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L = $6.022*10^{23}$enter image description here

If I have one atom of oxygen , it has exactly L value.

Now 2 atoms of a oxygen have 2L value.(Note that I mean 2O and not $O_2$here)

Well , if we take a molecule made up of 2atoms or combined i.e ($O_2$It also has L value. How is that possible ?It has bigger size and probably more atoms.

Reference :Take a look at the table enter image description here

enter image description here

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    $\begingroup$ What's an L value? I don't think it is a thing at all. $\endgroup$ – Ivan Neretin Dec 30 '20 at 18:46
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    $\begingroup$ Well, Avogadro number is just a number. Admittedly, a big one, but still a number. It has no magic powers. Substitute it everywhere with "thousand", and then the confusion must vanish. $\endgroup$ – Ivan Neretin Dec 30 '20 at 19:03
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    $\begingroup$ As of 2019 the Avogadro constant is no longer based on 12 g of C-12. It is now just a specific number: $6.02214076 \times 10^{23}$. $\endgroup$ – MaxW Dec 30 '20 at 19:15
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    $\begingroup$ @MaxW Be aware there is the tiny difference between the unitless Avogadro number and the Avogadro constant with unit 1/mol. $\endgroup$ – Poutnik Dec 30 '20 at 20:17
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    $\begingroup$ What is the title and author of that horrible textbook, and what is the intended audience? $\endgroup$ – Karl Dec 30 '20 at 20:40
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Sorry to see that this is your grade 10 textbook. This also means you have no choice but to use this book for passing public exams. Please don't use L or NA as a symbol.

If I have one atom of oxygen , it has exactly L value.

No. Atoms are discrete objects i.e., countable. If you have 1 atom of oxygen, it means 1 atom of oxygen. If you said 1 mole of oxygen atoms, then it is 6.022x10$^{23}$ atoms of oxygen.

Your L comes from Loschmidt number or constant. From Science Progress in the Twentieth Century (1919-1933) , 1933, Vol. 27, pp. 634-649

Loschmidt 's number, N, is defined as the number of atoms in gram-atom or the number of molecules in a gram-molecule.More than eighty different experimental determinations of this number have been made [i], and as it is a basic atomic constant its most probable value is of great importance in atomic physics. It is, therefore, the purpose of this article to outline the main methods by which Loschmidťs number has been evaluated, and to give some indication of current opinions of its most probable value

Footnote

1 This number is frequently referred to as " Avogadro's Number/' the term " Loschmidťs Number " being then reserved for the number of mole- cules in a cubic centimetre of a gas under standard conditions. Unfortunately, these designations are often interchanged. Avogadro's important hypothesis on the identity of the numbers of molecules in equal volumes of different gases at the same pressure and temperature was formulated in 1811, and is appropriately associated with his name ; but Avogadro made no quantitative estimate of either of the above-mentioned constants. The first actual estimate of the number of molecules in one cubic centimetre of a gas under standard conditions was made in 1865 by Loschmidt, and from this the number of molecules (atoms) in a gram molecule (atom) was later evaluated. From the quantitative view-point it thus seems preferable to speak of " Loschmidťs number per gram-molecule (atom)/' and of " Loschmidťs number per cubic centimetre," as is almost invariably done in the German scientific literature. This terminology avoids ambiguity, and has been adopted here.

You can see here your textbook is using the symbolism of 1930s or earlier.

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A mole is just a number, but you have to specify what you are counting

Aside from problematic issues about how your textbook is explaining things, the central misunderstanding arises because you need to specify what you are counting when you say there is a mole of something.

If you are counting oxygen atoms then there are two moles of oxygen atoms in one mole of dioxygen gas. There is no magic or confusion if you specify what it is you are counting.

This is not a problem unique to chemistry. If you count eggs, for example, you might use a dozen eggs (a dozen is just a signifier for 12 of anything) to make one (large) omelette. There is no confusion if you are clear about what you are counting, omelettes or eggs.

A mole is just a number, like a dozen (but a lot bigger). Indeed, you could have a mole (the number) of moles (the small burrowing mammal) but this would be enough matter to form a small planet.

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