Any constant in physics or chemistry comes from experimental design and consistency in measurements. For example, if you divide multiple of pressure and volume with the multiple of moles and temperature, you will get the constant R for any ideal gas.
However, given how small planck’s constant is, how can we assume its accuracy?
In: Physics
There are two questions hidden here: how can we measure Planck’s constant so accurately, and what’s up with these experiments not actually measuring Planck’s constant?
You lay the groundwork for the first question: we set up a scenario where some result will depend on Planck’s constant, then we measure all of the inputs and that lets us do the math.
In the case of Planck’s constant the most accurate experiment is known as a Kibble balance. The specifics are a bit deep, but it involves using a known current in a wire to match the mass of a known object. The setup allows us to measure some quantities that are pretty straightforward to get super precise, so the resulting value for Planck’s constant winds up being pretty precise, too.
In fact, this got to be so precise that the best Kibble balances in the world were able to measure the mass of a kilogram more precisely than we could maintain the mass of a kilogram. Until recently the kilogram was defined in terms of a platinum-iridium cylinder kept in Paris. The mass of this chunk of metal could not be “measured” since the mass is *defined*. Instead, an experiment that would normally measure the mass of the prototype kilogram would actually be measuring the measurement device: if the scales read 1.001 kg then that tells you that your scale is wrong by 1 gram, not that the artifact is a gram heavier than expected.
It used to be that most physical constants were defined this way. The meter was once defined in terms of the length of a rod of metal. The second was defined in terms of a 12,756 kilometer diameter sphere of rock hurtling through space (known to most people simply as home, or perhaps as “earth”). These sorts of definitions are easy to write, but they’re hard to maintain and transmit, especially when it turns out that the physical artifact varies over time. Rods of metal lengthen and contract with temperature or can wear down. Earth’s rotation slows by about 2 milliseconds per century and once you have clocks good enough to measure this fact it becomes awkward to say that each rotation of the planet is exactly 86,400 seconds and the seconds just get longer and shorter.
Over time we went from defining physical constants in terms of these artifacts to defining them in terms of fundamental physical constants. For example, there’s a specific form of EM radiation that a caesium atom can emit. This is perfectly repeatable, so far as we can tell, so we just declared that 9,192,631,770 cycles of that radiation shall be a second. That number may look random, but it was chosen so that a second with this definition is basically the same as previous definitions.
Armed with that definition of a second and the observation that the speed of light is constant we can go on to *define* the speed of light to be *exactly* 299,792,458 m/s. Like the earlier description of measuring the prototype kilogram, if you were to measure the speed of light and you got 300,000,000 m/s then that means that your measurement device is reading ever so slightly high (about 0.07%).
For a long time most of the units of SI were defined in terms of physical constants in this way, but the kilogram proved elusive. Nobody had managed to demonstrate a method of measuring a physical constant precisely enough that it exceeded the precision with which the mass of a morsel of platinum-iridium alloy could be maintained. The Kibble balance was the breakthrough here (building on a long line of similar devices) as it reached that level of precision. In 2019 SI formally redefined the kilogram: instead of saying “this lump of metal in Paris is a kilogram and everything else is measured relative to it” they declared that Planck’s constant has a value of 6.62607015 * 10^-34 J*s. The Joule is a kilogram m^2 / s^2 so once you know the value of a meter and second you can use this definition to derive the size of a kilogram.
This is what is meant when Planck’s constant is given as an exact value. It isn’t that we can measure Planck’s constant with infinite precision–such a thing would obviously be impossible. Instead it identifies that this is one of the constants that we build our unit system off of. As we get more and more precise experiments to measure Planck’s constant this manifests as more and more precise measurements of what a kilogram is.
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