick Soddy; the last discovered the disintegration of uranium. Wells already knew that radioactive elements released far more energy than any bombs based on chemical reactions. Although scientists like Soddy and Rutherford knew the nucleus of an atom contained enormous amounts of energy, they believed that energy was unavailable for human use.
Scientists of the time were well aware that the slow natural radioactive decay of elements like radium continues for thousands of years, and that while the rate of energy release is negligible, the total amount released is huge. His whole book was based on the premise that if the energy was released over a very short time, instead of thousands of years, it would be an incredible bomb.
(Based on the description of the book from the Wikipedia article.)
He made several accurate predictions of nuclear weapons.
He said, "a man could carry about in a handbag an amount of latent energy sufficient to wreck half a city." The critical mass of some fissile isotopes is small enough for a man to hold in his hands.
He predicted that cities would remain radioactive wastelands for many years after the battles were over.
He predicted the creation of nuclear weapons based on radioactive elements. But he did not go far enough to understand that only a small fraction of elements have fissile isotopes because no scientists in 1914 understood that.
He predicted that a single atomic weapon could destroy a city.
And even more noteworthy is that he predicted the military doctrine of Mutual Assured Destruction!
He predicted they would be deployed by bomber planes when military strategists of the time dismissed airplanes as mere toys with no military application.
He also made some interesting wrong predictions.
He believed atomic weapons would explode continuously for days.
He believed the presence of nuclear weapons would force humanity to come together in peace and create a world government.
He thought the atom bomb would work by merely accelerating the natural decay process so that the half-life of an element was mere days instead of centuries.
