Mole and Avogadro's Number: A Closer Look

[Rasalhague]

I'm a bit confused about the terms "mole", "Avogadro's number", "Avogadro's constant" and "the Avogadro constant" and the concept - or concepts - they represent. I'm puzzled about how "amount of substance" differs from "amount", and in what sense this is a dimension if - unlike other dimensions - its units can be used to measure anything.

The terms "mole" and "Avogadro's number" seem to be synonymous (interchangeable), at least for some authors, given statements such as these:

"Definition of the mole: if we measure out exactly 12g of carbon-12, then we have exactly 1 mole of carbon 12 atoms. There will be exactly an Avogadro's number of atoms in the pile" (Atkins & Jones: Chemical Principles, F43).

"Things to understand about Avogadro's number. It is a number, just as is dozen, and thus is dimensionless; you can think of Avogadro's number as the chemist's dozen." (General Chemistry Virtual Textbook http://www.chem1.com/acad/webtext/intro/MOL.html ).

"The Mole is like a dozen. You can have a dozen guitars, a dozen roosters, or a dozen rocks. If you have 12 of anything then you would have what we call a dozen. The concept of the mole is just like the concept of a dozen. You can have a mole of anything" ( http://library.thinkquest.org/19957/atomic/molebody.html ).

"A convenient name is given when there is an Avogadro's number of objects - it is called a mole. Thus in the above example there was a mole of pennies. 1 mole = N_A objects. The mole concept is no more complicated than the more familiar concept of a dozen : 1 dozen = 12 objects" ( http://www.iun.edu/~cpanhd/C101webnotes/quantchem/moleavo.html ).

Atkins and Jones distinguish between Avogadro's number, "a pure number", and Avogadro's constant (or the Avogadro constant - they use both terms), "a constant with units". They designate the latter concept N_A, unlike some of the sites I quoted above who use N_A as their symbol for the former (supposing those other sites make this distinction).

On the other hand, the Wikipedia article Avogradro constant takes a different approach. It says the Avogadro constant was "originally called" Avogadro's number. According to this article, "The change in name to Avogadro constant (N_A) came with the introduction of the mole as a separate base unit in the International System of Units (SI) in 1971, which recognised amount of substance as an independent dimension of measurement. With this recognition, the Avogadro constant was no longer a pure number but a physical quantity associated with a unit of measurement, the reciprocal mole (mol^−1) in SI units."

If a mole is a certain number, approximately 6.022 * 10^23, also called, according to some sources, Avogadro's number, and if - as the Wikipedia article defines it - the Avogadro constant is this same number times mol^-1, its inverse, then the Avogadro constant must equal 1... but that contradicts the statement that the Avogadro constant is not dimensionless. I'm guessing the contradiction arises because some of these definitions I've read conflict with each other. Which is the best or most thorough or more orthodox definition? Are there competing definitions currently in use, and if so could someone explain the differences to me? Or are they all somehow equivalent really?

For those who take "amount of a substance" as a dimension, would it still be correct to talk of a mole of anything (eggs, dollars, meters, radians, tens, troubles...?) or would a mole be limited, according to this definition, to amounts of physical items such as eggs or dollar bills, or would it be limited still further to counting only atoms, ions or molecules?

The Wikipedia article Amount of a subtance

http://en.wikipedia.org/wiki/Amount_of_substance

offers several rationals for having a unit for the amount of a substance. They mention the comparison with "standard batch size" such as reams of paper or dozens of eggs, but say a better analogy is the value of gold held by a bank. In their "rationale for preferring amount-of-substance to absolute numbers", they compare the mole to the astronomical unit. See in particular the paragraph:

"This is somewhat similar to the situation that existed in Solar system astronomy for a time, where one did not know very well the absolute distances of the planets to the Sun, but one did know quite precisely the ratios of these distances to each other and, in particular, their ratios to the distance from the Earth to the Sun. The latter distance became known as the astronomical unit, and one way to describe the situation is to say that one knew quite precisely all the distances in terms of the astronomical unit, while the length of the astronomical unit itself was known quite poorly. [...]"

Perhaps I'm taking this too literally, but surely the AU is a unit of length (however poorly its exact relationship to other units of length may be known), and economic value is measured in dollars (or euros or rupees, etc.); we can't say how many kilograms away the sun is, or how many AU of roosters or guitars there are, or how many dollars of radians make half a circle. If we can say how many moles there are of anything (including units of any other dimension), then a mole seems more like a pure number than a dollar does or an AU. More like pi, for example, since we could express any quantity as a multiple of pi, if not exactly, to as much accuracy as we choose.

