# Is amount of substance fundamentally a scalar quantity? (in the mathematical sense of scalar)

Reading the SI (and ISO) standard for units and quantities, I'm currently puzzled by something very subtle.

If I can see and understand why we talk about scalars, vectors, and tensors in the context of physics quantities (because we fundamentally talk about space-time, geometry, and systems of coordinates), it does not seem as trivial to me for the quantity "amount of substance" (unit mol) that originally comes from chemistry (and which does not seem related to vector spaces or coordinate systems).

Could someone clarify that from a very (very) pedantic standpoint (ideally with sources/citations that would talk about this question)?

Important addition regarding the context: According to BIPM/JCGM: units of measurements are "real scalar quantity, defined and adopted by convention, with which any other quantity of the same kind can be compared to express the ratio of the two quantities as a number". And "amount of substance", is one of the 7 base quantities of SI. And the standard also talk about scalars, vectors, and tensors. The question is raised in this context.

• pedantic standpoint: There is no such thing as “a scalar quantity”. There are “scalars under 3D rotations”, “scalars under Lorentz transformations”, etc. You have to specify the transformations being considered. For example, relativistic energy $\gamma mc^2$ is a scalar under 3D rotations but not under Galilean boosts, and it’s the temporal component of the energy-momentum four-vector under Lorentz transformations. Feb 6 at 23:31
• (However particle number is a scalar under any reasonable coordinate transformation – even if we are in a strongly accelerated frame where the Unruh bath becomes relevant the particle number does not change as particles and antiparticles are created symmetrically. And the "amount of substance" is just a convenient rescaling of the particle number.) Feb 6 at 23:34
• @SebastianRiese So if I have a particle-antiparticle pair, I have no “substance”? I doubt that that is how the OP thinks about “substance”. A mole of matter and a mole of antimatter is not zero moles to me. Feb 6 at 23:35
• But that's the only reasonable definition of a conserved quantity "particle number" (or baryon and lepton number or whatever). Of course the "amount of substance" stops being useful in situations where we have pair production as the energy scales break chemical bonds, ionizes atoms, etc. Feb 6 at 23:41
• Particles don't care: they just go about their business without calculating anything. Mathematical objects like scalars exist only in the mind. So, the question becomes, what aspect of reality are you attempting to model? Feb 7 at 0:15

## 2 Answers

The unit $$\mathrm{mol}$$ is dimensionless. It refers to a countable, integral number of a thing. It's like counting in 100s, except $$1 \mathrm{mol}$$ is larger. $$1 \mathrm{mol}$$ is an integer, equal to about $$1 \mathrm{mol} \approx 6.022 \times 10^{23} \, .$$ The number on the right is called the Avogadro Constant.

The $$\mathrm{mol}$$ is conventionally used in chemistry to count the number of molecules, or the number of atoms. That's really it. It's more convenient to count in numbers of moles because $$1 \mathrm{mol}$$ of common compounds and atoms are an easily imaginable mass-- i.e., $$1 \mathrm{mol}$$ of hydrogen atoms about $$1 \mathrm{gram}$$.

If you want to be pedantic, the number of moles of something may not be well-defined because the substance may be in a superposition of different particle-number states. But this is always ignorable in practical uses of the unit $$\mathrm{mol}$$.

If you want to be pedantic in a different way, the word 'scalar' is used in physics to refer to members of $$\mathbb R$$ or $$\mathbb C$$. In more advanced physics, the word 'scalar' is used as a description of a mathematical entity's transformation properties. For example, a 'scalar field' is a field which doesn't change value at any point as a result of spatial transformations which scale the coordinates. The points move around but internally the scalar value doesn't change, it just shifts around. On the other hand, a scalar density is a quantity which is defined to stretch/squeeze with the coordinate system, in such a way that if you integrate over a given volume, that integral will be preserved even under a coordinate transformation.

The reason that scalars and scalar densities have so much salience to us (i.e., they are intuitive) is because locally (which is what our classical brain is attuned to-- variations in spacetime are not large enough for us to really observe in a visceral and immediate sense), all spacetimes are Minkowskian (flat spacetime). And all of the isometric transformations you can do on a Minkowskian spacetime (Poincare transformations) happen to have no effect on scalars or scalar densities. I.e., if you and I are using different coordinates on the same shared space, then regardless of our coordinate choices, we agree on the value of scalars and scalar densities. (I have ignored Lorentz boosts, since we don't have classical intuition for those either).

I think the answer is somewhat context dependent.

Let $$N_X(t)$$ be the number of moles of substance $$X$$ in volume $$V$$ at time $$t$$". Then certainly $$N_X(t)$$ does not depend on how $$V$$ is oriented in space. In that sense, $$N_X(t)$$ is a scalar -- its numerical value does not change under rotations.

However, suppose we consider the reaction container from the perspective of an observer who is moving at near the speed of light, and is spacelike separated from the reaction vessel at time $$t$$. Then conceivably the number of moles in the reaction vessel may be different, if a reaction has been taking place changing the number of moles of $$X$$, or if there is a net flow of $$X$$ into or out of the container, due to the relativity of simultaneity. For this reason, number density is not a 4-scalar quantity in special relativity, but a component of a 4-vector.

I think given the context of the measurement standards you link, the relevant answer is the first one, not involving special relativity. Namely, the number of moles of a substance is a scalar because it is invariant under rotations.

• $N_{X}$ is a scalar, but formally speaking $N_{X}$ dimension is $N.L^{-3}$, $N$ being the dimension of "amount of substance in SI". It may be subtle but what about the quantity "amount of substance" (not per volume)? Feb 7 at 1:29
• @Vincent I don't understand. $N_X$ is the total number of moles of $X$ in some container; I haven't normalized it by a standard volume like "moles per cubic meter". Give me a physical example where you would have dimension $N$ instead of $N\cdot L^{-3}$ in your notation. Feb 7 at 2:11