# More atoms in a cup of water, than cups in an ocean?

#### bchandler

I've heard this before, and am just wondering if you guys think it's true. The saying is "There are more atoms in a cup of water than there are cups of water in the oceans of the world." I am pretty sure I heard this on some Science Channel show about atoms.

I know it would depend on the size of the cup, etc. Let's not be too logical about it :)

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#### Danger

Gold Member
Sounds right to me. Atoms are really small.

#### WarPhalange

The density of water is 1g/cm^3 or 1kg/m^3, right? So just use the mass of one water molecule to calculate how many there are per cm^3, and however many cm^3 there are in an average cup of water.

I'd say it's definitely more than 1 mol.

#### bchandler

The density of water is 1g/cm^3 or 1kg/m^3, right? So just use the mass of one water molecule to calculate how many there are per cm^3, and however many cm^3 there are in an average cup of water.

I'd say it's definitely more than 1 mol.
Well, that doesn't really tell you how many cups are in the ocean, though, does it? :)

#### Pythagorean

Gold Member

The volume of Earths oceans is 1.37 billion cubic kilometers. Translating this into measuring cups (standard 8 ounce cup), you get 22,617,336,000,000,000,000 cups of water - 22.6 million trillon, give or take a little. This would be over 3.2 billion cups (200 million gallons) for each and every person on the planet.
1 cup water = 235.6 cm^3 = .2366 liters = .2366 kg = 236.6 g

molecular mass of water is 18 g/mol, so:

(236.6 g) / (18g/mol) = 13 mols

13*6*10^23 =

---------------------------------------------------------------------------------

7.8*10^24 molecules of water in a cup of water

compared to

2.2*10^19 cups of water in the ocean

----------------------------------------------------------------------------------
CONCLUSION

that's a factor of about 3.5*10^5 = 350000

so there's over a quarter million times more molecules than cups!

(notice, I ignored that it was saltwater, but I don't think you can overcome that factor of a quarter million with the adjustments)

#### Chi Meson

Homework Helper
Woooooooooo HOOOOOOO!

Now That's science, Yeah Baby! Yeah!

ehm

Gold Member

#### lisab

Staff Emeritus
Gold Member
I know it would depend on the size of the cup, etc. Let's not be too logical about it :)
I think your confusion is based on the word "cup." It's an English unit of volume, and is well-defined (as seen in Pythagorean's calcuation) as 0.2366 L.

#### Moonbear

Staff Emeritus
Gold Member
Excellent answer, Pythagorean! Though, you spoiled all my fun of planning to hand bchandler a measuring cup and telling him to go to the beach and find out.

#### Pythagorean

Gold Member
Woooooooooo HOOOOOOO!

Now That's science, Yeah Baby! Yeah!

ehm
well, that's the theoretical half. Sounds like Moonbear has an experimentalist lined up for us -_-

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