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I understand that the one mole of atoms is equal to $6.02\times10^{23}$ atoms. However, when I used various resources to learn about stoichiometry, they showed me a balanced equation:

$$\ce{2H2O->2H2 + O2}$$

and said "2 moles of water" decomposed to form "2 moles of molecular hydrogen" and "1 mole of molecular oxygen". It is my understanding that there are only 2 molecules of water, not $2 \times 6.02 \times 10^{23}$ molecules of water and so on with the other parts of the equation as well. Could someone please clear up my confusion?

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Balanced equations represent ratios of reacting chemical species, not absolute quantities. What we mean when we say $$\ce{2H2 +O2->2H2O}$$ is not literally that two molecules of hydrogen react with one of oxygen, but rather that two units of hydrogen react. These units could be moles or molecules or something else entirely.

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    $\begingroup$ I'll just add that the units can't be weights or masses. The units ultimately have to be proportional to the number of molecules. $\endgroup$ – MaxW Jul 24 '18 at 1:49
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The balanced equations are just ratios. Read your example closely, it states that:

2 moles of water decomposes to form 2 moles of hydrogen and 1 mole of oxygen.

Note that all of the units are in moles and the statement does NOT say:

2 moles of water decomposes to form 2 molecules of hydrogen and 1 molecule of oxygen.

The mole unit is just a multiplier. The question could have alternatively said:

2000 molecules of water decomposes to form 2000 molecules of hydrogen and 1000 molecules of oxygen.

or

$12.04\times10^{23}$ molecules of water decomposes to form $12.04\times10^{23}$ molecules of hydrogen and $6.02\times10^{23}$ molecules of oxygen.

and the implication of either would be the same. In all cases the ratio of water:hydrogen:oxygen is 2:2:1 respectively.

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