# Dimensional analysis problem

I am asking this question here because i think this is fundamentally linked to physics as it revolves around around dimensional analysis and physical quantities.

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 (or any unit for amount of substance) on simplification if mole (or any other unit for amount of substance as stated before) is taken as the unit for amount of substance.

In short, the all expressions for amount of substance should have the unit for amount of substance on simplification (if necessary). In this question i will use the SI unit for amount of substance, i.e., mole for all purposes.

If $A$ = Amount of substance (in moles), $m$ = Mass of the substance in a particular unit (normally in grams), $M$ = Mass per unit amount of substance (normally in grams/mole)

then,

\begin{align}\tag1\label1 A = \frac{m}{M} \end{align}

If the substance is in molecular form, 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 may be the numerical part in the amount of substance. The formulas given there were strange. One of the formulas for amount of substance was as follows:

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}\tag2\label2 n = \frac{m}{x} \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 $\ref1$ and $\ref2$, 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?

• Yes equation 2 is incorrect with the interpretation you gave to all the quantities. If x is the molecular weight, n is the number of molecules not moles. – octonion May 30 '17 at 13:24
• Your equation is dimensionally correct if "n" is the number of moles, "m" is the mass in grams, and "x" is the molecular weight in g/mole. – David White May 30 '17 at 18:17
• @DavidWhite That's a misunderstanding. Molecular weights are always in daltons (dimensionless); the quantity in g/mol you're looking for is the molar mass $M$ used in (1). – Emilio Pisanty May 30 '17 at 18:19
• I've provided an answer but frankly I think this should be migrated to Chemistry - chemists can also do dimensional analysis, you know. – Emilio Pisanty May 30 '17 at 18:19
• @emilio pisanty, the "old" nomenclature defined molecular weight as g/mole. I recognize that definitions tend to change with time. – David White May 30 '17 at 18:23

Your equation (1) is completely correct: if you have a mass $m$ of a substance whose mass per unit amount of substance is $M$ (the molar mass, a quantity in kilograms per mole), then you have an amount of substance $$A = \frac mM,$$ in moles.
To make sense of your equation (2), you use the atomic/molecular mass $x$ of the species in question; this is formally a dimensionless quantity which is normally notated in daltons / atomic mass units, and it is related to the molar mass via $M=xM_u$, where $M_u=1\:\mathrm{g/mol}$ is the molar mass constant. With this convention, if you have a mass $m$ of your substance, then the quantity $$\frac{m}{x}=AM_u$$ is in kilograms, and it makes very little sense.
When people calculate the $^*$number of moles $n$ (where $^*$ means that the quantity should not be used, to borrow some notation from linguistics), what they normally do is divide the mass $m$ of the sample by the mass of one mole of the substance in question, $^*\,y=M\times1\:\mathrm{mol}$, giving you $$^*\,n=\frac{m}{y} = \frac{m}{M\times1\:\mathrm{mol}} = \frac{A}{1\:\mathrm{mol}},$$ i.e. the numerical value of the amount of substance when expressed in moles. This usage, while widespread, is incorrect, and it will only cause you grief down the line.
Similarly, any resource that's giving you molecular masses in grams has no idea what they're talking about (or at least of how to talk about it) and you should give those resources a wide berth. You should use the molar mass (in kilograms per mole) or the molecular mass (in daltons, i.e. dimensionless), but unless they're selling you masses down in the $10^{-26}\:\mathrm{kg}$'s, give or take, they're completely mishandling their units.