Chemical engineers define one mole as the amount of a substance which possess as many entities as $12\ \mathrm g$ of $\ce{^{12}C}$.
The number of atoms in $12\ \mathrm g$ of $\ce{^{12}C}$ is $6.022 \times 10^{23}$ which is a constant by its definition.
Now come to the relevant definition which is given in my textbook that defines one mole of a substance as the atomic mass, molecular mass or formula mass in grams.
Is this definition correct?
Answer
The mole is a base unit as specified in the Système international d’unités (SI) by the bureau international des poids et mesures. Its decisive definition is that published in French:
La mole est la quantité de matière d’un système contenant autant d’entités élémentaires qu’il y a d’atomes dans 0,012 kilogramme de carbone 12 ; son symbole est « mol ».
Lorsqu’on emploie la mole, les entités élémentaires doivent être spécifiées et peuvent être des atomes, des molécules, des ions, des électrons, d’autres particules ou des groupements spécifiés de telles particules.
La mole est une unité de base du Système international d’unités.
The proposal was brought forth by the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC) and the International Organisation for Standardization (ISO). As with all SI texts, the decisive French version has a semi-official English translation:
The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in $0.012$ kilogram of carbon 12; its symbol is “mol”.
When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
(The third point is not translated.)
This definition is more or less identical with the one in your first paragraph.
In practice and well within experimental error, this means that your later definition will hold true for any substance. I.e. take $1~\mathrm{mol}$ of an entity and the combined mass of that mole will be the same numerical value in grams as a single entity has in atomic mass units ($\mathrm{u}$). It is not the correct definition, and for any entity that is not carbon-12 the masses will differ slightly (but well within margin of your macroscopic experimental error) but it is good enough for most contexts.
A redefinition of the SI units is being discussed and will likely be adopted at the 26th General Conference of Weights and Measures in autumn 2018. This would redefine the mole in a way that the Avogadro constant is defined to be numerically exactly $6.02214 \cdot 10^{23}~\mathrm{mol^{-1}}$ (with a few further digits appended to the end of the number that yet need agreement). This would mean that the new definition of the mole would be along the lines of:
The mole, mol, is the unit of amount of substance of a specified elementary entity, which may be an atom, molecule, ion, electron, any other particle or a specified group of such particles; its magnitude is set by fixing the numerical value of the Avogadro constant to be equal to exactly $6.02214X \cdot 10^{23}$ when it is expressed in the unit $\mathrm{mol^{-1}}$.
Currently, the Avogadro constant must be measured experimentally giving a value of $6.022140857(74)~\mathrm{mol^{-1}}$; the digits in brackets express the numerical uncertainty.
This will mean that $1~\mathrm{mol}$ of carbon-12 atoms will no longer have the mass of exactly $12~\mathrm{g}$ (but again, it will be well within experimental error for everybody not practising theoretical physics).
To answer the follow-up question you asked in the comments: The coefficients in chemical equations such as
$$\ce{Zn + 2 HCl -> ZnCl2 + H2}$$
are always and exclusively to be understood as ratio coefficients. Thus, instead of thinking one atom or one mole of zinc, think amount $n$ of zinc and amount $2n$ of $\ce{HCl}$.
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