## Main Question or Discussion Point

1 mole of a substance equals the amount of grams needed for that substance to have 6.0221413e+23 (Avogadro's number) atoms in it, isn't it?

In order to determine how many grams one mole of a substance is, I've learned that you just need to check the atomic mass number on your periodic table, take that number and put "gram" at the end.

The thing I don't understand is, how come it's that simple?

The atomic mass number is the number of protons and electrons (nucleons) in the atom, right? So how come 6.0221413e+23 times the mass of all nucleons ALWAYS amounts to the number of nucleons in grams for any substance?

For example, carbon has an atomic mass number of 12 so 1 mole of carbon equals 12 grams of carbon.

Hydrogen has an atomic mass number of 1 so 1 mole of carbon equals 1 gram of hydrogen.

Is this some great coincidence? Or is there a link that I'm not seeing?

I'm not sure I understood what a mole is and my research on it just confuses me, so there's why I'm asking. Thank you for reading and (hopefully) helping!

Borek
Mentor
The thing I don't understand is, how come it's that simple?
Because we selected amu value to be so.

Because we selected amu value to be so.
Can you please explain how they did?

1 amu is the mass of 1 nucleon. The mass of a nucleon can't be chosen, it's fixed, it's a fact of nature that we can merely observe.

This source explains that scientists couldn't directly measure the mass of a nucleon so they created the relationship:

1 amu = 1/6.0221415 x 10^23 grams

I understand that 1 mole of 12 amu equals 12 grams following this relationship

6.0221415 x 10^23 = N = 1 mole

1 amu = 1/N g

N x 12 x 1 amu = N x 12 x 1/N g

<=> N x 12 x 1 amu = 12 x 1 g

<=> N x 12 amu = 12 g

But what I still don't get is, how did they know that the mass of a nucleon was equal to 1/6.0221415 x 10^23 grams ?

The gram is a fixed quantity too, so they couldn't just arbitrarily choose that relationship to fit their needs, right?

1 mole of a substance equals the amount of grams needed for that substance to have 6.0221413e+23 (Avogadro's number) atoms in it, isn't it?

In order to determine how many grams one mole of a substance is, I've learned that you just need to check the atomic mass number on your periodic table, take that number and put "gram" at the end.

The thing I don't understand is, how come it's that simple?

The atomic mass number is the number of protons and electrons (nucleons) in the atom, right? So how come 6.0221413e+23 times the mass of all nucleons ALWAYS amounts to the number of nucleons in grams for any substance?

For example, carbon has an atomic mass number of 12 so 1 mole of carbon equals 12 grams of carbon.

Hydrogen has an atomic mass number of 1 so 1 mole of carbon equals 1 gram of hydrogen.

Is this some great coincidence? Or is there a link that I'm not seeing?

I'm not sure I understood what a mole is and my research on it just confuses me, so there's why I'm asking. Thank you for reading and (hopefully) helping!

Welcome to PF You are a critical thinker, and your question is a nice one and it proves that you don't just take knowledge as constant information but a "why" question always pops in your mind Avogadro's is a selected number, it's not a coincidence at all that the mass of Avogadro's number is equal to the mass number, Avogadro's was precisely selected because they certainly knew that the mass of his number for any element atoms will be equal to its mass number, how come? That seems so weird!! But it's actually very simple, let's do some math to get to the bottom of this.
But first lets just point to some impotent things
Atoms consist of protons, neutrons and electrons
1 - protons and neutrons roughly have equal masses
2 - electrons' mass is negligible

So the mass of an atom = mass of neutrons + mass of protons = a.m.u x Mass number
Mass of "n" atoms = a.m.u x mass number x n
If n = the reciprocal of a.m.u then Mass of " n " atoms = Mass number
So mass of Avogadro's number of atoms = mass of proton or neutron x mass number x Avogadro's number = a.m.u x mass number x 1/a.m.u = mass number
So it's not a coincidence, Avogadro's number was selected to be the mass of one a.m.u reciprocal so that one mole of any substance becomes equal to its mass number, thanks to Avogadro, chemical calculations are very Simple.

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Borek
Mentor
So the mass of an atom = mass of neutrons + mass of protons = a.m.u x Mass number
That's not true.

Mass of neutron is 1.674927351×10-27 kg

Mass of proton is 1.672621777×10-27 kg

Mass of alpha particle (helium nucleus) is 6.64465675×10-27 kg

Alpha particle is two neutrons and two protons, so its mass should be

2*1.674927351×10-27 + 2*1.672621777×10-27 = 6.695098256×10-27

Where is the missing 0.050441506×10-27 kg?

