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Why do we take the mass of 1 mole of an atom to be equal to its gram atomic mass and the mass of 1 molecule to be equal to its gram molecular mass? I want a mathematical rigorous proof. There are other questions on this site dealing with this question but none of them have the "proof".

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  • $\begingroup$ @Mithoron This is a self answered question. I did not like the answers in the question mentioned and there were many answers there and my answer probably would not get noticed there so i decided to post it separately. $\endgroup$
    – MrAP
    Commented Feb 8, 2017 at 22:42

1 Answer 1

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$\def\p{\mathrm{p^+}}\def\n{\mathrm{n}}$ Mass of one proton = $m_\p$ and mass of one neutron = $m_\n$. (Masses are in unified atomic mass unit-$\mathrm{u}$)

C-12 isotope consists of 6 $\p$ and 6 $\n$. Therefore, mass of C-12 isotope is equal to 6 $m_\p$ + 6 $m_\n$. Since $m_\p$$m_\n$, the mass of C-12 can be replaced by 12 $m_\p$. (electrons are not taken into account since they have negligible mass)

1 $\mathrm{u}$ is defined as 1/12 of the atomic mass of C-12 isotope.

⟹1 $\mathrm{u}$ = 1/12 × (6 $m_\p$ + 6 $m_\n$)

⟹ 1 $\mathrm{u}$ ≈ 1/12 ×12 $m_\p$

⟹ 1 $\mathrm{u}$ ≈ 1 $m_\p$ ...equation(1)

Now,

12 g of C-12 has $\pu{N_A}$ particles and constitutes 1 mole of it. (according to the definition of 1 mole)

⟹ 1 mole of (6 $\p$ + 6 $\n$) have a mass of 12 g.

⟹ 1 mole of (12 $\p$) have a mass of 12 g.

⟹ 1 mole of $\p$ have a mass of 1 g

⟹ 1 mole of $\p$ × 1 $m_\p$ = 1 g

⟹ 1 mole of $\p$ × 1 $\mathrm{u}$ = 1 g

⟹ 1 mole of $n$ $\p$ × 1 $\mathrm{u}$ = $n$ g

Therefore, mass of 1 mole of $n$ $\p$ is equal to $n$ g

Now, the final part,

Let an atom have $x$ $\p$ and $y$ $\n$ . The atom can be considered to be $(x+y)$ $\p$ for our purposes.

1 mole of $(x+y)$ $\p$ have a mass of $(x+y)$ g.

Thus the mass of 1 mole of atoms of an element is equal to its gram atomic mass.

Molecular mass of an element or a compound is equal to the sum of the atomic masses of each type of it's constituent atoms. Thus the mass of 1 mole of molecules of an element or a compound is equal to the gram molecular mass.

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  • $\begingroup$ the way you get the mass of carbon-12 is just wrong. $m_p \ne m_n$ and you ignored the binding energy. $\endgroup$
    – MaxW
    Commented Feb 8, 2017 at 23:12
  • $\begingroup$ @MaxW Firstly, I have clearly said that the mass of a proton is approximately equal to the mass of a neutron and secondly, why should i consider binding energy? $\endgroup$
    – MrAP
    Commented Feb 8, 2017 at 23:15
  • $\begingroup$ The point is that $N_A$ atoms of $\ce{^12_6C = 12 \text{grams}}$ by definition. $\endgroup$
    – MaxW
    Commented Feb 8, 2017 at 23:22
  • $\begingroup$ I have stated this in my answer. $\endgroup$
    – MrAP
    Commented Feb 8, 2017 at 23:29
  • $\begingroup$ Why is my answer downvoted? $\endgroup$
    – MrAP
    Commented Feb 9, 2017 at 15:20

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