The Size of Atoms and Molecules: Incredible Claim?

[Dekan]

Today I was reading a book, in which this passage occurs:

 

"Air is made up of molecules, which are themselves made up of groups of atoms.

A molecule is almost unbelievably small....to give an example:

 

Take a small box with a capacity of 1 cubic inch, and fill it with ordinary air.

 

If we release 10 million molecules every second, how long will it take for the box to empty itself completely?

 

A second - a minute - a month?

 

No - 50 million years!"

 

This answer - 50 million years - strikes me as not credible. But I don't know how to do the maths to check it!

 

Can anyone do the maths, and prove or disprove it please?

 

Thanks, Dekan

Edited April 25, 2011 by Dekan

[insane_alien]

okay, in 1 cubic inch of air(at STP, standard conditions) there are 0.722mmol

 

this translates to 4.348*10^20 molecules

 

so if we divide this number by 10 million we will get the number of seconds to empty the container.

 

10million is 10^7 so we get 4.348*10^13 seconds

 

so thats 1.379 million years.

 

so yes, it is wrong.

[lemur]

idk, but this example doesn't refer to the size of the atoms and molecules themselves. It refers to the number of molecules in a given volume at standard temperature and pressure (I assume you mean standard temperature and pressure at least). You can maybe argue that the molecules can't be any bigger than the volume of the container divided by the number of molecules in it, but how do you know how much smaller they might be than that?

Edited April 25, 2011 by lemur

[SMF]

I think this bit should go into the Other Sciences, Cool Facts thread. SM

[Dekan]

okay, in 1 cubic inch of air(at STP, standard conditions) there are 0.722mmol

 

this translates to 4.348*10^20 molecules

 

so if we divide this number by 10 million we will get the number of seconds to empty the container.

 

10million is 10^7 so we get 4.348*10^13 seconds

 

so thats 1.379 million years.

 

so yes, it is wrong.

 

 

Thanks Insane_Alien for your reply.

 

You've cut the length of time down to a million years and a bit.

 

Even that seems incredibly long.

 

I thought someone would prove it was something slightly more plausible. Like 10 years, or 100 years at the outside.

 

But that's Modern Physics. It requires the willing suspension of disbelief.

 

 

Thanks again, Dekan

Edited April 25, 2011 by Dekan

[insane_alien]

well there isn't really much you can do about it.

 

there are a LOT of molecules in a cubic inch of air. enough that 10million doesn't really make that much of a difference.

 

luckily, most flows and transport phenomenon are a lot faster than 10million molecules per second so you don't have to wait a few million years to pour a cup of tea.

[swansont]

Take a different example if you wish. A grain of salt will have somewhere around 10^18 atoms/ions in it. The underlying principle is that Avogadro's number is really, really big.

[Dekan]

Take a different example if you wish. A grain of salt will have somewhere around 10^18 atoms/ions in it. The underlying principle is that Avogadro's number is really, really big.

 

The salt example appreciated. Chosen by design, one suspects.

Cum grano salis: what many of the claims of modern Science should be taken with?

 

Vale, Dekan

 

 

[swansont]

The salt example appreciated. Chosen by design, one suspects.

Cum grano salis: what many of the claims of modern Science should be taken with?

 

Vale, Dekan

 

 

No, not so much. Use a grain of sand. The precious metal in a ring you might wear. A drop of water.

 

Avogadro's number is still really big.

[lemur]

Take a different example if you wish. A grain of salt will have somewhere around 10^18 atoms/ions in it. The underlying principle is that Avogadro's number is really, really big.

I'm not good with dividing exponents. Do you subtract them to get the quotient? E.g. 10^23/10^18 = 100,000? If so, what would the volume and/or weight of 100,000 grains of salt be, approximately?

[swansont]

I'm not good with dividing exponents. Do you subtract them to get the quotient? E.g. 10^23/10^18 = 100,000? If so, what would the volume and/or weight of 100,000 grains of salt be, approximately?

 

Yes, you subtract exponents when dividing. 10^23 atoms would be somewhere around 1 cm on a side as a solid crystal, almost twice that on a side if it were grains.

[lemur]

Yes, you subtract exponents when dividing. 10^23 atoms would be somewhere around 1 cm on a side as a solid crystal, almost twice that on a side if it were grains.

Are you saying 100,000 grains of salt can fit into 2cm^2?

[insane_alien]

http://www.wolframalpha.com/input/?i=1*10^18+NaCl+molecules

 

typical grain of salt, 97 micrograms and a cube 0.356 mm on a side

 

http://www.wolframalpha.com/input/?i=1*10^23+NaCl+molecules

 

100000 grains of such, a cube 1.65 cm per side, thats 4.49cm^3 at least, more with packing densities.

