Size and Scale Put in Perspective: The Universe Isn't Really That Big -- and Atoms Aren't Really That Small

[metacogitans]

Edit: mixed up Bohr radius with diameter, fixed some numbers (sort of), as other posters have pointed out the estimate of number of atoms placed side by side the length of a millimeter was wrong, and was based on Bohr radius.

 

If we keep the length of a meter in mind for comparison when looking at units expressed in scientific notation, it starts to dawn us that lengths such as the Bohr radius of a hydrogen atom, (5.2x10^-11, which is 52pm) are really not that small.. not very small at all. In fact, it's large enough we can visualize it mentally:

 

If put in a straight line side by side, there'd only be ~10 million hydrogen atom diameters in a millimeter. Looking at a millimeter, having in mind that it's only 10 million hydrogen atom diameters long, and 10 million not being that large of a number, everything starts to just seem.. simple -- and that's just hydrogen, with its 1 proton; every other element would have an even smaller number of diameter-lengths in one millimeter (lets forget for a moment about deuterium and whatever the name is for +2 helium without any electrons).

 

Realistically however, in one cubic millimeter of the air in front of you, there'd be even less than 10 million atoms. [Edit: as other posters have pointed out, this is wrong ] (perhaps someone could provide some more accurate estimates of how many atoms there are in a cubic millimeter of air, or a wood desk or other common object).

Let's have a paragraph just to conceptualize and point out that 10 million is not that large of a number; 1,000 can be verbally counted to in 5-10 minutes; 1,000 dots can be plotted on a piece of paper in 4; and there is only one thousand thousands in a million. To put the number one million in perspective visually, here's a picture of 1,000,000 pennies: https://c1.staticflickr.com/1/93/235878348_d72f315683.jpg

Less than 10 million atoms in one millimeter is a small enough number you'd almost think you'd be able to visibly see them; however, you wouldn't be able to see individual atoms anyways for purely mechanical reasons: the photoreceptors in your eyes are, themselves, made of many atoms; to trigger an action potential in an optical nerve, the electromagnetic radiation required has to, in almost all cases, come collectively from many atoms (excluding hypothetical scenarios with extreme conditions). Action potentials do not clear and reset fast enough either to be able to differentiate individual atoms from one another - and even after that, the processing of visual stimuli done subconsciously in the visual cortex would lose track of the stimulus produced by a lone atom; the stimulus would be melded in with (or simply lost amidst) other activity taking place.

 

Thinking of the figure "<10 million atomic diameters in a millimeter" a little bit more, one might then start to think about the size of a biological cell, how many atoms wide a cell is, and how then a large number of cells can fit in the length of a millimeter: assuming we'd have to knock at least few figures off 10 million, we're only in the thousands to hundreds range for number of cells that would fit placed side by side in one millimeter (which likely isn't far off from reality), which then begs the question "why can't we see individual cells?" I think part of the answer lies still in the mechanical limitations of the eye, such as there being a finite amount of photoreceptors and optical nerves, so a 'perfect image' of something is always an impossibility at any scale, and our vision is really nothing more than a quagmire of blurry nonsense splashing against our photoreceptors, which sputter out electrical noise down the axon of an optical nerve, where it then haphazardly blast electrochemicals at other neurons across synapses, adding collectively to a sum of neuronal noise in the visual cortex. Only with the aspect of 'mind' (arising out of the semantics in neural connections' structural organization), is the noise made coherent into a rough image of something. The subject of how the brain processes visual stimuli is actually quite interesting on its own; there are a lot of other mechanical limitations to the macroscopic structure of the eye as well, with visible consequences that aren't usually noticed unless specifically pointed out; I'll link to more about it in a follow up post.

 

Anyways, the scale of the atomic world put in perspective sort of makes us lose that awing, unimaginably deep and complex feeling as an aspect of the microcosm. But what about the macrocosm? We should still be okay in having that, 'small, insignificant, mystifying feeling' about how big the universe is, right? I'd like to think we should, and with there even maybe being a whole multi-verse, I don't think we should lose that magic of being mystified by the cosmos. However, with the convenience of the metric system and being able to conceptualize scale with exponents of 10, the macroscopic cosmos is entirely fathomable mathematically, especially if you keep the numbers laid out in front of you:

• The estimated radius (according to the featured link by Google for the search 'diameter of known universe') of 46.5 billion light years is only 4.40×10²⁶ meters.
• 4.46x10²⁶ means we're only dealing with around 27 figures, when concerning our unit of measurement the 'meter'. In kilometers, just 24 figures; and with a thousand kilometers per unit, or 'megameters', we knock 3 more figures off to have just 21 figures. By itself, a number in the 20s is very easy to work with mentally, even toddlers are quite capable of counting to 20, or conceptualizing 20 of something; it's not as though we are dealing with some obscenely large number with 10 to the power of a several-hundred figure number; even 10^{999999999999999999,} which is much larger than 4.46x10^26, is easy to write out and work with using scientific notation. Given the potential of mathematics and how insanely tedious calculations can be, things like the size of the universe and the scale of the subatomic world start to seem kind of boring, really. 'Mathematics in practical measurements' compared to 'arithmetic which seeks to harness the potential of mathematics' makes the practical world seem small and perhaps even a little bit claustrophobic for a mathematician who dabbles in the philosophical.

Edited November 17, 2015 by metacogitans

[Strange]

Realistically however, in one cubic millimeter of the air in front of you, there'd be even less than 20 million atoms. (perhaps someone could provide some more accurate estimates of how many atoms there are in a cubic millimeter of air, or a wood desk or other common object).

