Why is weight of 1 mole of substance equal to atomic/molecular mass in grams?

Why is weight of 1 mole of substance equal to atomic/molecular mass in grams?

According to me, it happens because mole has been defined in such a way. It is defined as the numbers of particles in $\pu{12 g}$ of $\ce{^{12}C}$. If it were $\pu{24 g}$ instead of $\pu{12 g}$, then the weight of 1 mole of substance would equal 2 times the atomic/molecular mass in grams.

Please correct me if I am wrong somewhere (or maybe everywhere).

Answer

Why is weight of 1 mole of substance equal to atomic/molecular mass in grams?

According to me, it happens because mole has been defined in such a way.

Yes! That is correct.

It is defined as the numbers of particles in 12 g of C12. If it were 24 g, instead of 12 g, then the weight of 1 mole of substance would equal 2 times the atomic/molecular mass in grams.

Also correct, assuming that the definition of unified atomic mass units (amu) remained the same. @Martin's answer is correct, but we can also arrive at the same conclusion using a simple dimensional analysis approach.

First we need the definition of an amu:

$$\pu{1 atom}~ \ce{^12C} = \pu{12 amu}$$

Now take the real definition of a mole:

$$\pu{1 mol}~ \ce{^12C} = \pu{12 g}$$

Now, divide the first equation by the second:

$$\frac{\pu{1 atom}~ \ce{^12C}}{\pu{1 mol}~ \ce{^12C}} = \frac{\pu{12 amu}}{\pu{12 g}}$$

Cross-multiply and reduce:

$$1~\frac{\pu{g}}{\pu{mol}~ \ce{^12C}} = 1~\frac{\pu{amu}}{\pu{atom}~ \ce{^12C}}$$

What this tells us is that the ratio of g/mol to amu/atom is exactly one - and we made sure it would work out that way by carefully choosing how we defined moles and amus.

Since the masses of all elements are defined in terms of amu, which are ultimately based on the mass relative to carbon-12, this ratio holds for all any atom or molecule.

Let's take it further and put in your hypothetical doubling of the number of grams of carbon-12 per mole:

$$\frac{\pu{1 atom}~ \ce{^12C}}{\pu{1 mol}~ \ce{^12C}} = \frac{\pu{12 amu}}{\pu{24 g}}$$

Now when we cross-multiply and reduce, we get:

$$2~\frac{\pu{g}}{\pu{mol}~ \ce{^12C}} = 1~\frac{\pu{amu}}{\pu{atom}~ \ce{^12C}}$$

Since we kept the definition of an amu the same, but changed that of the mole, what we see is that 1 mole of carbon-12 using these units would have to weigh 24 g (the molar mass), even though its atomic mass would still be 12.

That's a lot harder to keep track of mentally, and so it is a good thing for us that the definitions of amus and moles were chosen so carefully!