Atom

Theory that atoms exist, i.e. matter is not continuous.

Much before atoms were thought to be "experimentally real", chemists from the 19th century already used "conceptual atoms" as units for the proportions observed in macroscopic chemical reactions, e.g. $2H_2+O_2\leftrightarrow H_{2}O$. The thing is, there was still the possibility that those proportions were made up of something continuous that for some reason could only combine in the given proportions, so the atoms could only be strictly consider calculatory devices pending further evidence.

Subtle is the Lord by Abraham Pais (1982) chapter 5 "The reality of molecules" has some good mentions. Notably, physicists generally came to believe in atoms earlier than chemists, because the phenomena they were most interested in, e.g. pressure in the ideal gas law, and then Maxwell-Boltzmann statistics just scream atoms more loudly than chemical reactions, as they saw that these phenomena could be explained to some degree by traditional mechanics of little balls.

Confusion around the probabilistic nature of the second law of thermodynamics was also used as a physical counterargument by some. Pais mentions that Wilhelm Ostwald notably argued that the time reversibility of classical mechanics + the second law being a fundamental law of physics (and not just probabilistic, which is the correct hypothesis as we now understand) must imply that atoms are not classic billiard balls, otherwise the second law could be broken.

Pais also mentions that a big "chemical" breakthrough was Isomers suggest that atoms exist.

Very direct evidence evidence:

Less direct evidence:

Subtle is the Lord by Abraham Pais (1982) page 40 mentions several methods that Einstein used to "prove" that atoms were real. Perhaps the greatest argument of all is that several unrelated methods give the same estimates of atom size/mass:

Subtle is the Lord by Abraham Pais (1982) mentions that this has a good summary of the atomic theory evidence that was present at the time, and which had become basically indisputable at or soon after that date.

On Wikimedia Commons since it is now public domain in most countries: commons.wikimedia.org/w/index.php?title=File:Perrin,_Jean_-_Les_Atomes,_F%C3%A9lix_Alcan,_1913.djvu

An English translation from 1916 by English chemist Dalziel Llewellyn Hammick on the Internet Archive, also on the public domain: archive.org/details/atoms00hammgoog

Subtle is the Lord by Abraham Pais (1982) page 85:

However, it became increasingly difficult in chemical circles to deny the reality of molecules after 1874, the year in which Jacobus Henricus van't Hoff and Joseph Achille Le Bel independently explained the isomerism of certain organic substances in terms of stereochemical properties of carbon compounds.

so it is quite cool to see that organic chemistry is one of the things that pushed atomic theory forward. Because when you start to observe that isomers has different characteristics, despite identical proportions of atoms, this is really hard to explain without talking about the relative positions of the atoms within molecules!

TODO: is there anything even more precise that points to atoms in stereoisomers besides just the "two isomers with different properties" thing?

Small microscopic visible particles move randomly around in water.

If water were continuous, this shouldn't happen. Therefore this serves as one important evidence of atomic theory.

The amount it moves also quantitatively matches with the expected properties of water and the floating particles, was was settled in 1905 by Einstein at: investigations on the theory of the Brownian movement by Einstein (1905).

This suggestion that Brownian motion comes from the movement of atoms had been made much before Einstein however, and passed tortuous discussions. Subtle is the Lord by Abraham Pais (1982) page 93 explains it well. There had already been infinite discussion on possible causes of those movements besides atomic theory, and many ideas were rejected as incompatible with observations:

Further investigations eliminated such causes as temperature gradients, mechanical disturbances, capillary actions, irradiation of the liquid (as long as the resulting temperature increase can be neglected), and the presence of convection currents within the liquid.

The first suggestions of atomic theory were from the 1860s.

Tiny uniform plastic beads called "microbeads" are the preferred 2019 modern method of doing this: en.wikipedia.org/wiki/Microbead

Original well known observation in 1827 by Brown, with further experiments and interpretation in 1908 by Jean Baptiste Perrin. Possible precursor observation in 1785 by Jan Ingenhousz, not sure why he wasn't credited better.

Was the first model to explain the Balmer series, notably linking atomic spectra to the Planck constant and therefore to other initial quantum mechanical observations.

This was one of the first major models that just said:

I give up, I can't tie this to classical physics in any way, let's just roll with it, OK?

It still treats electrons as little points spinning around the nucleus, but it makes the non-classical postulate that only certain angular momentums (and therefore energies) are allowed.

Bibliography:

Bagic jump between orbitals in the Bohr model. Analogous to the later wave function collapse in the Schrödinger equation.

Causality and quantum jumps are incompatible.

Refinement of the Bohr model that starts to take quantum angular momentum into account in order to explain missing lines that would have been othrwise observed TODO specific example of such line.

They are not observe because they would violate the conservation of angular momentum.

Introduces the azimuthal quantum number and magnetic quantum number.

TODO confirm year and paper, Wikipedia points to: zenodo.org/record/1424309#.yotqe3xmjhe