When and Why Was the Definition of the Meter Changed?

[Dale]

I understand that, but it isn’t the issue I am getting at. The basic rules for working with dimensionful are that you can only add quantities with the same dimension and you can multiply quantities of different dimensions to make quantities with new dimensions. You can always do a series expansion to express more complicated functions as sums and products.

What I am interested in is the idea of “amount of substance” as a dimension. Does it behave that way? Specifically, is there any use case for adding moles of different substances. Does “amount of substance” in general behave as a dimension under addition, or is each “amount of substance X” a separate dimension.

 

[Orodruin]

What I am interested in is the idea of “amount of substance” as a dimension. Does it behave that way? Specifically, is there any use case for adding moles of different substances. Does “amount of substance” in general behave as a dimension under addition, or is each “amount of substance X” a separate dimension.

In SI it is the same dimension.

Specifically, is there any use case for adding moles of different substances.

This to me sounds no different from ”is there any use case for adding m/s of velocity components in different directions?”

 

[Dale]

This to me sounds no different from ”is there any use case for adding m/s of velocity components in different directions?”

Correct. For which you gave the standard formula $\sqrt{v_x^2+v_y^2}$ which when expanded clearly is the addition of lots of terms of dimension L/T.

Is there a similar use case for adding moles of different substances?

I don’t know of one, but chemistry isn’t my thing. Again, I don’t care if it is a straight addition or inside a more complicated function.

 

[Orodruin]

which when expanded clearly is the addition of lots of terms of dimension L/T.

It is the addition of two terms of dimension (L/T)^2 and then taking the square root of that.

To me, the more fundamental aspect of dimensional analysis is the application of the Buckingham pi theorem and that works also for chemical reactions and substance amounts. Consider a case with an initial amount $n_A$ of A and $n_B$ of B and we consider a reaction $A + B \leftrightarrow C$ and you want to know the equilibrium amount of C $n_C$. Let $\pi_1 = n_C/n_A$ and $\pi_2 = n_B/n_A$. The Buckingham pi theorem then results in
$$
n_C = n_A f(n_B/n_A).
$$
This is no different from
$$
v = v_x f(v_y/v_x),
$$
which would be the general form of speed from dimensional analysis.

 

[cmb]

I admit I was a floating voter here, but this post suggests to me that the dimensions of a mole, if it is to make any sense, must be different for every substance.

If you have a mole of oranges, then either you have a dimensionless number of you have a unit of orange.

This SI unit of "a number of whatever thing you are talking about" seems to me neither one thing nor the other.

What's the counterargument?

Did my post #46 not convince you, with its simplicity?

 

[Orodruin]

Did my post #46 not convince you, with its simplicity?

Your post is wrong according to the SI definition.

I agree that it would be more natural for amount of substance to be dimensionless, but it is not, at least not in the SI definition.

 

[cmb]

Your post is wrong according to the SI definition.

I agree that it would be more natural for amount of substance to be dimensionless, but it is not, at least not in the SI definition.

What is the precise 'definition' you think I am wrong about?

I am reading https://www.bipm.org/utils/common/pdf/SI-statement.pdf where it says;-
"The mole has been redefined with respect to a specified number of entities (typically atoms or molecules)"

That sounds like they want it to be dimensionless, to me (clearly not 'specifically' atoms or molecules).

 

[Orodruin]

What is the precise 'definition' you think I am wrong about?

I am reading https://www.bipm.org/utils/common/pdf/SI-statement.pdf where it says;-
"The mole has been redefined with respect to a specified number of entities (typically atoms or molecules)"

That sounds like they want it to be dimensionless, to me (clearly not 'specifically' atoms or molecules).

Please read the actual SI brochure. In particular section 2.3.3 (page 136 for the English version).

 

[cmb]

Please read the actual SI brochure. In particular section 2.3.3 (page 136 for the English version).

"An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. "

That is saying to me it has to be some form of fundamental particle which are indistinguishable from each other.

"If you have a mole of oranges, then either you have a dimensionless number of you have a unit of orange."