Welcome to PF Thank you. So the mass of an atom = mass of neutrons + mass of protons = a.m.u x Mass number
Mass of "n" atoms = a.m.u x mass number x n
If n = the reciprocal of a.m.u then Mass of " n " atoms = Mass number
So mass of Avogadro's number of atoms = mass of proton or neutron x mass number x Avogadro's number = a.m.u x mass number x 1/a.m.u = mass number
So it's not a coincidence, Avogadro's number was selected to be the mass of one a.m.u reciprocal so that one mole of any substance becomes equal to its mass number, thanks to Avogadro, chemical calculations are very Simple.
I think I understand the logic behind it:

Having n times the mass of an atom is practical in that it enables us to cancel out the number of nucleons in the equation (they're equal to 1)

mass of Avogadro's number of atoms = [strike]mass of proton or neutron[/strike] x mass number x [strike]Avogadro's number[/strike]

My only issue now is that this should be correct:

Avogadro's number = 1/mass of proton or neutron

But it isn't by my calculations

mass of proton = 1.6726×10^-24 g (according to wiki)

6.0221413x10^23 != 1/1.6726×10^-24

(the answer to 1/1.6726×10^-24 is 5.9787157718...x10^23 , going by my calculator).

Did I make a mistake in my calculations? Or did I actually not understand the logic that you explained?

My reply is not accurate 100%

Borek
Mentor
how did they know that the mass of a nucleon was equal to 1/6.0221415 x 10^23 grams ?
They didn't KNEW it, they DEFINED it as 1g/NA.

What they really did was they defined first 1 mole to be the number of atoms in exactly 12 grams of C-12. Selection of C-12 and 12 is arbitrary (actually in the past it was O-16 and 16 g). That allowed them to determine NA. Once they had NA, they calculated amu value - and this way it was guaranteed to work the way it does.

Thank you. I think I understand the logic behind it:

Having n times the mass of an atom is practical in that it enables us to cancel out the number of nucleons in the equation (they're equal to 1)

mass of Avogadro's number of atoms = [strike]mass of proton or neutron[/strike] x mass number x [strike]Avogadro's number[/strike]

My only issue now is that this should be correct:

Avogadro's number = 1/mass of proton or neutron

But it isn't by my calculations

mass of proton = 1.6726×10^-24 g (according to wiki)

6.0221413x10^23 != 1/1.6726×10^-24

(the answer to 1/1.6726×10^-24 is 5.9787157718...x10^23 , going by my calculator).

Did I make a mistake in my calculations? Or did I actually not understand the logic that you explained?
I was thinking about that too, and I also did some calculations and I find out that there something going wrong, as I said before my reply isn't accurate 100% but it got the main logical idea I hope someone on the forum could edit a correction, I did my best • 1 person
Borek
Mentor
Simple summation won't work for two reasons, First, mass of the nucleus is lower than the sum of masses of all nucleons by so called binging energy. Second, molar mass is the weighed average of masses of all isotopes present. That's why it was important to choose an isotope (C-12) and not an element (C) for the definition.

They didn't KNEW it, they DEFINED it as 1g/NA.

What they really did was they defined first 1 mole to be the number of atoms in exactly 12 grams of C-12. Selection of C-12 and 12 is arbitrary (actually in the past it was O-16 and 16 g). That allowed them to determine NA. Once they had NA, they calculated amu value - and this way it was guaranteed to work the way it does.
EDITED POST

You're saying that they came up with Avogadro's number first and then calculated the amu value?
So the number of Avogadro isn't actually the reciprocal of the mass of a nucleon (chosen to make the mass of the nucleon cancel out in the equations so that the atomic number would remain)?

If that's correct, then how come their arbitrary choice (of the number of atoms in 12 grams of C-12) causes the mass of the atom to equal the atomic number in grams?

I'm sorry, I'm really doing my best, but I'm not sure I understand what you mean...

Borek
Mentor
You're saying that they came up with Avogadro's number first and then calculated the amu value?
Yes.

So the number of Avogadro isn't actually the reciprocal of the mass of a nucleon
Exactly.

If that's correct, why did they define Avogadro's number?
They defined mole, not the Avogadro's number. Once they defined mole, it was enough to count atoms in a mole (remember, exactly 12 g of C-12) to determine NA.

