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This evening my six year old asked me

"Would a piece of paper look as big as this room to an atom?, or bigger?" ('this room' being a small sized bedroom)

A friend suggested it would probably look as big as the state of Connecticut. Then I figured well really the question is not defined enough, because really a piece of paper would look as big as a piece of paper.

Let us instead ask

"If you blew up an atom to the size of a human, and you expanded a piece of 8.5" x 11" paper by the same amount, how big would the piece of paper be?

I found someone's estimate of the number of molecules in a piece of paper:

$$2,247 \times 10^{23} \text{molecules}/\text{sheet of paper}$$

I guess all I need do now is multiply an 8.5" x 11" sheet of paper by that number. Only think is I don't have a clue how to work with that number!

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    $\begingroup$ I'm voting to close this question as off-topic because it isn't about concepts in physics $\endgroup$ Jul 29, 2015 at 7:07
  • $\begingroup$ @JohnRennie Where should it be if not in physics?, Math? $\endgroup$ Jul 29, 2015 at 20:49
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    $\begingroup$ It doesn't belong on Math. Folks closed it because it really comes down to the fact that you don't know how to work with the large number $10^{23}$. I think if the question were edited to ask more about the physical intuition of how large the paper would be, say compared to other objects, after being blown up by the expansion factor, then it could be re-opened. Since you've already accepted an answer though this probably isn't really an issue. $\endgroup$
    – DanielSank
    Jul 30, 2015 at 5:43
  • $\begingroup$ :( Strange tho, that it's closed cause someone doesn't know something. Isn't that the point of asking a question? SE can be a strange place sometimes o.O :( $\endgroup$ Jul 30, 2015 at 22:50

2 Answers 2

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A typical atom is roughly a few times $10^{-10} \text{m}$ wide. A piece of paper is say $(1/4) \text{m}$ wide. Therefore the ratio of the width of an atom to the width of a piece of paper is around $10^9$. A piece of paper is roughly the same width as a human, so $10^9$ is also a rough guess for the ratio of the width of a human to the width of an atom.

The ratio of the width of a bedroom to the width of a person is say roughly $10$.

Therefore, to an atom, a piece of paper would look like a square $10^8$ bedrooms to a side. That's one hundred million bedrooms to a side, and $10^{16}$ bedrooms total area.

If we take a piece of paper and scale it up by $10^9$ we get $155,000 \, \text{miles}$ to a side, which is 20 times the diameter of Earth.

Atoms are really small.

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  • $\begingroup$ This is a great answer. How does it compare with Ultima's and WillO's below? They both come up numbers of a different magnitude. $\endgroup$ Jul 29, 2015 at 5:11
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    $\begingroup$ @AgentZebra Those other answers are comparing the volumes of various objects while in my answer I've stuck to the widths of the objects. For example, I pointed out that a piece of paper or a person is about $10^9$ times wider than an atom. That means that a person is about $10^9 \times 10^9 \times 10^9 = 10^{27}$ times bigger in volume than is an atom. I stuck to the width comparison because I think it's easier to intuitively envision than it is to intuitively envision volume comparisons. For example, I can say the scaled up paper is the width of 20 Earths and you can sort of picture that. $\endgroup$
    – DanielSank
    Jul 29, 2015 at 5:41
  • $\begingroup$ @AgentZebra No problem. I don't think I've ever had an accepted answer on a negatively scoring question before :P It's too bad the folks who down-voted the question didn't bother to leave a comment indicating what they think is wrong with it. $\endgroup$
    – DanielSank
    Jul 29, 2015 at 21:13
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For a 6 year old, you might want to focus on thickness instead of length, as the numbers get too big with length.

A ream of paper (500 sheets) is a bit over an inch thick, say $3.5 \, \text{cm}$, so one sheet is $3.5/50 \, \text{mm}$, or $.07 \, \text{mm}$, which is $7 \times 10^{-5} \text{m}$.

An atom has diameter $0.1 \, \text{nm}$ to $0.5 \, \text{nm}$ (source). Paper is mostly carbon, some oxygen some hydrogen. Focus on carbon, which has diameter $.22 \, \text{nm}$ or $2.2 \times 10^{-10} \, \text{m}$ (source).

So, one sheet is about $300,000$ carbon atoms thick. If a carbon atom were the size of a basketball a sheet of paper would be about 60 miles high (and quite a bit larger than that lengthwise).

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  • $\begingroup$ This answer suffered many punctuation, grammar, and formatting errors. Please take a look at the source (by hitting the edit button) so you can see how to make nice things like in-text hyperlinks. Also please do consider proper grammar in the future; it really does make answers easier to understand. $\endgroup$
    – DanielSank
    Jul 29, 2015 at 6:06

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