You can't have a mole of oranges. But even if you could, what you say there is already answered in my original post. It is the functional operator "of" which you are ignoring. "Of" is a mathematical operator here, resulting in the product of a dimensionless number and the characteristic of the thing that 'of' operates on.

 

[Orodruin]

"An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. "

That is saying to me it has to be some form of fundamental particle which are indistinguishable from each other.

"If you have a mole of oranges, then either you have a dimensionless number of you have a unit of orange."

You can't have a mole of oranges. But even if you could, what you say there is already answered in my original post. It is the functional operator "of" which you are ignoring. "Of" is a mathematical operator here, resulting in the product of a dimensionless number and the characteristic of the thing that 'of' operates on.

You are missing the point entirely. And no, you are also wrong about the ”of”. If you read section 2.3.3 properly you will find that the mole, being one of the SI base units, has its own independent physical dimension N. The mole simply is not dimensionless in SI. Your original post suggested that a mole of ”something” had dimensions of [something]. An electron in itself is not a physical quantity, it is a physical concept and it is not associated to any particular physical dimension - at least not in the SI definition.

You are also taking completely unrelated quotes of mine out of context without using the quotation feature. This is a strongly misleading and quite dishonest thing to do.

 

[cmb]

You are missing the point entirely. And no, you are also wrong about the ”of”. If you read section 2.3.3 properly you will find that the mole, being one of the SI base units, has its own independent physical dimension N. The mole simply is not dimensionless in SI. Your original post suggested that a mole of ”something” had dimensions of [something]. An electron in itself is not a physical quantity, it is a physical concept and it is not associated to any particular physical dimension - at least not in the SI definition.

You are also taking completely unrelated quotes of mine out of context without using the quotation feature. This is a strongly misleading and quite dishonest thing to do.

Please read https://www.bipm.org/utils/common/pdf/SI-statement.pdf which is clearly there to add clarity to the matter.

 

[Orodruin]

Please read https://www.bipm.org/utils/common/pdf/SI-statement.pdf which is clearly there to add clarity to the matter.

Again, please read the SI brochure, which is the actual relevant document. The document you are linking to says nothing about the physical dimensions of the base quantities as those were not updated in the SI update.

The relevant passage reads:

Each of the seven base quantities used in the SI is regarded as having its own dimension.

Amount of substance is a base quantity in the SI and therefore has its own dimension. The mole is a unit of amount of substance and therefore has this physical dimension. You are simply in the wrong here. However, I do not blame you for thinking it would be more appropriate for the mole to be dimensionless. This is a matter of definition as has been pointed out in this thread as well as in the SI brochure and my main argument (see posts 18, 21, 23, 56, 59) in this thread has consequently been that it is more natural to have amount of substance as a dimensionless quantity.

 

[Orodruin]

The brochure also has this to say about the Avogadro constant:

The Avogadro constant NA is a proportionality constant between the quantity amount of substance (with unit mole) and the quantity for counting entities (with unit one, symbol 1). Thus it has the character of a constant of proportionality similar to the Boltzmann constant k.

It is pretty clear from this statement that amount of substance does not have the same dimensions as counting entities since the unit is mole and mole by definition has its own physical dimension, whereas counting entities have unit one.

It is interesting to note that they explicitly note the similarity to the Boltzmann constant as that is also something that should be redundant (and indeed is put to 1 in natural units).

 

[Dale]

Please read https://www.bipm.org/utils/common/pdf/SI-statement.pdf which is clearly there to add clarity to the matter.

The full SI brochure that was linked to early is completely clear on the matter. In the SI system the mol is unambiguously defined to have the dimension of amount of substance. It isn’t a dimension that I think is a good one to introduce, but that is unambiguously the official SI approach.

The clarity statement you cite here does not even address the topic of dimensionality of the mol.

 

[PeroK]

Did my post #46 not convince you, with its simplicity?

You are saying that we take "quantity" In the sense of "number of things" as a physical dimension?

Then the whole debate boils down to whether we ask "why" or "why not"?

 

[vanhees71]

Yes sure, so my idea that the dimension of a quantity depends on the system of units used, is not wrong after all. The SI is one specific system of units with 7 base units. So we have 7 basic dimensions within this system of units.

