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Background:

Amount of substance is a fundamental physical quantity which has mole (mol) as it's SI unit. Therefore all expressions for amount of substance should have the unit mole on simplification.

If $A$ is the amount of substance (normally in moles), $m$ is the mass of the substance in a particular unit (normally in grams), $M$ is the mass per unit amount of substance (normally in grams/mole), then \begin{align} A &= \frac{m}{M}\tag1\label1 \end{align}

If the substance is a molecule then $M$ is known as molar mass of the substance.

But in my textbook and in many websites on the internet, I have encountered the phrase number of moles. I think it refers to the amount of substance or maybe the numerical part in the amount of substance. The formulas given there were strange.

If $n$ is the number of moles, $m$ is the mass of the substance in grams and $x$ is the atomic weight (for atoms of elements) or molecular weight (for molecules of elements and compounds), then,

\begin{align} n &= \frac{m}{x}\tag2\label2 \end{align}

This equation is not dimensionally correct if I am right.

My questions:

First of all I would like to ask whether amount of substance and number of moles refer to the same thing, or is it that amount of substance has a unit along with a numerical value whereas number of moles does not have a unit and represents the numerical value in the magnitude of the amount of substance.

Secondly, between equations $\eqref1$ and $\eqref2$, which one is completely correct (both in meaning and dimension)?

Thirdly, since in some place I have encountered gram atomic/molecular mass in place of molar mass, I would like to know what are the differences between both and do they have the same units or different units?

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"Amount of substance" is one of the seven base quantities of the SI [1]. Its dimension is noted with the symbol N, and its base unit is the mole (mol). Similarly "length" is another base quantity with dimension noted L and with base unit the meter (m).

"Number of moles of [substance X]" rigorously means the numerical value (thus dimensionless) associated with the corresponding "amount of [substance X]" expressed in moles, but from my experience it is often misused for "amount of [substance X]" (thus of the dimension of an "amount of substance"). - Think about how you would respond to the question: "How many meters are there between your house and the nearest bakery?" You could answer "35" or "35 meters", both would be grammatically correct; but strictly speaking the question expects just a numerical answer.

Note that according to the SI, a number of entities (molecules, atoms,..) is dimensionless and the unit of a dimensionless quantity is simply the number one, 1 [1]. My way of reconciling these two views (an "amount of substance" of dimension N and of unit the mole vs. a number of entities, dimensionless and of unit 1) is that the "amount of substance" refers to a combination of an amount (a number) and a given type of entity of a given substance, whereas a number of entity simply refers to the number itself. Why not consider an amount of substance as a dimensionless quantity? I think it is a reminiscence of history, at the time when chemists had to measure amount of substance without knowing how this substance was structured (but I can be wrong).

Regarding the equations, I agree with this other answer by @desc (equation 1 is correct, equation 2 isn't). Importantly: "atomic weight" is dimensionless (it is a ratio between a mass and a reference mass) and should not to be mistaken with "atomic mass". [2]

Finally, regarding the units "gram-atom" or "gram-molecule", these are outdated units which, according to SI were connected with "atomic weights" and "molecular weights" [2]:

Following the discovery of the fundamental laws of chemistry, units called, for example, "gram-atom" and "gram-molecule", were used to specify amounts of chemical elements or compounds. These units had a direct connection with "atomic weights" and "molecular weights", which are in fact relative masses. "Atomic weights" were originally referred to the atomic weight of oxygen, by general agreement taken as 16. But whereas physicists separated the isotopes in a mass spectrometer and attributed the value 16 to one of the isotopes of oxygen, chemists attributed the same value to the (slightly variable) mixture of isotopes 16, 17 and 18, which was for them the naturally occurring element oxygen. Finally an agreement between the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC) brought this duality to an end in 1959/60. Physicists and chemists have ever since agreed to assign the value 12, exactly, to the so-called atomic weight of the isotope of carbon with mass number 12 (carbon 12, $^{12}$C), correctly called the relative atomic mass $A_r$($^{12}$C). The unified scale thus obtained gives the relative atomic and molecular masses, also known as the atomic and molecular weights, respectively. [2]

  1. https://www.bipm.org/en/publications/si-brochure/chapter1.html

  2. https://www.bipm.org/en/publications/si-brochure/mole.html

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  • $\begingroup$ Your answer is trying to reconcile sloppy terminology with actual meaning. While I find the question you pose, "How many meters are there between your house and the nearest bakery?", okay for a casual conversation, I would regard it as wrong (or very sloppy) in the scientific context, where you should ask "What is the distance between ...?". You are quite correct though that the phrase 'number of moles' has incorrectly been used synonymous with 'amount of substance'. $\endgroup$ – Martin - マーチン Jan 15 at 17:45
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Your question is mainly semantic. The amount of a substance and number of moles are both unitless quantities (and the same thing). A mole is simply defined as "Avogadro number" of something. That is, 1 mole of (say) carbon atoms is about $6.02\times10^{23}$ carbon atoms. Both have a dimension of 1 (the number one); a mole is conveniently represented as a unit of measurement but doesn't actually hold any physical meaning. Using moles (and derivative quantities, such as molar mass which is the mass of "Avogadro number" items of something) is only a way to avoid extremely large or small numbers.

Another important point is that the variables in an equation only have the meaning you attach to them, so though conventions as to the meaning of each letter in your equations do exist I will answer to your definitions.

Equation (1) is correct - dividing the mass of a sample with its molar mass will give the number of moles.

Equation (2) is incorrect - diving the mass of a sample with the weight of a single item (atom or molecule) will give the number of items, not the number of moles.

Two notes:

  • Common convention is that $n$, and not $A$, is the number of moles. So the most conventionally correct equation is $$n = {m \over M}$$ where $n=[\mathrm{mol}]$, $m=[\mathrm g]$ and $M=\left[\mathrm{g \over mol}\right]$ (brackets are used to denote the unit of measurement of quantities).
  • The reason the second equation usually works is a double error - the atomic "weight" is usually given in number of nuclear particles (protons and neutrons) and not in actual weight units. This number is very close to the molar weight in $\mathrm{g \over mole}$ (on purpose, the definition of a mole is meant to be this way), but these are different quantities. Many chemistry books tend to confuse them. The actual atomic or molecular weight (in grams) of a single particle is very small and it is not likely you will actually use it much.
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    $\begingroup$ Amount of substance is not a dimensionless quantity. $\endgroup$ – MrAP May 29 '17 at 5:42
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    $\begingroup$ This answer is not in accordance with the International System of Units (SI). Therefore, for all intents and purposes of this site, this answer is wrong. $\endgroup$ – Loong Jan 15 at 17:13

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