Edited May 1, 2011 by insane_alien

[lemur]

http://www.wolframal...+NaCl+molecules

 

typical grain of salt, 97 micrograms and a cube 0.356 mm on a side

 

http://www.wolframal...+NaCl+molecules

 

100000 grains of such, a cube 1.65 cm per side, thats 4.49cm^3 at least, more with packing densities.

 

That's a smaller volume than I would expect for so many salt grains. So then if there are 10^18 salt molecules in a grain, that compares to how large a volume of salt grains? 97 micrograms x 10^18 = 97 X 10^15 grams? Or 97 X 10^12 kilograms? Or 97 X 10^9 metric tons? It sounds like there's a mountain of salt-grains worth of atoms in just one grain of salt. I wonder what the smallest grain of salt possible is, a single molecule?

[swansont]

That's a smaller volume than I would expect for so many salt grains. So then if there are 10^18 salt molecules in a grain, that compares to how large a volume of salt grains? 97 micrograms x 10^18 = 97 X 10^15 grams? Or 97 X 10^12 kilograms? Or 97 X 10^9 metric tons? It sounds like there's a mountain of salt-grains worth of atoms in just one grain of salt. I wonder what the smallest grain of salt possible is, a single molecule?

 

~10^18 atoms is 97 micrograms. You don't multiply them.

[lemur]

~10^18 atoms is 97 micrograms. You don't multiply them.

I know. That was the amount of atoms in a grain of salt. I was trying to make an analogy to consider how many atoms were in a grain of salt by thinking about how big a pile of salt would be with as many grains as there are atoms in a single grain.

[insane_alien]

97 million tonnes

 

or enough to form a cube 355m on a side if in a solid block

 

orders of magnitude are funny things until you get your head around them.

 

and it IS difficult to get your head round them because a lot of stuff occurs outside the range of normal human perception. its easy to think of a n object a few tens of centimeters across but trying to think of an object a few pico meters across is very very difficult and seems impossible until you understand it.

 

its likely the reason for the disbelief at 1 million years to pump 1 cubic inch of air at only a few millions of molecules per second. the mind isn't good at handling these scales at first because nothing it encounters under normal circumstances is even close to it.

 

the mind thinks of 10 million as BIG it also classes the number of atoms in a cubic inch of air as BIG so it thinks BIG/BIG = small

 

but what it really is is MASSIVELY BIG/BIG = BIG

[lemur]

97 million tonnes

 

or enough to form a cube 355m on a side if in a solid block

 

orders of magnitude are funny things until you get your head around them.

I would guess that this is roughly equivalent to the volume of a mall. So if you filled a mall with salt and pictured the mall as a grain of sand with you the size of a microbe standing next to it, an actual grain of salt would be analogous to an atom relative to the mall as a figurative grain of salt? The amazing thing, imo, is the idea that all those grains of salt could spontaneously lock together to form a boulder the size of a mall and that such a boulder wouldn't crack when dropped on a pile of similar boulders. I guess at that scale, gravity exerts force at a magnitude similar to the moon or some other low-gravity situation. Now that this salt-grain vs. mall analogy has been established, maybe it would be interesting to make an analogy between Earth-gravity at the scale of a grain of salt and the effect of gravity on a boulder the size of a mall.

 

edit: just for fun I looked up the volume of Earth to compare with the size of a human. Wiki gives it as 1.08321×10¹² km^3.

^{So if you add 9 to the superscript 12 to get the number of cubic meters, it would be 1.08x10^21, which would be about 600,000 times less than avogadro's number, right? So if atoms were cubic meters, a mole of them would fill up 600,000 Earths? What is 600,000 times bigger than the Earth, then? The sun?}

Edited May 1, 2011 by lemur

[mississippichem]

I would guess that this is roughly equivalent to the volume of a mall. So if you filled a mall with salt and pictured the mall as a grain of sand with you the size of a microbe standing next to it, an actual grain of salt would be analogous to an atom relative to the mall as a figurative grain of salt? The amazing thing, imo, is the idea that all those grains of salt could spontaneously lock together to form a boulder the size of a mall and that such a boulder wouldn't crack when dropped on a pile of similar boulders. I guess at that scale, gravity exerts force at a magnitude similar to the moon or some other low-gravity situation. Now that this salt-grain vs. mall analogy has been established, maybe it would be interesting to make an analogy between Earth-gravity at the scale of a grain of salt and the effect of gravity on a boulder the size of a mall.

 

The electromagnetic force that holds ions together in salt, or atoms together in sand is WAY stronger than gravity. I'm comparing apples to oranges a bit here, but the gravitational constant is: [math] 6.67 \times 10^{-11} [/math] while the Coloumb constant (the proportionality constant that goes into calculating the force between two charges) is [math] 8.99 \times 10^{9} [/math].