What is the precise 'definition' you think I am wrong about?

I am reading https://www.bipm.org/utils/common/pdf/SI-statement.pdf where it says;-
"The mole has been redefined with respect to a specified number of entities (typically atoms or molecules)"

That sounds like they want it to be dimensionless, to me (clearly not 'specifically' atoms or molecules).

In the SI $N_A$ has the dimension $1/\text{mol}$, $N_A \simeq 6 \cdot 10^{23}/\text{mol}$.

 

[Orodruin]

The SI is one specific system of units with 7 base units. So we have 7 basic dimensions within this system of units.

It is a convenience that base units have their own physical dimension. A system of units could, for example, specify several base units of the same dimension. For example, in natural units it may be convenient for some purposes to deal with eV and in some other cases with 1/m (many such cases appear in neutrino oscillations - your neutrino energies are typically in GeV and your baselines in km).

In the SI NANAN_A has the dimension 1/mol1/mol1/\text{mol}, NA≃6⋅1023/molNA≃6⋅1023/molN_A \simeq 6 \cdot 10^{23}/\text{mol}.

Note the difference between the unit ”mol” and the physical dimension ”amount of substance”. For example, I could use units 1/fmol to write down Avogadro’s constant, but it would still have the physical dimension of 1/amount of substance. Both mol and fmol are units of the physical dimension amount of substance.

 

[cmb]

I am not really sure why it is much of a debate?

I mean, if a mole is not dimensionless, OK, so ... what is its dimension? You can't say it has a dimension and then not know what it is!

I'll put it in a slightly more mathematical way. Notwithstanding the fundamental quantum properties, which SI doesn't consider, the essential principle of all the SI units, except the mole, is that you can pick any positive real number of that thing.

In fact, going one step further and taking quantum on board, it is physically unreal to have a precise integer or real number quantity of the other SI units. It is impossible to have exactly 'one meter' or '273.15K'.

Meanwhile, you can only ever have an exact real number of moles.

So the mole is clearly fundamentally different to the other units that have physical dimensions.

If the mole had a physical dimension, it would be impossible to have exactly one mole of stuff. OK, it might be difficult but it is not physically unreal to imagine exactly one mole of substance.

 

[Dale]

what is its dimension?

It is “amount of substance”.

So the mole is clearly fundamentally different to the other units that have physical dimensions.

It is also fundamentally different because it doesn’t tie into the second in any way, as all of the others do. But different $\ne$ dimensionless. It is weird, but it is defined to be dimensionful.

If the mole had a physical dimension, it would be impossible to have exactly one mole of stuff.

No if about it. It is dimensionful by definition. Your argument here is irrelevant.

It is also wrong. Charge is now something for which you can have an exact value also. You can have exactly 1 C of charge, and we can already count individual electrons reliably. By your argument charge could not be a dimension either.

Also, in the previous SI definition the IPK was exactly 1 kg.

 

[Orodruin]

I mean, if a mole is not dimensionless, OK, so ... what is its dimension? You can't say it has a dimension and then not know what it is!

Its dimension is ”amount of substance”, just like the dimension of a meter is ”length”. This is a matter of convention and in the SI it is defined like that. Please read the SI brochure.

Do you understand the fact that what is dimensionful or not is a matter of definition? As is what dimension something has. In natural units, only a single base dimension remains and $c = \hbar = k_B=1$ are dimensionless, the dimension length is the same as the dimension time, which are both the same as the dimension 1/mass = 1/temperature.

The dimension of a quantity is a matter of definition, you cannot obtain the dimensionality of a quantity from reasoning alone without first defining your base dimensions. You are free to argue that you would find particular definitions more natural than the SI definitions, but this in no way changes the SI definitions unless you become part of the standardisation committee and convince the other members. In fact, arguing that the mole should be dimensionless has been the subject of several posts of mine in this thread, but that does not change the fact that the mole - as defined in SI - is dimensionful with dimension amount of substance.

 

[cmb]

From what is said above, I am willing to be persuaded.