There is a plan to redefine the mole, and express it in terms of NA - that is, to define NA as being equal to some arbitrary value. But so far Avogadro's constant is not defined, but determined.

epenguin
Homework Helper
Gold Member
Some of the answer to your queries is in the link below. The thing to realise is scientists had got the mass proportions in which atoms combine and could work with moles about a century before they could get the mass of an atom (and so Avogadro's number). In traditional school courses they used to emphasise these laws of combinations and drilled you with endless calculations concerning them. Telling us really anything known about atoms was left till University. But now they reckon that's boring so they tell it earlier. But it sounds like they haven't told you how things like the mass of the atom are found out. The teachers in my time were instructed not to do that, to be scientific you had to grasp the reason for believing something. Now you can see that dealing with 1022, say, atoms that is something you can see and weigh and so verify these laws of constant proportion in combinations etc. that were in fact the original experimental basis for atomic theory and all chemistry in a school lab. Hence the boring calculations - 'chemistry is not all fireworks' was a saying of one of my teachers - which students still have to do - it's called 'stoichoimetry' - and that we get many questions on here and that have an undeserved reputation for difficulty.

You asked, almost, how did they determine the mass of an atom or molecule in grams also opposed to its relative mass I.e. the mass of one relative to another which is what I was just talkng about. There were several ways (all interesting stories) but the most key one historically was this: although the atoms and molecules are too small to see in any ordinary simple setup, these atoms are in constant motion and give kicks to other particles such as the fat globules of milk which are big enough to see by microscope moving as a result of these kicks ('Brownian motion'). Studying that, in brief, they got to Avogadro's number. The theory of it was the work of Einstein.

https://www.physicsforums.com/showpost.php?p=4680146&postcount=7

Yes.

Exactly.

They defined mole, not the Avogadro's number. Once they defined mole, it was enough to count atoms in a mole (remember, exactly 12 g of C-12) to determine NA.

There is a plan to redefine the mole, and express it in terms of NA - that is, to define NA as being equal to some arbitrary value. But so far Avogadro's constant is not defined, but determined.
I think you were answering me while I was editing my post, so you didn't see the change, but I'm still confused on:

how come their arbitrary choice (of the number of atoms in 12 grams of C-12) causes the mass of the atom to equal the atomic number in grams? How can something arbitrary fit so perfectly?

I don't understand how they could "define" the mass of a nucleon and not "know" it, given that grams are a fixed quantity of mass and the mass of a nucleon is a fixed quantity in nature that they could merely observe not choose, too...

If you're not seeing my problem, here's an example:

I "define" that the ball I'm holding has a mass of 2 kg. But when I lay it on my balance, it says that it weighs 9.81 newton here on earth. If the acceleration on earth equals 9.81 then the ball must have a mass of 1 kg (regardless of what I "defined"/chose earlier), it's not me who "defines" its mass, its mass is a fact of reality which I "know" because I calculated it.

I've read it, but it defines avogadro's number as the reciprocal of the mass of a nucleon (on the second page) which our calculations and Borek contradicts. :/

They didn't KNEW it, they DEFINED it as 1g/NA.
yhPscis said:
So the number of Avogadro isn't actually the reciprocal of the mass of a nucleon
Exactly.
Wait, if 1 amu equals 1g/Avogadro's number (as defined), then Avogadro's number equals 1g/1 amu, right?
And 1g/1amu is the reciprocal of 1 amu, right?
And 1 amu is 1/12 of the mass of a C-12 atom, so it's the mass of one nucleon.
So that definition does say that Avogadro's number is the reciprocal of the mass of 1 amu, right?

Ugh, this is so confusing :(

Borek
Mentor
I don't understand how they could "define" the mass of a nucleon and not "know" it, given that grams are a fixed quantity of mass and the mass of a nucleon is a fixed quantity in nature that they could merely observe not choose, too...
It is not mass of a nucleon that is defined, it is a UNIT used to measure it (amu) that is defined.

It is defined to be 1/12 of the mass of the C-12 atom. Close to the mass of the nucleon, but different.

It is not mass of a nucleon that is defined, it is a UNIT used to measure it (amu) that is defined.

It is defined to be 1/12 of the mass of the C-12 atom. Close to the mass of the nucleon, but different.
But if 1 amu does not equal the mass of a nucleon, then the mass of the nucleon can't be canceled out in the equaltion:

1 amu = 1 g/NA

NA = 1g/1 amu

Mass of atom = number of nucleons x mass of 1 nucleon

NA x Mass of atom = NA x number of nucleons x mass of 1 nucleon

1g/1 amu x mass of atom = 1g/1 amu x number of nucleons x mass of 1 nucleon

You can't say:

Mass of nucleon/amu x number of nucleons = 1g x number of nucleons

So

NA x mass of atom = number of nucleons g

So how come having an Avogadro number (NA) of atoms of an element equals the number of nucleons of that element in grams anyway?

