Chemistry - What exactly is a mole?
Solution 1:
The first thing to realize is that "mole" is not a mass unit. It is simply a quantity - a number - like dozen or gross or score. Just as a dozen eggs is 12 eggs, a mole of glucose is $6.02 \times 10^{23}$ glucose molecules, and a mole of carbon atoms is $6.02 \times 10^{23}$ carbon atoms. "Moles" are only associated with mass because individual objects have mass, and thus a mole of objects also has a certain mass.
So why $6.02 \times 10^{23}$? What's so special about Avogadro's number? Well, nothing, really, it just makes calculations work out nicely. Avogadro's number is defined as the number of $\ce{^{12}C}$ atoms which weigh 12 g. So it's effectively a ratio: how many times larger is a gram than an atomic mass unit? If you have one atom of $\ce{^{12}C}$, it weighs 12 amu. If you have $6.02 \times 10^{23}$ of them, they weigh 12 g. If you have one molecule of glucose, it weighs 180 amu (or thereabouts). If you have $6.02 \times 10^{23}$ of them, they weigh 180 g (or thereabouts). -- This is just like if one egg weighs 60 g (on average), then a dozen of them weigh 720 g (on average), and if one cup of flour weighs 120 g (on average), then a dozen cups of flour weigh 1440 g (on average). The only difference is that dozen is defined forwards ("a dozen is twelve"), whereas a mole is defined "backwards" ("a dozen is the number of 60 g eggs that are in a collection of eggs that weighs 720 g.")
This convenience definition - pegging the value of Avogadro's number directly to the difference in scale between the amu and the gram - is what is probably throwing you. Moles are not a mass unit, but the definition is intimately tied to mass units. The equivalence in numbers at the atomic scale (amu) and at the macroscopic scale (grams) can also result in chemists playing fast and loose with terminology, quickly working back and forth from atomic to macroscopic scale, without a necessarily clear distinction between the two.
Solution 2:
Perhaps a hardware analogy might help. Let's say you're in the business of putting things together using nuts, bolts, and washers. For the sake of argument, let's say that whenever you join something you always need one bolt, two washers, and one nut. Oh, and you're going to be bolting together a lot of stuff.
So, you're going to need a lot of nuts, bolts, and washers, but that's okay, the hardware store sells them by the pound.
The bolts have a mass of 20 carats, the nuts are 5 carats and the washers are 2 carats. What's a carat? Doesn't matter really because you don't know how much stuff you're going to be joining - just "a lot".
So, you go to the hardware store and you buy 20 pounds of bolts, 5 pounds of nuts and 4 pounds of washers.
Now, you have no idea how many nuts, bolts and washers you have, but you do know that you have the same number of nuts as you have bolts and you have twice as many washers. So long as you convert the mass of the item in carats to the weight you buy in pounds, you can always make sure you get the items in the proportions you want.
Down the track a bit, you find out that there are 6,000 carats to the pound. That means 20 pounds of bolts is actually 6,000 bolts, and 5 pounds of nuts is also 6,000 nuts, and 4 pounds of washers is 12,000 washers.
So, in this analogy, a lot (aka a "mole") of items is 6,000 or 6 x 103
If you relocate to a country with a metric system and their hardware stores sell the same bolts, nuts, and washers but by the kilogramme (heresy!), are you in the least bit phased? Nah. You buy 20 kg of bolts, 5 kg of nuts and 4 kg of washers. Now you have 2.2 x 6,000 lots of joiner uppers. So, you're still in business.
The idea with atoms is basically the same, when you convert the mass in amu (or the dalton as I believe is preferred now) to grammes, then the mole is NA or Avogadro's number 6.02 x 1023.
The precise definition of exactly what is an amu (Da) is a fine point, important, but tangential to the concept of the mole which is what you're struggling with. Interestingly, and equally tangentially, Avogadro's number may well be used to redefine what a kilogramme is: https://www.youtube.com/watch?v=ZMByI4s-D-Y
Solution 3:
Atoms combine on a particle by particle basis. That is, by numbers of particles.
Laboratory and industrial operations rely on measuring masses.
The mole is the bridge between these. It allows us to "count out" a particular number of atoms by massing the substance.
Solution 4:
The first point is that it is a definition. In practical terms it enables us to have a convenient unit to express concentrations as $\pu{mol dm^{-3}}$ without having to use huge numbers. More importantly it allows us to know how many atoms molecules etc. are in a given mass of substance.
A mole is defined as the amount of substance, n, that contains as many objects (atoms, molecules, ions for example) as there are atoms in 12 grammes of carbon-12. This number has been determined by experiment and is approximately $6.02214~10^{23}$. This is Avogadro's number.
If a sample has N atoms or molecules then the amount of substance it contains is $n=N/N_A$ where $N_A$ is Avogadro's constant and is $6.02214~10^{23} \pu { mol^{-1}}$. Thus $\pu {1 mol}$ contains $6.02214~10^{23}$ atoms , molecules or ions etc.
It follows that
The mass of one mole of a substance equals its relative molecular mass expressed in grams,
thus 18 g of water (for simplicity using O = 16; H = 1 instead of exact masses) or 78 g of benzene contains Avogadro's number of molecules, similarly 32 g of sulphur, 200g of mercury, all have Avogadro's number of atom.
Solution 5:
A mole is a quantity of discrete objects, not a dimension for measuring one thing. In the same way that a dozen objects is 12 or a score of objects is 20, a mole of objects is about $6.02214 \times 10^{23}$.
Technically, we shouldn't say "1 mole of glucose". We should instead say "1 mole of glucose molecules". The same goes for any other object: "1 mole of water" should be "1 mole of water molecules", and so on. If we want to speak of raw elements, then we'd be talking about a mole of atoms of that element, rather than a mole of molecules, but the principle is the same. By convention, we usually omit that "molecules" part, but if we really want to be strict about things we should say it.
Avogadro's Number was chosen so that the mass of a mole of molecules (of the same thing) happens to equal $x$ grams, where $x$ is the molecular mass of that thing. Originally, it was defined so that 1 mole of hydrogen (which, remember, is really "1 mole of hydrogen atoms") weighs 1 gram. Nowadays, we define it so that one mole of Carbon-12 (again, "1 mole of Carbon-12 atoms") weighs 12 grams, but the proportions still work out, more or less. The folks behind SI are working on pinning this to a mathematical constant.