A When and Why Was the Definition of the Meter Changed?

[Orodruin]

The SI is not supposed to provide "natural units" but well-defined precise units that can be reproduced everywhere (by assumption of the cosmological principle even everywhere in the entire universe) to be used FAPP under everyday circumstances.

According to the definition 1 mole is the amount of substance consisting of a specific number of entities (relevant degrees of freedom I'd translate it). That's why the Avogradro number in the SI has the dimension 1/mol, i.e., you have $\simeq 6 \cdot 10^{23}$ entities per mole.

The dimension of Avogadro's number is 1/N, not 1/mol. The mole is a unit for quantities of dimension N.

But this is completely irrelevant to the issue of whether the mole should be dimensionful or not, it is the same whether or not [mol] = N or [mol] = 1.

 

[vanhees71]

Well, in the SI the electric charge has a dimension, though the natural dimension is 1. Read the official text: The Avogadro number in the SI has the dimension 1/mol. In natural units the Avogadro number is simply the above quoted number, i.e., it's dimensionless.

The same is true for angles: I'm not sure what's the status in the SI. I remember there was some debate concerning angles and solid angles, i.e., whether you should write rad or sr in the sense of units. If you do so, angles and solid angles get a dimension of rad or sr, respectively though the natural measure is again dimensionless.

You can, in principle, drive it to the extreme of using Planck units (in various variants around in the literature), which is only not done, because the Gravitational Constant is so difficult to be measured accurately. Then everything is dimensionless, and you have no more units for any quantity.

 

[Orodruin]

The Avogadro number in the SI has the dimension 1/mol

I suggest you read the official document where it is made clear that mol is a unit of dimension N (amount of substance), it is not a dimension in and of itself. Saying that something has dimensions of mol is like saying that a distance has dimensions of meters (it does not, it has dimensions of length L). This is described in section 2.3.3 of the SI brochure. Units are not the same thing as physical dimension although the concepts are somewhat related.

The same is true for angles: I'm not sure what's the status in the SI. I remember there was some debate concerning angles and solid angles, i.e., whether you should write rad or sr in the sense of units. If you do so, angles and solid angles get a dimension of rad or sr, respectively though the natural measure is again dimensionless.

Angles are dimensionless in the SI so the situation is not equivalent. Again, there is a distinction between the physical dimension and the units used to describe quantities of those dimensions.

Then everything is dimensionless, and you have no more units for any quantity.

This is not entirely true. You can still express a meter in Planck units. It would just be a number used to relate to other numbers, much like mol would be a number used to relate to other numbers if you define amount of substance to be dimensionless.

 

[vanhees71]

Interesting, I guess I've to read the bruchure again.

But why then do they write
$$N_A=6.xxx \cdot 10^{23} \frac{1}{\text{mol}}$$
oif $\text{mol}$ had the dimension of $N$ (which dimension is in fact 1). That's very confusing. Maybe I get something wrong here.

But let's take the "natural units" used in HEP physics. There you have $\hbar=c=k_{\text{B}}=1$. Then everything is measured in principle using only one unit, e.g., GeV. In addition for some quantities one uses fm. The conversion is simply $1 fm \simeq \frac{0.197}{\mathrm{GeV}}$.

Maybe I'm using the expression "dimension" wrong, but which dimension a quantity takes, depends on the system of units used, i.e., in the HEP natural units masses, energies, momenta, and temperatures have the same dimension. The same holds for lengths and times. Velocities are dimensionless.

Another example is electromagnetism, where the quantities have different dimensions depending on whether you use Gaussian/Heaviside Lorentz or SI units. In Gaussian or Heaviside units the components $\vec{E}$ and $\vec{B}$ of the electromagnetic field-strength tensor take the same dimension, while they are different in the SI. The reason is that in the SI an additional unit for electric charge, C (or equivalently electric current, the Ampere) is introduced, which enforces the introduction of one more conversion factor, $\mu_0$ in addition to $c$, which is used in Gaussian and Heaviside units es well.

In Planck units all quantities would have the same dimension, namely 1, i.e., all quantities are dimensionless.

Of course, you can also specify "dimensions" independent from units. Is it this sense the SI brochure uses the word "dimension"? Than it's clear that I used the wrong meaning in context of the SI.

 

[Dale]

Somewhat tangentially related to the recent discussion. Perhaps a chemist can answer.