So, we can say so many times pi roosters (or so many dozens of roosters, moles of roosters?) and so many times pi meters (dozen meters) and so many times pi dollars (dozens of dollars), but not so many dollars of meters - which leaves me wondering in what sense the value analogy used by the Wikipedia article is preferable to the dozen analogy, except for the technical matter of whether the quantity is literally counted one, two, three...

 

[Bohrok]

I think Wikipedia explained it pretty well in the excerpt you included from Avogadro constant.
Avogadro's number is just a number: 6.022×10²³
Avogadro's constant has the reciprocal mole (mol^−1) unit attached to the number. I suppose in this case you can see more significance/purpose in Avogadro's number, or giving it a kind of definition, by attaching a unit to it, making it Avogadro's constant: 6.022×10²³ of something/mol of something.
Someone else could probably explain it with the kind of depth you've been studying it and give a better answer.

"Mole" and "Avogadro's number" are interchangeable since each is just a number, actually the same number.

 

[Rasalhague]

I think Wikipedia explained it pretty well in the excerpt you included from Avogadro constant.
Avogadro's number is just a number: 6.022×10²³
Avogadro's constant has the reciprocal mole (mol^−1) unit attached to the number. I suppose in this case you can see more significance/purpose in Avogadro's number, or giving it a kind of definition, by attaching a unit to it, making it Avogadro's constant: 6.022×10²³ of something/mol of something.
Someone else could probably explain it with the kind of depth you've been studying it and give a better answer.

"Mole" and "Avogadro's number" are interchangeable since each is just a number, actually the same number.

Thanks Bohrok! So was I right to conclude that the Avogadro constant = mol mol^-1 = 1? Under what circumstances can 1 be regarded as not dimensionless; or to put it another way, how do we do dimensional analysis on equations involving moles? Do they even follow the usual rules of dimensional analysis? I found a couple more definitions in undergraduate physics textbooks.

"A mole of any substance is the amount of that substance that contains Avogadro's number N_A of atoms or molecules defined as the number of carbon atoms in 12 g of carbon-12: N_A = 6.022 * 10²³. If we have n moles of a substance, then the number of molecules is N = nN_A" (Tipler & Mosca: Physics for Scientists and Engineers, 538).

"One mole of a given gas is the amount of gas with an amount in grams equal to the atomic or molecular weight of the gas, e.g. 1 mol of helium gas has mass 4 g. One mole of gas always contains Avogadro's number of molecules: N_A = 6.022 * 10²³. The total number of molecules in a container, N, can be written in terms of the number of moles, n, as: N = nN_A" (Fishbane et al.: Physics for Scientists and Engineers, 468-469).

In these definitions, it seems that N_A is dimensionless, in contrast to Wikipedia's notation. In which case, are all of the other quantities in the relation N = nN_A being treated as dimensionless by these authors, or should we regard N and n as both having the dimension "amount of substance"? I suspect they're all meant to be dimensionless though, given that the unit of "amount of substance" is the mole, whereas both N and n are numbers of individual molecules.

How would this relation be rewritten, or reinterpreted, using the conventions of the Wikipedia article Avogadro constant

http://en.wikipedia.org/wiki/Avogadro_constant

where N_A has dimension inverse moles?

The Wikipedia article Mole dismisses criticisms to the concept of the mole being a unit like the meter or the second, including the criticism that "the mole is simply a shorthand way of referring to a large number." Is the article then in disagreement with the definitions I quoted and with your statement that the mole and Avogadro's number are both simply a number? Would the Wikepedia writer, and/or the SI, disagree with those who say you can have "a mole of anything", including a mole of meters, etc.? It continues:

"The second misconception, that the mole is simply a counting aid, has even found its way into elementary chemistry textbooks. These books and others often contend that the mole is defined in terms of the Avogadro constant, rather than the other way around, [...]"

http://en.wikipedia.org/wiki/Mole_(unit)

The distinction here seems to be that the mole is defined as the number of atoms of carbon 12 in 12 grams of it (approximately 6.022... * 10²³), whatever that number may turn out to be exactly, rather than being defined as exactly 6.022... * 10²³. But I'm still struggling to see how uncertainty alone would make a mole qualitatively any less of a number than pi.