But the remaining hold up I have is that the definition says; "The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles."

"any other particle or specified group of particles"

So if I have "a mole of electrons", it is physically a half of "a mole of pairs of electrons".

I find this 'unphysical', so can you persuade me that the SI definition does NOT allow me to pick and choose a specified grouping of particles, as the definition seems to say?

 

[Orodruin]

From what is said above, I am willing to be persuaded.

But the remaining hold up I have is that the definition says; "The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles."

"any other particle or specified group of particles"

So if I have "a mole of electrons", it is physically a half of "a mole of pairs of electrons".

I find this 'unphysical', so can you persuade me that the SI definition does NOT allow me to pick and choose a specified grouping of particles, as the definition seems to say?

This is just diverting from the issue at hand. This is another issue you have with the SI, it is completely unrelated to the dimensionality of the mole.

As has already been pointed out in this thread, that two quantities (such as amount of substance of electrons and amount of substance of electron pairs) have the same physical dimension does not necessarily mean that they are comparable or relatable (although in this case $n_1 = 2 n_2$), it is only a necessary condition, not a sufficient one.

 

[Orodruin]

Or, as a teacher of mine once put it (although not about the SI): "It is not silly, it is a definition. Although it may be a silly definition."

 

[cmb]

This is just diverting from the issue at hand. This is another issue you have with the SI, it is completely unrelated to the dimensionality of the mole.

As has already been pointed out in this thread, that two quantities (such as amount of substance of electrons and amount of substance of electron pairs) have the same physical dimension does not necessarily mean that they are comparable or relatable (although in this case $n_1 = 2 n_2$), it is only a necessary condition, not a sufficient one.

It is not diverting from the issue at hand that I am seeing.

If I can use SI units and get twice as much of a thing for the same value, then it is unphysical and that doesn't make sense to me.

You can't do that with any other units. You can't have 10 metres "of timber" OR 10 meters "of 2 m timber". It is still 10 m of timber.

 

[Orodruin]

It is not diverting from the issue at hand that I am seeing.

If I can use SI units and get twice as much of a thing for the same value, then it is unphysical and that doesn't make sense to me.

You can't do that with any other units. You can't have 10 metres "of timber" OR 10 meters "of 2 m timber". It is still 10 m of timber.

You are confusing units and dimensions. Your first statement is "10 m of timber", not "10 m of 1 m timber", the second of those statements makes no sense whatsoever and is what is akin to your second statement.

It is also not twice as much of "a thing". It is 1 mol of electrons or 0.5 mol of electron pairs. An electron pair is not the same thing as an electron.

Edit: Let me also note again that all of these things are as they are by definition. You might like that definition or not, but you will not be able to reinterpret the definition to mean what you would like it to mean or so that it "makes sense" to you.

 

[vanhees71]

From what is said above, I am willing to be persuaded.

But the remaining hold up I have is that the definition says; "The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles."

"any other particle or specified group of particles"

So if I have "a mole of electrons", it is physically a half of "a mole of pairs of electrons".

I find this 'unphysical', so can you persuade me that the SI definition does NOT allow me to pick and choose a specified grouping of particles, as the definition seems to say?

It's not "unphysical". To the contrary, it's very physical ;-))).

If I interpret it right, the mole is introduced into physics for specifying many-body systems in terms of macroscopic quantities.

Now macroscopic quantities can be described as a coarse-grained view on the macro-system in terms of some averaging over many microscopic degrees of freedom, which we are unable to resolve, because we simply can not describe all the $\sim 10^{24}$ degrees of freedom of a macroscopic system.

Now the question is, how to determine the "macroscopically relevant" degrees of freedom and the underlying "relevant microscopic" degrees of freedom over which I'm averaging to get the effective description of the macroscopic observables.

Let's consider only the most simple case of thermal equilibrium. The most simple way to describe macroscopic matter in terms of the socalled grand-canonical ensemble. Let's take a somewhat "exotic" example, where the arguments become quite drastically clear: the case of strongly interacting matter under extreme conditions, which is investigated using ultrarelativistic heavy ions in various accelerators (SPS/LHC@CERN, RHIC@BNL, GSI/FAIR@Darmstadt,...) and via observations of neutron stars in (multi-messenger) astronomy (em. waves over a large scale of wavelengths, gravitational waves with LIGO/VIRGO).