For mass it makes sense to add a kg of glucose and a kg of NaCl to get a total mass. Would you ever add a mol of glucose to a mol of NaCl to get a total amount of substance?

Or if you add 1 mol of Na and 1 mol of Cl would you ever say you had 2 mol of anything?

 

[vanhees71]

Well, my chemistry is quite rusty, but wouldn't I get some Na, Cl but also NaCl? I'd say I've less than 2 moles of substance, depending on the conditions. For full equilibrium the question, how many moles I get is answered by the mass-action law.

Then the issue with mass is also not that trivial. According to relativity mass is not conserved, i.e., if you have an exothermic (endothermic) reaction your total mass gets smaller (larger) by the amount $\delta Q/c^2$ (the true meaning of the most misunderstood but most famous formula of physics $E=mc^2$). In chemistry that's of course usually negligible, not so in nuclear reaction like fission!

 

[PeroK]

Somewhat tangentially related to the recent discussion. Perhaps a chemist can answer.

For mass it makes sense to add a kg of glucose and a kg of NaCl to get a total mass. Would you ever add a mol of glucose to a mol of NaCl to get a total amount of substance?

Or if you add 1 mol of Na and 1 mol of Cl would you ever say you had 2 mol of anything?

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?

 

[PeroK]

Well, my chemistry is quite rusty, but wouldn't I get some Na, Cl but also NaCl? I'd say I've less than 2 moles of substance, depending on the conditions. For full equilibrium the question, how many moles I get is answered by the mass-action law.

Then the issue with mass is also not that trivial. According to relativity mass is not conserved, i.e., if you have an exothermic (endothermic) reaction your total mass gets smaller (larger) by the amount $\delta Q/c^2$ (the true meaning of the most misunderstood but most famous formula of physics $E=mc^2$). In chemistry that's of course usually negligible, not so in nuclear reaction like fission!

I'm not convinced. In principle you can add lengths or masses. A physical process may not support simple addition, but that's not the issue. Another example would be relativistic velocity addition. It's not simple addition, but you can manipulate velocities mathematically regardless of what's moving.

You can't in principle add moles of different things, which suggests (to me anyway) it's not the same unit in each case.

 

[Orodruin]

But why then do they write

NA=6.xxx⋅10231molNA=6.xxx⋅10231mol​

N_A=6.xxx \cdot 10^{23} \frac{1}{\text{mol}}
oif molmol\text{mol} had the dimension of NNN (which dimension is in fact 1). That's very confusing. Maybe I get something wrong here.

In the SI Avogadro’s number is dimensionful. If you would instead make amount of substance dimensionless, 1 mol would be exactly the number that the SI currently defines as the avogadro number’s measured value in 1/mol. The Avogadro number is then just a conversion factor with value 1 just like c in natural units but it is still 1 = 6.xxxe23 / mol.

Maybe I'm using the expression "dimension" wrong, but which dimension a quantity takes, depends on the system of units used, i.e., in the HEP natural units masses, energies, momenta, and temperatures have the same dimension. The same holds for lengths and times. Velocities are dimensionless.

Sure, it is a matter of convention what you give physical dimension to. The argument here is that it is more natural not to give amount of substance a physical dimension contrary to the SI convention. Much similar to it being natural to have dimensionless velocities in natural units.

Of course, you can also specify "dimensions" independent from units. Is it this sense the SI brochure uses the word "dimension"? Than it's clear that I used the wrong meaning in context of the SI.

The SI brochure first defines all of the units and then define the physical dimensions used by stating that each base unit has its own independent physical dimension. This was by no means necessary. The SI could just has well just have defined meters and seconds to be different units for length, which would make velocities dimensionless but have c as a dimensionless conversion factor.

 

[Orodruin]

Somewhat tangentially related to the recent discussion. Perhaps a chemist can answer.

For mass it makes sense to add a kg of glucose and a kg of NaCl to get a total mass. Would you ever add a mol of glucose to a mol of NaCl to get a total amount of substance?

Or if you add 1 mol of Na and 1 mol of Cl would you ever say you had 2 mol of anything?

That two numbers have the same physical dimension is a prerequisite for an addition to make sense. However, there is no guarantee that having the same physical dimension implies that the sum makes sense. For this, we need modelling.

Example: The x- and y-components of velocity $v_x$ and $v_y$, respectively. The sum $v_x + v_y$ makes little physical sense. However, $\sqrt{v_x^2 + v_y^2}$ does have a physical meaning as the total speed.

 

[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.