 

[symbolipoint]

Basically it's just a number, much the way "dozen" is a number; the main difference is that the value is different. Dozen means 12; mole means 6.022*10²³

 

[Rasalhague]

Basically it's just a number, much the way "dozen" is a number; the main difference is that the value is different. Dozen means 12; mole means 6.022*10²³

So you reckon the Wikipedia writer(s) is making a mountain out of a molehill here? Sorry, I couldn't resist... I'm trying to get my head around how a dimension ("amount of substance") can be defined in terms of a base unit that's just a number. On the one hand, we can't use "the dozen" or "the pi" as a unit of length, mass or time because dozen and pi are dimensionless. On the other, we can multiply the meter, kilogram or second by a dozen (i.e. twelve), or a score (20) or pi (3.141...) without changing their dimension. In these two ways, meters, kilograms and seconds seem different from dozens and scores and pi. So where does the mole belong? What qualities of a base unit does it have? How is mole defined as the measure of amount without that definition also allowing dozen and pi and 1 and 1000 to be dimensional quantities of dimension "amount". Is it correct to say a mole of meters = 6.022*10²³ m? Or do we need to balance the dimensional books somehow?

 

[Ygggdrasil]

You can convert a quantity in moles into a dimensionless quantity by multiplying by Avogadro's number.

e.g. 0.5 mol * (6.02x10²³ mol^-1) = 3.01x10²³

In this respect, Avogadro's number is simply a conversion factor to convert moles and dimensionless quantities.

 

[Rasalhague]

You can convert a quantity in moles into a dimensionless quantity by multiplying by Avogadro's number.

e.g. 0.5 mol * (6.02x10²³ mol^-1) = 3.01x10²³

In this respect, Avogadro's number is simply a conversion factor to convert moles and dimensionless quantities.

Ah, thanks. So, if I've understood this right, your "Avogadro's number" is equal to Wikipedia's "the Avogadro constant" and Atkins & Jones's "the Avogadro constant", and to mol^-1 * "Avogadro's number" as defined by Atkins & Jones, Tipler & Mosca and Fishbane et al.

Atkins & Jones use the symbol N_A for what they call the Avogadro constant, as does Wikipedia, whereas Tipler & Mosca and Fishbane et al. use this symbol for what A&J, T&M and F call "Avogadro's number".

In T&M and F's equation N = n N_A, are all three quantities dimensionless? That was my first guess. Alternatively, N and n might both have dimension "amount of substance". I don't know. But when A&J present an equation using the same symbols, it seems that N_A has dimension inverse amount; N is the number of objects (dimensionless?), and n is the number of moles (dimension "amount of substance").

 

[Rasalhague]

Hmm... So how exactly is "conversion factor" defined. Given that 1 day = 24 hours, would we say that the "conversion factor" for converting days into hours is 24, or would it be 24 day^-1?

 

[alxm]

Yes they're all dimensionless.

Note that mole is not a dimension! It's a quantity - just like dozens. A "dozen" isn't a thing or a measure of a thing - it's an amount of a thing. So dozens, thousands, scores and moles are all quantities, whereas inches, pounds and degrees are units. Moles just happen to be a convenient number for dealing with chemistry, because it's the number of things, each having a chemical weight of 1 u, that you need to have 1 gram of that substance.

You could also write N = n * N_D
Where N_D is the 'dozen constant', i.e. 12. and n is then the number of dozens you have, and N is the ordinary number.

Or, if you wanted to you could do a "unit analysis" (even though they're not properly called units) like:
[things] = [dozens]*[things/dozen]

 

[Ygggdrasil]

Hmm... So how exactly is "conversion factor" defined. Given that 1 day = 24 hours, would we say that the "conversion factor" for converting days into hours is 24, or would it be 24 day^-1?

You have the right idea. Mathematically, if we multiply any number by 1, we still have the same number. Now, since 24 hours = 1 day, we can write 24hrs/day = 1. So, if we want to convert 0.5 days into hours, we can do the following calculation:

0.5 days = 0.5 days * 1 = 0.5 days * (24 hrs / 1 day) = 12 hrs

Similarly, 6.02x10²³ (atoms) = 1 mol, so we can use the conversion factor 6.02x10²³ (atoms)/mol = 1. (the atoms is in parenthesis because it's an example. It could also refer to the number of ions, molecules, birds, etc. Technically this is the unit, but usually a number of particles is just represented without units as a dimensionless quantity).Note: I would not pay any attention to the difference between Avogadro's number and Avogadro's constant. Even if they have a different meaning, it is absolutely useless to our understanding of chemistry to know what that meaning is. As long as you get the general idea of what a mole means and why working with particles in units of 6.02x10²³ is useful, you're all good.

