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I'll start by saying that I'm more oriented towards the math community and I hope to write a good question.

Context

I'm taking a quantum mechanics course where the professor is using a convention called Natural units where he puts

$$c=1\qquad \hslash=1\qquad G=1\qquad\frac{1}{4\pi\varepsilon_0}=1\qquad k=1$$

Now, to do this operation it is technically possible to assign to $1$ only a maximum of $7$ universal constants, given that the fundamental units of measurement of the international system are $7$ (second, meter, kilogram, ampere, kelvin, mole, candela).

Question

Strictly speaking, it would therefore be possible to assign $1$ to two other universal constants, which in this case however must depend only on the moles and/or the candela (since the conversion between these first $5$ quantities is solved autonomously without the use of moles and candela).

In my ignorance, I simply looked up the list of universal constants on Wikipedia and looked for those that could depend on mole and candela.

  • For the mole I saw that there are several constants that depend to some extent on the moles (trivially the ideal gas constant)
  • For the candle there are no constants associated with this quantity

This seems a bit strange to me, because it would mean that there are no physical phenomena that relate to this quantity to create derived quantities.

How come this phenomenon occurs?

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The mole is just a pure number, approx 6.02E23, so there is no physical constant it is based on.

The candela is based on the light that the human eye perceives, and so is a function of our biology, not anything fundamental about the universe.

Both of these units are defined and standardized because they are useful for science and commerce, in chemistry for moles, and in standardizing light bulbs and other devices for candela (you'll more often see its derived units, lux and lumens). Not because they are foundational to physics.

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  • $\begingroup$ Ok, yes, but for moles there are relationships that correlate different quantities that depend on constants that have to do with moles (for example $R\approx 8.314 J\operatorname{mol^{-1} K^{-1}}$). I was wondering why there were no constants that depend on lumens in equations that connect different physical quantities. Even if they depend on the perception of the human eye if it is a measurable quantity, right? $\endgroup$ Commented Apr 10 at 12:26
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    $\begingroup$ @MathAttack, Re, "why there [are] no constants that depend on lumens in equations that connect different physical quantities. We use lumens to report how effective a lighting appliance is at lighting up a room or other space. Physicists aren't interested in how bright a light source appears to human eyes. Physicists want to know how much power it emits, and maybe, how much power in any given band of wavelengths. The label on a laser won't tell you how many lumens it puts out. The label tells you how many watts. $\endgroup$ Commented Apr 10 at 14:32
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    $\begingroup$ P.S., The label on a light bulb also typically mentions "watts," but that's for an entirely different reason. The wattage of a light bulb tells you how much electrical energy it consumes. Back when I was a kid, and nearly all residential lighting was incandescent, we used the wattage of a light bulb as a proxy for its brightness, but today, pretty much all light bulb labels will tell you both how much power it uses (watts), and how bright it appears (lumens). $\endgroup$ Commented Apr 10 at 14:35

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