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Counting # of atoms don't match in rounding? The total number of atoms contain in 1.0L as 1000g of liquid water must equal the total number of atoms contained in a gas released by 1000g of liquid water.

A mole of water molecules contains 2 moles of hydrogen atoms and 1 mole of oxygen atoms. Avogadro's number is 6.023×10^23 molecules/mole.

One mole of water =6.023×10^23 molecules of water.

1 molecule of water = 3 atoms of water. One mole of water =3×6.023×10^23=1.807×10^24 atoms. Now

Atomic mass of H2O = 16 + 2 = 18g/moles Number of moles = Mass Given/(Atomic Mass) or Molecular Mass

Number of moles = 1000g/18

Number of moles = 55.556

55.56 moles of water as liquid = 55.56 x 3×6.023×10^23 = 1.0039x10^26

"Total number of atoms = 1.0039x10^26 "

The mass of 1.0L as 1000g or 55.56 moles of water produces 1866 L of gas (1244L of H2 and 622L of O2).

Number of moles of hydrogen = 1244/22.4 = 55.54

Number of moles of oxygen = 622/22.4= 27.76

Atoms contain in 55.54 moles of hydrogen = 55.56 x 6.023×10^23= 3.346x10^25

Atoms contain in 2x 55.54 moles of hydrogen = 2x55.56 x 6.023×10^23= 6.69x10^25

Atoms contain in 27.76 moles of oxygen = 27.76 x 6.023×10^23=1.67x10^25

"Total number of atoms = 8.36 x10^25 "

Where did I make the error in rounding?

i didn't mention STP but it was presumed

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    $\begingroup$ Always try to approach the task first by the general algebraic solution and plug in the numbers in the end. It has several benefits. 1/ It allows you better grasp on underlaying principles. 2/ It allows both of you and readers to more easily spot a solution error or conceptual mistake. 3/ It gives the Q/A higher and more permanent value to be reused/applied for similar questions. 4/ It follows the site policy to focus on explaining principles and procedures, rather than solving of particular tasks with the value for 1 person only. 5/ You will avoid the intermediate rounding errors. $\endgroup$ – Poutnik Nov 3 '20 at 14:22
  • $\begingroup$ Note the purpose of the site is not proof reading. Is there any principle or concept you have trouble to apply ? $\endgroup$ – Poutnik Nov 3 '20 at 14:23
  • $\begingroup$ Note that moles of oxygen atoms and moles of oxygen gas are not the same thing. Same with moles of hydrogen atoms and moles of hydrogen gas. Both of those gases are diatomic. $\endgroup$ – Zhe Nov 3 '20 at 14:32
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    $\begingroup$ Avogadro number is $\pu{6.02214076E23}$. (exactly by definition) $\endgroup$ – Poutnik Nov 3 '20 at 14:35
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    $\begingroup$ Use MathJax for eventual formatting for formulas or expressions. E.g., instead of 0.123456x10^78, write $\pu{0.123456E78}$, what will look $\pu{0.123456E78}$. $\endgroup$ – Poutnik Nov 3 '20 at 14:38
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This is an extended comment...

I assume the statement that you are trying to prove is:

The total number of atoms contain in 1.0L as 1000g of liquid water must equal the total number of atoms contained in a gas released by 1000g of liquid water.

It is a rather general statement. The statement doesn't say anything about temperature, pressure, volume, or composition of the gas. I'm guessing that you're trying to use gaseous hydrogen and oxygen since water doesn't have a vapor pressure of 1 atmosphere at STP.

Avogadro's number is $6.023 \times 10^{23}$ molecules/mole.

You must be using an old book/reference. $\mathrm{N_A} \approx 6.022 \times 10^{23}$ had been used for quite a while. But in 2019 the SI base units were redefine and the value now is exactly $6.02214076 \times 10^{23}$.

Atomic mass of H2O = 16 + 2 = 18g/moles

Number of moles = Mass Given/(Atomic Mass) or Molecular Mass

Number of moles = 1000g/18

Number of moles = 55.556

The calculation for number of moles of water has a massive mistake. You calculated the mass of water to only 2 significant figures but end up with 5 significant figures for the moles of water. Significant figures are a crude way of doing error propagation, but proper use prevents exactly this kind of mistake. You could easily lookup the molecular weight of water which is 18.015 g/mol to 5 significant figures.

You then go on to talk about "55.56 moles of water as liquid" and then "Atoms contain in 55.54 moles of hydrogen". This is just sloppy.

Number of moles of hydrogen = 1244/22.4 = 55.54

Here again you've used an old value, 22.4, for the volume of a gas at STP. Since 1982, STP is defined as a temperature of 273.15 K (0 °C, 32 °F) and an absolute pressure of exactly $10^5$ Pa (100 kPa, 1 bar). This gives a molar volume of 22.711 liters/mole.

Now in general I agree with user Poutnik's advice. Try to write out an algebraic formula if you can. However if the formula is going to be large and messy, then look at all the given values and decide how many significant figures should be in the answer. I'd use two extra significant figures in intermediate calculations to prevent rounding errors as well as possible.

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