 

[pzona]

I think this has been explained thoroughly, but I just wanted to let you guys know I like the "dozen" analogy. I can't believe I haven't heard that one before.

 

[Rasalhague]

If amount of a substance is expressed as some quantity of moles of the substance, then it seems that amount of a substance is a ratio of dimensionless numbers (pure numbers)--the number of items (molecules, etc.) divided by the number of atoms of carbon-12 in 12g of carbon-12. This suggests that "amount of a substance" would also be dimensionless. But even if amount of a substance was defined as a dimension in the SI, wouldn't any quantity of moles still be dimensionless, since it would be a ratio between two quantities which have the same dimension as each other (like an angle measured in radians is the ratio between the length of the arc and the length of the radius)?

Could mole, Avogadro's number or the Avogadro constant be added to the table of dimensionless quantities here, or would that be wrong?

http://en.wikipedia.org/wiki/Pure_number

 

[Rasalhague]

Thanks for all your replies. Yes, I like the dozens too: I hope that's all there is to it! Some of these definitions, and in particular the Wikipedia articles, give me the impression that there's still something I'm missing though... If the difference between a mole and Avogadro's number was the difference between a dozen and twelve, that's pretty trivial! What's so unique about the mole, as opposed to "the one", "the pi" or "the 3 times 10^8" or any of those dimensionless physical constants--they're all more-or-less handy units of amount? In SI units, if we multiply a number by a meter, the result has dimension length. If we multiply a mole and a meter, the result has dimension length, or does it? That would make moles dimensionless, but isn't it part of what it means to be a base unit that the unit possesses a unique dimension and defines that dimension. If the dimension of a thousand monkeys is amount, and the dimension of a thousand meters is amount times length, and any quantity can be measured in moles, does the definition of the mole as the base unit of amount of substance redefine all dimensionless quantities as quantities of dimension amount of substance? Is "amount of substance" like the identity element of dimensional analysis: amount * length = length? Hmm, but I guess not, or there wouldn't be a distinct concept of inverse amount...

 

[alxm]

What's so unique about the mole, as opposed to "the one", "the pi" or "the 3 times 10^8" or any of those dimensionless physical constants--they're all more-or-less handy units of amount?

Yup. Look at it from the chemical perspective: Once upon a time, they just didn't know how many atoms there were in any given amount of matter. However - given the theory that stuff was made out of atoms in various combinations, they could figure out how much these different elements weighed, relative each other. If you split water into hydrogen and oxygen gas, you got a weight-proportion of 2:32. (or 1:16)

So given your unit of atomic weight, obviously there had to be an Avogadro's number telling us how many atomic units there were in a gram. Determining that number is a lot more difficult though!

In SI units, if we multiply a number by a meter, the result has dimension length. If we multiply a mole and a meter, the result has dimension length, or does it?

Yup.

That would make moles dimensionless

Yup. No more dimensions than 'kilo' or 'milli'.

but isn't it part of what it means to be a base unit that the unit possesses a unique dimension and defines that dimension.

Well, but moles doesn't. Avogadro's number is just a number. It's a 'unit' in the sense that SI includes it among the units. But it's not really a unit in the sense that it's a measure of anything in particular. It's just a quantity. A mole of meters is a distance, a mole of bicycles are still bicycles.

 

[Rasalhague]

Before I read alxm's last post, I made some guesses which I think must be partly mistaken... I wrote:

Is it perhaps that a dimensionless quantity (any dimensionless quantity?) can be given the dimension "amount of substance" by dividing the quantity by the number of atoms in 12g of carbon-12, called variously a mole or Avogadro's number, approximately 6.022 * 10²³, and thereby expressing the quantity as a number of moles? And the quantity has dimension "amount of substance" when expressed in terms of some number of moles. But if expressed as a number of atoms, e.g. ten atoms or a dozen atoms, or in any units other than a mole, then the same quantity is considered dimensionless?

And to convert a quantity expressed in terms of moles (an SI "amount of substance"), into a quantity expressed in terms of the number of individual items that make it up (a pure number, a dimensionless quantity), the number of moles of the substance is multiplied by the conversion factor, Avogadro's constant: 1 mol / 1 mol = 1 = approx. 6.022 * 10^23 mol^-1. (Or would it be more correct to say that the number 6.022... * 10^23 is the conversion factor?)