Why didn't they use the reciprocal of the mass of a nucleon instead of Avogadro's number although the former fits the equation perfectly (and the latter does not)?

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epenguin
Homework Helper
Gold Member
Reflect, all measurements are comparisons.

It is essential to have some standard to compare that back to, and since the nineteenth century the standards have been internationalised. And it is good to make them as convenient as reasonably possible.

When you ask in #what a coincidence! how come it worked out that simple it's like saying hey a litre of water turns out to weigh exactly one kilo, incredible coincidence! Reflect, all measurements are comparisons.

It is essential to have some standard to compare that back to, and since the nineteenth century the standards have been internationalised. And it is good to make them as convenient as reasonably possible.

When you ask in #what a coincidence! how come it worked out that simple it's like saying hey a litre of water turns out to weigh exactly one kilo, incredible coincidence! Oh, I understand that people look for the most convenient way to measure things. Before it was redefined as 1/1000 of the mass of the international prototype of the kilogram in France, the gram was defined as the mass of one cubic centimeter of water (according to wiki), so I guess it makes sense that one liter of water turns out to have a mass of 1 kilogram. It's a logical extension of that arbitrary definition of the gram.

But it doesn't seem to make sense that if you take NA amount of atoms of a substance, you turn out to have the number of nucleons in grams of that substance because, as shown in my first post on this page, the equations don't fit... the logic behind that practice seems erroneous...

How come this logic is used despite that the equations don't fit? Why don't they use the mass of a nucleon instead of Avogadro's number?

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SteamKing
Staff Emeritus
Homework Helper
For example, carbon has an atomic mass number of 12 so 1 mole of carbon equals 12 grams of carbon.
That's the definition. Before 1961, oxygen was defined as having 1 mole O = 16 grams, but now
1 mole O = 15.999 grams. Is the oxygen from before 1961 different from post-1961 oxygen? No, it isn't. We've just changed the weight slightly.

Hydrogen has an atomic mass number of 1 so 1 mole of [STRIKE]carbon[/STRIKE] hydrogen equals 1 gram of hydrogen.
This is not entirely correct. 1 mole of H-1 = 1.007 825 0 grams, due to 1 mole of C-12 = 12 grams exactly, by definition.

Is this some great coincidence? Or is there a link that I'm not seeing?

I'm not sure I understood what a mole is and my research on it just confuses me, so there's why I'm asking. Thank you for reading and (hopefully) helping!
No. As pointed out, it is by design.

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

Borek
Mentor
So how come having an Avogadro number (NA) of atoms of an element equals the number of nucleons of that element in grams anyway?
In general - it doesn't. It does only approximately. Have you read my earlier post about mass of the helium nucleus? Not only mass of the nucleus doesn't equal sum of the masses of the nucleons involved, but the difference is different for every isotope.

Why didn't they use the reciprocal of the mass of a nucleon instead of Avogadro's number although the former fits the equation perfectly (and the latter does not)?
C-12 was an arbitrary choice, as was the earlier choice of O-16. I don't remember exact reasoning behind, for sure it can be googled - but the choice was not made by a bunch of monkeys, so I trust their reason.

Besides - which nucleon? Proton, or neutron? The difference is around /1830, so quite large.

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In general - it doesn't. It does only approximately. Have you read my earlier post about mass of the helium nucleus? Not only mass of the nucleus doesn't equal sum of the masses of the nucleons involved, but the difference is different for every isotope.

C-12 was an arbitrary choice, as was the earlier choice of O-16. I don't remember exact reasoning behind, for sure it can be googled - but the choice was not made by a bunch of monkeys, so I trust their reason.

Besides - which nucleon? Proton, or neutron? The difference is around /1830, so quite large.
Oh, I see, I mistakenly thought it didn't matter which nucleon because the difference of mass between both nucleons was negligible.

So, If I understood it well, by using Avogadro's number, you only get an approximation of the mass, but they use that method anyway because, although it's not completely accurate, it's accurate enough to achieve the chemical reactions you want in a lab.

Thanks for your explanations (and your patience) Borek
Mentor
So, If I understood it well, by using Avogadro's number, you only get an approximation of the mass, but they use that method anyway because, although it's not completely accurate, it's accurate enough to achieve the chemical reactions you want in a lab.
No, you are still making some mistake somewhere. These are not approximate masses, they are highly accurate.

Please remember we measure atomic mass. We don't calculate atomic mass by summing masses of nucleons, quite the opposite, we deduct their number from the measured atomic mass and from the spectroscopically determined nucleon charge.

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