# Why is the mass of 1 mole of an atom/molecule equal to its gram atomic/molecular mass?(proof) [duplicate]

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".

• @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.
– MrAP
Commented Feb 8, 2017 at 22:42

$$\def\p{\mathrm{p^+}}\def\n{\mathrm{n}}$$ 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.

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