But from what you say, it seems more like the mole and any quantity measured in moles or inverse moles, has no dimension, in spite of the textbooks that list its dimension as "amount of a substance" and the strong wording of the Wikipedia articles which insist that "amount of substance" is a true "physical quantity" or "independent dimension of measurement", somehow qualitatively different from absolute number--althought I don't understand their arguments.

http://en.wikipedia.org/wiki/Mole_(unit)
http://en.wikipedia.org/wiki/Amount_of_substance
http://en.wikipedia.org/wiki/Avogadro's_constant

Avogadro's number is just a number. It's a 'unit' in the sense that SI includes it among the units. But it's not really a unit in the sense that it's a measure of anything in particular. It's just a quantity. A mole of meters is a distance, a mole of bicycles are still bicycles.

Do the objects that make up the substance have to all be of the same kind or defined in the same way; could we, say, have a mole of "cycles and cyclists", or a mole of "hydrogen atoms and helium atoms", or a mole of "hydrogen atoms and hydrogen molecules, H₂"? What does the official definition mean when it says: "Note that this definition specifies at the same time the nature of the quantity whose unit is the mole"; does it perhaps mean that only quantities of a physical substance should be expressed in moles (atoms and bicycles, as opposed to meters and desires), or does it mean that the quantity must be made up of discrete items (material or immaterial, but not continuous)?

http://books.google.com/books?id=I3...onal&printsec=frontcover#v=onepage&q=&f=false

 

[MagicTopQuark]

I think you are putting way too much into this (not necessarily a bad thing). When used in dimensional analysis it is simply a number that can be applied to something. Like if I wanted to convert 36 grams of water to molecules of water:

           (1 mole H20)     (6.022×10^23 molecH20)
(36gH2O) × ------------- × ------------------------  = 1.2044 × 10[SUP]24[/SUP] molecH2O
             (18gH20)           (1 mole H20)

In this case it is simply a number. Almost always it is just a number. I have not come across a case yet where, in practice, Avogadro's Number has a dimension.

You have the understanding of it. Now just do it.

 

[ideasrule]

Who cares about dimensions or dozens or whatever? Keep it simple. A mole is 6.03*10^23. I don't particularly care what units it has or how it's defined; it's just 6.03*10^23.

 

[f95toli]

But from what you say, it seems more like the mole and any quantity measured in moles or inverse moles, has no dimension, in spite of the textbooks that list its dimension as "amount of a substance" and the strong wording of the Wikipedia articles which insist that "amount of substance" is a true "physical quantity" or "independent dimension of measurement", somehow qualitatively different from absolute number--althought I don't understand their arguments.

Remember that the SI is a practical system of units. There is not necessarily any deep "philosophical" reason for why a unit is considered a base unit (as is clearly demonstrated by the Candela). If a unit is A) Of great practical importance and B) can not be derived from other units (which the Mole can not) then it is a base unit. Ultimately, the Mole is a unit because the people who decide these things (I know some of them) voted that way.

Or in other words, the reason the Mole is a unit is that chemists wants to be able to talk about e.g. "2 Mole" of something which in turn means that we need a universal definition of what they mean, more importantly we ALSO need a way of calibrating measurement equipment so that 2 Mole in one lab is the same amount as 2 Mole in another lab (in reality the Mole is never realized directly; the measurement are made using other SI units and Avogadro's constant, but that is a more technical point).

 

[guskiv]

Rasalhague, I find this your thread by searching "Avogadro's constant = 1". My question here was why does the Mathematica7 produce me the answers: SI[AvogadroConstant]=1. and SI[Mole]=6.02214*10^23 ? It was astonishing for me. Until now I have represented the mole as molecular weight in grams. For example, I know from diabetes that 1 mmol/L=18.02 mg/dL for blood glucose concentration. Seemingly, as it may seem strange, the SI unit "mole" can be represented either mass or number and both representations are correct if we don't mix theirs.

 

[Vulgar]

I like how this guys explains moles, he starts from the ideal gas law and after 10 to 15 minutes he arrives at moles, the first 20 minutes of these lecture are about ideal gas law, avogadro's number and moles:

http://www.youtube.com/watch?v=YxGHbnwqd14&feature=BF&list=SPFE3074A4CB751B2B&index=22