# A SI meter definition changed?

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#### Orodruin

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Note: If we wanted "nice" numbers, then we would probably define a reasonably sized length unit such that the speed of light would be $10^9$ of that length unit per second ... Oh wait! That is within 2% of a foot, can't have that ...

#### Mister T

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I admire that goal, and I think that the old definition of Avogadro's number was the last piece of the SI to still exemplify that, without the ugly numbers.
Avagadro's Number looks a lot less ugly to me now than it did before. Now it's an integer. It used to have an uncertainty to it, that seems more ugly to me.

#### maline

Note: If we wanted "nice" numbers, then we would probably define a reasonably sized length unit such that the speed of light would be $10^9$ of that length unit per second ... Oh wait! That is within 2% of a foot, can't have that ...
Bwa Ha Ha! You caught on to my sinister plot for Imperial supremacy!

#### maline

Avagadro's Number looks a lot less ugly to me now than it did before. Now it's an integer. It used to have an uncertainty to it, that seems more ugly to me.
Well yes... meaningful quantitative statements do tend to have uncertainty... only tautological ones don't.

#### killinchy

I prefer 'Avogadro's Constant (Na) = 6.022E23/mol

the "mole" is not defined as a number; it is defined as an 'amount of substance' (symbol, n)

and 1 mol of anything is the amount of that thing that has 6E22 entities.

Somebody mentioned that the mole is a conversion unit. It surely is. It is a miracle constant. It instantly converts atomic mass numbers into grams. From the micro world to the macro world. What is the value of this constant? Who gives a damn'? (OK, it's the inverse of the atomic mass unit expressed in grams)

If the value of Avogadro's constant were 42/mol, a chemist's life would be horrible. The poor chemist would have to deal with one number if he/she is thinking about atoms and molecules, and a different number if he/she were in the lab with bottles of stuff. It doesn't bear thinking about.

#### Orodruin

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the "mole" is not defined as a number; it is defined as an 'amount of substance' (symbol, n)
Nobody has said otherwise. It has been argued that it would be more natural to define it as a number or symbol.

#### vanhees71

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I prefer 'Avogadro's Constant (Na) = 6.022E23/mol

the "mole" is not defined as a number; it is defined as an 'amount of substance' (symbol, n)

and 1 mol of anything is the amount of that thing that has 6E22 entities.

Somebody mentioned that the mole is a conversion unit. It surely is. It is a miracle constant. It instantly converts atomic mass numbers into grams. From the micro world to the macro world. What is the value of this constant? Who gives a damn'? (OK, it's the inverse of the atomic mass unit expressed in grams)

If the value of Avogadro's constant were 42/mol, a chemist's life would be horrible. The poor chemist would have to deal with one number if he/she is thinking about atoms and molecules, and a different number if he/she were in the lab with bottles of stuff. It doesn't bear thinking about.
You should update your notion of how the SI defined today. Don't worry, the change officially got into effect only in May this year :-).

#### vanhees71

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Nobody has said otherwise. It has been argued that it would be more natural to define it as a number or symbol.
But it IS defined as a number:

The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly $6.02214076 \cdot 10^{23}$ elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit $\text{mol}^{−1}$ and is called the Avogadro number.[7][49] 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.

For details about the new SI, see the Wikipedia article

https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units

#### Orodruin

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But it IS defined as a number:

The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly $6.02214076 \cdot 10^{23}$ elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit $\text{mol}^{−1}$ and is called the Avogadro number.[7][49] 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.

For details about the new SI, see the Wikipedia article

https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units
It is not defined as a number in the sense we typically use in the meaning of having no physical dimension. Although the Avogadro number is defined by its numerical value and represents the number of entities in a mole by definition, amount of substance (and hence the unit mole) has its own physical dimension within SI. A mole is therefore not dimensionless and neither is Avogadro's number (even if it is called "number").

The argument that some (including myself) have made in this thread is that it would be more natural not to give substance amount its own physical dimension and having the mole be a symbol like % or making it actually represent a dimensionless number (which would in essence be $N_A$).

Edit: The physical dimensions of the base units are also discussed in the SI brochure section 2.3.3:
Physical quantities can be organized in a system of dimensions, where the system used is decided by convention. Each of the seven base quantities used in the SI is regarded as having its own dimension.
It is clear from this that the mole (which is a unit of the base quantity amount of substance) has non-trivial physical dimension and therefore is not just a number under the current SI definition. The argument made here is that it would be more natural to define it as being dimensionless.

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#### vanhees71

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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.

#### Orodruin

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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

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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

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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

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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.

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#### Dale

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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?

"SI meter definition changed?"

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