As we know after some decades of research, what happens in an ultrarelativistic heavy-ion collision is that a very dense and hot fireball of collectively moving strongly interacting matter exists. You need a lot of different signals measured to achieve this (momentum-distributions of hadrons, relative abundancies of different hadron species, the distribution of hard probes like jets, open heavy flavor, quarkonia, and dileptons and photons).

It turns out that the hadronic spectra in the low-(transverse-)momentum region can be described by the assumption that after some short "formation" time of $\lesssim 1 \text{fm}/c$ a blob of quark-gluon plasma is formed, which is close to local thermal equilibrium, and its further evolution can be well described by (viscous) relativistic fluid dynamics.

Now I've already made a point by calling this "early stage" of the fireball evolution a "quark-gluon plasma". I already made an assumption about the "relevant microscopic degrees of freedom". The reasoning is that the coupling constant of QCD, the underlying fundamental theory describing the strong interaction among quarks and gluons, becomes small for high-energy collisions, and in a very hot and dense medium the particles have large energies and momenta and rattle around with high-energy collisions. That's why in a first naive attempt to understand what's going on, one had assumed that almost massless quarks and massless gluons are the relevant microscopic weakly interacting degrees of freedom in this hot and dense medium and thus a description as a nearly perfect relativistic gas of massless quarks and gluons might be a good, though rough, description.

Now, as lattice-QCD calculations at finite temperature has revealed that's not quite true, but there's still substantial coupling between the quarks and gluons at the temperatures of $\sim 500 \; \text{MeV}$ reached in the early phases of the fireballs of matter created in heavy-ion collisions, and that there's a quite sharp cross-over transition at temperatures of around $T_{\text{pc}} \simeq 150 \; \text{MeV}$.

The interpretation is that above this "pseudo-critical" temperature the relevant microscopic degrees of freedom are rather quark- and gluon-like quite massive quasi-particles, i.e., something similar as the constitutent (valence) quarks inside hadrons but not anymore sharply bound into usual hadrons. At the cross-over transition one has a strong decrease in pressure, energy density (divided by $T^3$ or $T^4$, respectively), which shows a strong decrease in "relevant degrees of freedom". This is thus interpreted that at this point something like hadrons is formed, but also these hadrons show some "medium modifications", i.e., they are also quasi-particles with some mass and width.

Now it's clear that the macroscopic quantity "amount of substance" when defined via the "number of microscopic constituents" depends on the state of this substance, which determines which microscopic degrees of freedom are relevant to describe the thermodynamics of the (equilibrated) medium. While in the early hot stages of the fireball evolution in the medium created in heavy-ion collisions the relevant microcopic degrees of freedom are the quark- and gluon-like quasiparticles, in the later colder stages of the fireball evolution the relevant macroscopic degrees of freedom are hadron-like quasiparticles (with medium-modified properties of mass and width). So in the QGP-phase to get the thermodynamics right you have to consider quark- and gluon-like degrees of freedom and thus you'd define "amount of substance" in terms of the corresponding "particle numbers/densities" counting these QGP-degrees of freedom, while in the later hadronic phase you'll rather count hadrons and hadron resonances. This keeps track of the drastic changes at the corresponding cross over.

The same holds for the treatment of most systems. E.g., take the air around us. It consists mostly of nitrogen and oxygen, but of course not in atomar but molecular form. Here the air is described best as a (nearly) ideal gas consising of $\mathrm{O}_2$ and $\mathrm{N}_2$ molecules, which thus make up the relevant microscopic degrees of freedom. You can even describe them as rigid rotators since the vibrational molecular modes are not yet excited at usual room temperature. This of course changes at higher temperatures, and at a certain point of very hot densities you dissociate the molecules to atoms and finally the atoms into a plasma of atomic nuclei and electrons. Always you change your description from one kind of relevant degrees of freedom to another. You can even think further, going back to the very early stages of the big bang: There at some point the atomic nuclei resolve into protons and neutrons and even earlier you rather had a QGP!

