Which experiments prove atomic theory?

I would say that one experiment that demonstrates the atomic nature of things is the observation of Brownian motion. But it is not the experiment itself that convinces that things are made of atoms, rather its theoretical explanation given by Einstein in one of his 1905 papers (actually Einsteins work for his PhD was on the subject of atomic theory and there are several publications in the period 1903-1905). Of course there is also the observation of Rayleigh who calculated Avogadro’s number by the distance from which he could make out the figure of Mount Everest, assuming that light is scattered by atoms and that is why far away objects look fuzzy (1,2). Also scattering experiments demonstrated the atomic nature of things.

(1) Rayleigh, On the transmission of light through an atmosphere containing small particles in suspension, in Scientific Papers by Lord Rayleigh Vol. 4, pp. 247–405, New York: Dover, 1899/1964.

(2) P. Pesic, Eur. J. Phys. 26, 183 (2005).

(3) Patterson, G. Jean Perrin and the triumph of the atomic doctrine (2007) Endeavour, 31 (2), pp. 50-53.

I once heard Uhlenbeck give a lecture on this to high school students over the Christmas break at the Rockefeller Univ. years ago. He recounted a published argument he attributed to Einstein around 1905 (I think), which was that atoms were real if you could count the number of them/mole (Avogadro's number) many different independent ways, and you always got more or less the same answer. So Brownian motion, gas law, counting with an atomic force microscope, X-ray diffraction, spreading oil film, and many other possibilities would all count as subarguments to the main argument, i.e. that atoms were real. If anyone knows the reference(s), I would appreciate them.

This was a live question at the time. For example Mach, who died in 1916, was apparently an atomic skeptic.

No experiments prove any theory. Experiments can only refute theories.