 

So let's do a little calculation:

 

The magnitude of gravitational force is given by: [math] F= G \frac{m_{1}m_{2}}{r^{2}} \hat{r} [/math]

 

"G" is the gravitational constant, the two "m's" are the masses in question, r is the radial distance between the two masses and the "r" with the funny looking hat equals 1 (don't ask, ).

 

So the gravitational force between two sodium atoms (one nanometer apart) is:

 

[math]F = (6.67 \times 10^{-11} N \cdot m^{2} \cdot kg^{-2}) \frac {(3.82 \times 10^{-26} kg)(3.82 \times 10^{-26} kg)}{(1 \times 10^{-9} m)^{2}} [/math]...I get...[math] 9.73 \times 10^{-44} N [/math]. If you don't have a feel for Newtons, let me tell you that is a vanishingly small force.

 

Alright, so now lets look at the magnitude of electric repulsion between two positively charged sodium atoms (at one nanometer apart): We will use Coloumb's Law.

 

[math] F = k \frac {q_{1}q_{2}}{r^{2}} \hat {r} [/math]

 

Everything is the same here, except that "k" is now the Coloumb constant and the "q's" are charges instead of massses. The net charge on a sodium ion is [math] 1.60 \times 10^{-19} C [/math], so...

 

[math] F = (8.99 \times 10^{9} N \cdot m^{2} \cdot C^{-2}) \frac {(1.60 \times 10^{-19} C)(1.60 \times 10^{-19} C)}{(1 \times 10^{-9} m)^{2}} [/math]...here I get...[math] 2.3 \times 10^{-10} N [/math].

 

That is about [math] 2 \times 10^{33} [/math] times more force than we saw in the gravity calculation!

 

Like I said, I'm comparing mass to charge, and therefore apples to oranges but you should still be able to see that gravitational interactions between things on the molecular scale can't even hold a candle to electrostatic interactions at the molecular scale. That's why chemists can safely ignore gravity, and why the salt crystal stays intact when being dropped onto another salt crystal. Even though the calculation would be with the Earth's gravity, electrostatics still win out.

Edited May 1, 2011 by mississippichem

[insane_alien]

I would guess that this is roughly equivalent to the volume of a mall.

 

really? i've never seen a mall that big. biggest one i've seen is about 500m long and maybe 150m wide that would mean it'd have to be 600 meters tall.

 

i think it'd have to be a very very large mall. wolfram alpha lists it as being about 8% of the water in sydney harbour.

[lemur]

Like I said, I'm comparing mass to charge, and therefore apples to oranges but you should still be able to see that gravitational interactions between things on the molecular scale can't even hold a candle to electrostatic interactions at the molecular scale. That's why chemists can safely ignore gravity, and why the salt crystal stays intact when being dropped onto another salt crystal. Even though the calculation would be with the Earth's gravity, electrostatics still win out.

The reason your comparisons is like apples to oranges, imo, is that it's counterintuitive to think in terms of gravitational attraction between objects at our scale because we simply don't experience gravity at that level. The gravity we experience is at the level of objects' weight at sea level (unless you've been to the moon and experienced that gravity). So if you want to construct an experience-based intuitive comparison between gravitational and electrostatic force, it makes more sense to compare it to the amount of force holding some object to the ground. E.g. you could say that the charge holding molecules of salt together is analogical to the amount of gravity holding a bus to the ground, or a mall, mountain, etc. Or you could give it in numbers of such objects, e.g. 100 Mt. Everests stacked on top of each other. I think a better analogy would be to compare it to the strength of a small ferromagnet and then say how big a cube made out of such magnets would be if it had the same amount of total bonding force as a grain of salt.

 

 

 

really? i've never seen a mall that big. biggest one i've seen is about 500m long and maybe 150m wide that would mean it'd have to be 600 meters tall.

 

i think it'd have to be a very very large mall. wolfram alpha lists it as being about 8% of the water in sydney harbour.

A mall was just the biggest visualizable thing I could think of. Aircraft carrier might work too, or cruise ship, but I don't get to stand next to one of those as often as a mall, so it's harder to visualize the volume and how much salt would fit in one.

Edited May 1, 2011 by lemur

[insane_alien]

A mall was just the biggest visualizable thing I could think of. Aircraft carrier might work too, or cruise ship, but I don't get to stand next to one of those as often as a mall, so it's harder to visualize the volume and how much salt would fit in one.

 

aircraft carriers and cruise ships would also be too small.imagine a cube made out of eiffel towers thats going to be about right.

[lemur]

aircraft carriers and cruise ships would also be too small.imagine a cube made out of eiffel towers thats going to be about right.

8 50ft high malls stacked on top of each other?