So "amount of substance" in the sense of the mole of the SI is defined via the number of "relevant microscopic degrees of freedom", which of course depends on the (thermal) state used to describe it.

Of course the transition between different regions of effective relevant degrees of freedom is particularly interesting and investigated in terms of the phase diagram of the medium. The change between different degrees of relevant effective degrees of freedom indicate transitions like the cross over transition between QGP and hadronic matter, which is expected to become a true first-order transition at higher net-baryon densities as achieved in heavy-ion collisions at lower beam energies, with the first-order transition line in the phase diagram ending in a critical point, where the phase transition becomes 2nd order.

 

[Dale]

I find this 'unphysical', so can you persuade me that the SI definition does NOT allow me to pick and choose a specified grouping of particles, as the definition seems to say?

No, the SI definition clearly allows you to do just that. You are certainly within your rights to dislike mol. And as far as “unphysical” goes, it is a convention and all conventions are inherently unphysical in some sense simply by virtue of the fact that they are a convention.

 

[Dale]

In fact, arguing that the mole should be dimensionless has been the subject of several posts of mine in this thread, but that does not change the fact that the mole - as defined in SI - is dimensionful with dimension amount of substance.

I agree with this completely. The SI is nothing more than a convention, and as such the BIPM is entirely within their rights to define their convention in any way they wish.

However, I also realize that I am not required to use their convention. Nothing forces me to use SI units. I could adopt a convention where all of the units had the same size and names as the SI units, but the mol is dimensionless and the rad is dimensionful. Such a convention is every bit as valid as any other convention. It just is not the SI, it is my own private convention. Nothing is wrong with that, the BIPM is not divinely appointed and I can use any convention I like as long as I don’t confuse my private convention with the SI convention.

 

[cmb]

I remain ready to be persuaded but still underwhelmed with the arguments.

I asked someone at work today and he came up with a third, even simpler opinion. A 'mole' has units of 'mass'. It is a measurement of substance, therefore that substance can be measured as mass rather than needing to make up another unit. The 'mole' is not a unit that is 'independent' of another unit in the SI system, it is only a convenience.

Is there an argument against this? I do not 'need' moles to describe 'a mole of carbon 12', I can just say 12g of carbon 12. Useful as it may be, it may be described as a derivative unit not a fundamental unit.

Thoughts?

 

[Orodruin]

I remain ready to be persuaded but still underwhelmed with the arguments.

Honestly, what you think is irrelevant. The SI defines amount of substance as a physical dimension.

I asked someone at work today and he came up with a third, even simpler opinion. A 'mole' has units of 'mass'. It is a measurement of substance, therefore that substance can be measured as mass rather than needing to make up another unit. The 'mole' is not a unit that is 'independent' of another unit in the SI system, it is only a convenience.

Yes it is, by definition. It certainly is not mass. You might define a system of units where that is the case, but it will not be the SI.

Is there an argument against this? I do not 'need' moles to describe 'a mole of carbon 12', I can just say 12g of carbon 12. Useful as it may be, it may be described as a derivative unit not a fundamental unit.

No it may not, not if you want to use the SI.

Thoughts?

I think that you have still not understood that amount of substance is a physical dimension in the SI by definition and that what physical dimensions exist is a matter of definition. As it is a matter of definition, it is not up for questioning if you want to use the SI. It may be that you disagree with the conventions taken in the SI, you are free to do that, just do not pretend that you are using the SI if you insist on having amount of substance dimensionless. You cannot argue from fundamental principles what should be dimensionless and what not.

 

[cmb]

In that case, this is semantics and lies in the ambiguity of language. It is not metrology nor physics. I prefer my answer in #46.

 

[Dale]

still underwhelmed with the arguments.

There is no such thing as an argument stronger than “by definition”.

If you are unwilling to accept even a “by definition” argument then you are completely unwilling to be persuaded by any logic. There is nothing the rest of us can do about that.