 

That seems a little low. A quick estimate gives me a number more than 10 billion times larger.

[pzkpfw]

In any case, this seems more about the elegant systems of math we've got that allow us to efficiently express these quantities.

 

I've always liked how I can easily write on a small piece of paper a number that's "more" than the number of atoms in the visible Universe.

[swansont]

 

Realistically however, in one cubic millimeter of the air in front of you, there'd be even less than 20 million atoms. (perhaps someone could provide some more accurate estimates of how many atoms there are in a cubic millimeter of air, or a wood desk or other common object).

 

Strange is right. You're off by about 9 orders of magnitude. An ideal gas occupies 22.4 liters per mole (at STP), or 0.0446 moles per liter. There are 1000 cubic cm in a liter, and 1000 mm^3 per cm^3. So that's 4.46 x 10^-8 moles. That's around 2.7 x 10^16 molecules. (around twice that in terms of number of atoms)

 

Funny how, with these numbers being so easy to visualize, you'd be off by a factor of a billion in a simple estimate.

[geordief]

I have been meaning to ask a similar question but from the opposite standpoint.

 

It goes like this .Regardless of whether the scales can be written down in a mathematical way does the scale involved create a qualitative (and justifiable) shift in how we view the universe?

 

It seems to me that no matter what scale is agreed to be quantifiable that this is only an opening gambit and really there is no limit to the scale that possibly applies.

 

That "no limit" is what I ,personally find myself reconciled to and ,as I said I wonder if this should bring about a qualitative change in a view of the universe (maybe described as an acceptance of the "unknowable") or whether we are just talking about the same ,just bigger .

 

To answer the OP ,then I would say "yes it really is that big (bigger) ,yes it really is that small (and smaller) and (if you think about it ) then be very afraid"

[Ophiolite]

The OP is an interesting perspective on the scale of the universe and its contents. I followed the argument and attempted to imagine the quantities and dimensions being portrayed.

 

As pzkpfw implies this can be appreciated intellectually, but in my gut I don't really understand it and cannot actually imagine it. I accept that some people can, but it is beyond me.

 

As per the Hitchiker's Guide to the Galaxy I believe that "Space is big. Really big. You just won’t believe how vastly hugely mindbogglingly big it is. I mean you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space."

[metacogitans]

 

That seems a little low. A quick estimate gives me a number more than 10 billion times larger.

Here was my thinking behind that: if the diameter of a hydrogen atom is indeed 5.2x10-11 meters, and placed side by side the length of a millimeter, would number something like 18.8 million, this number cubed would be ~6.6 billion. However, assuming that would be of unrealistic density, I guessed it would be much less.

 

Now however, using avogadro's number instead of just putting numbers together, I'm not so sure about the math in my original post or the diameter of a hydrogen atom being 5.2x10-11 meters... Taking the volume of one gram of liquid hydrogen H2 which is roughly 0.011 cubic meters at room temperature, or 11,000,000 cubic millimeters, and roughly using avogadro's number 6*10^23, we get 50,000,000,000,000,000 atoms per cubic millimeter of liquid hydrogen -- quite a bit larger than 6.6 billion.

And considering the volume of gaseous H2 hydrogen, 22.4 liters per gram at room temperature and 1atm, which would be 22.4 million cubic millimeters, that's still 26,000,000,000,000,000 atoms in a cubic millimeter..

 

I need to investigate how they calculated the diameter of a hydrogen atom then, and where the number 5.2x10-11 meters came from. I'm guessing they're somehow including the reach of the atom's electromagnetic field (but wouldn't that technically be infinite? Just negligible after a certain distance?)

 

Edit: Found out where I got 5.2x10-11 meters is the Bohr radius. So the diameter would be twice that 10.4x10^-11 meters; I'm going to correct that in my original post.

Also, how was Bohr radius calculated, and why does an estimate using Bohr radius differ so much from what Avogadro's number gives you?

Edited November 17, 2015 by metacogitans

[swansont]

 

I need to investigate how they calculated the diameter of a hydrogen atom then, and where the number 5.2x10-11 meters came from. I'm guessing they're somehow including the reach of the atom's electromagnetic field (but wouldn't that technically be infinite? Just negligible after a certain distance?)

 

It's basically the Bohr radius (52.9 pm), which is the most probable distance the electron will be from the nucleus.

 

Your estimate of the number in a solid ignores that you can more efficiently pack spheres than placing them side-by-side and "cubing" that. If you offset rows by the radius you can get more into the volume.

The packing fraction you describe gives you 4/3 πr^3 per volume of 8r^3, or a ratio of 0.5236. Almost half that volume is free space. The maximum close-packing in a lattice is about 74%, almost half again more.

 

https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres

Also, how was Bohr radius calculated, and why does an estimate using Bohr radius differ so much from what Avogadro's number gives you?

 

The distance of the orbit for the Bohr model, and the most probable distance from the QM calculation.

 

The reason you got a different number is you did the math wrong. 18.8 million cubed is not 6.6 billion. (10^6)^3 is 10^18

[metacogitans]

 

The reason you got a different number is you did the math wrong. 18.8 million cubed is not 6.6 billion. (10^6)^3 is 10^18

You're right. I took 18.8^3 and asssumed that's how many "millions" there'd be; It's been too long since I've actually been in a math class . I'm a bit rusty. It's a good thing I'm not trying to be a physicist for a living; I'd get booed straight to the unemployment line.

Edited November 17, 2015 by metacogitans

[metacogitans]

What an embarrassing thread. Don't drink and math. I suppose it would have been a lot better if I didn't butcher it every time numbers tried coming out of my mouth

Edited November 25, 2015 by metacogitans