If A Water Molecule Was An Apple

[Mintaka]

If a water molecule was an apple, and there are 170 trillion water molecules in a droplet of steam,

 

what kind of volume of space would our apples fill, on this scale?

 

Would 170 trillion apples fit into a greyhound coach, for example?

 

Or would it be more like a football stadium?

[modest]

An apple is 277,090,909 times larger than a molecule of water. You can find this by dividing the diameter of a typical apple (0.0762 meters) by the diameter of a water molecule (2.75 x 10^-10 meters).

 

Because an apple is 277,090,909 times larger than a molecule of water, you would multiply the size of a steam droplet by 277,090,909 to get the scaled up size.

 

A steam droplet is about 0.00001 meters in diameter, so multiply: 0.00001 meters by 277,090,909 and the answer will be the width (or diameter) of the spherical collection of apples scaled up.

 

~modest

[Mintaka]

An apple is 277,090,909 times larger than a molecule of water. You can find this by dividing the diameter of a typical apple (0.0762 meters) by the diameter of a water molecule (2.75 x 10^-10 meters).

 

Because an apple is 277,090,909 times larger than a molecule of water, you would multiply the size of a steam droplet by 277,090,909 to get the scaled up size.

 

A steam droplet is about 0.00001 meters in diameter, so multiply: 0.00001 meters by 277,090,909 and the answer will be the width (or diameter) of the spherical collection of apples scaled up.

 

~modest

 

wow, thanks, that's over 1.5 miles in diameter....

 

roughly how many apples would it contain?

[modest]

wow, thanks, that's over 1.5 miles in diameter....

 

Yup

 

roughly how many apples would it contain?

 

I trust Craig's calculation of 170 trillion in the other thread which is to say that there are 1.7 x 10¹³ water molecules per 0.00001 meter diameter water droplet or 1.7 x 10¹³ apples per 2.77 kilometer diameter apple droplet)

 

The volume of the apple droplet would be about 1.1139 x 10¹⁰ meters cubed making the density about 1572 apples per cubic meter which seems to indicate that there would be very little space between apples (or, very little space between water molecules in water).

 

~modest

[Mintaka]

Yup

 

 

 

I trust Craig's calculation of 170 trillion in the other thread which is to say that there are 1.7 x 10¹³ water molecules per 0.00001 meter diameter water droplet or 1.7 x 10¹³ apples per 2.77 kilometer diameter apple droplet)

 

The volume of the apple droplet would be about 1.1139 x 10¹⁰ meters cubed making the density about 1572 apples per cubic meter which seems to indicate that there would be very little space between apples (or, very little space between water molecules in water).

 

~modest

 

 

This is amazing - I don't understand 1.7 x 10¹³ apples in my mind, how many billions or trillions is that number please?

[modest]

This is amazing - I don't understand 1.7 x 10¹³ apples in my mind, how many billions or trillions is that number please?

 

Woops! I misspoke. It is 17 trillion. To convert 1.7 x 10¹³ to a written out number, you move the decimal over to the right 13 times, so it is 17,000,000,000,000 or 17 trillion.

 

Index Notation and Powers of 10

 

As a comparison, the US national debt is about 14 trillion dollars.

 

~modest

[Boerseun]

As a comparison, the US national debt is about 14 trillion dollars.

...which is, of course, a hell of a lot of apples.

[Boerseun]

I wonder...

 

I we were to hold a "coin-laying" fundraiser to clear the US debt, at $14 trillion, and everybody lays a penny, how long the line will be?

 

At $14,000,000,000,000, which will be 1 400 000 000 000 000 pennies. At a penny a pop, let's make it easy and say a penny takes up 1cm. A bit more, but the line will wiggle a bit, okay, so let's stick to a cm penny. There are 100cms to a meter. And there are 1000 meters to a kilometer. Which means a 100 000cms/km. So we need to chop five zeros off - giving us a line 14 000 000 000kms long if we were lay pennies to clear the US debt. That's a line of pennies 14 billion kilometers long.

 

Alternatively, we can stack pennies. Let's say a penny is a millimeter thick. Covering the US debt will then require a stack of pennies 1 400 000 000kms high. That's a stack 1.4 billion kilometers in hight. Or, put another way, 3856 individual stacks of pennies from the Earth to the moon at perigee...

 

...which can buy a hell of a lot of apples.

[Turtle]

...

I trust Craig's calculation of 170 trillion in the other thread which is to say that there are 1.7 x 10¹³ water molecules per 0.00001 meter diameter water droplet or 1.7 x 10¹³ apples per 2.77 kilometer diameter apple droplet)

 

The volume of the apple droplet would be about 1.1139 x 10¹⁰ meters cubed making the density about 1572 apples per cubic meter which seems to indicate that there would be very little space between apples (or, very little space between water molecules in water).

 

~modest

 

actually, there would be quite a bit of space for the apples. not so sure about water molecules though. anyway, i thought right off of fuller - as in bucky, not more full - but he proved himself a little daft to me and as he is still dead & not up on the latest, i'll take a bite from a wiki.

 

Sphere packing @wikipedia

...

Random close pack

If we attempt to build a densely packed collection of spheres we will be tempted to always place the next sphere in a hollow between three packed spheres. If five spheres are assembled in this way, they will be consistent with one of the regularly packed arrangements described above. However, the sixth sphere placed in this way will render the structure inconsistent with any regular arrangement. (Chaikin, 2007). This results in the possibility of a random close packing of spheres which is stable against compression.

 

When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%. Recent research predicts analytically that it cannot exceed a density limit of 63.4%[2] This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres (i.e. line segments or disks) will yield a regular packing. ...

[Qfwfq]

...so it is 17,000,000,000,000 or 17 trillion.

 

Index Notation and Powers of 10

 

As a comparison, the US national debt is about 14 trillion dollars.

OK, so the next time the IMF bugs the U. S. about controlling its public debt, you can point out it's less than a dollar per apple, which ain't all that bad, after all.

[Mintaka]

I wonder...

 

I we were to hold a "coin-laying" fundraiser to clear the US debt, at $14 trillion, and everybody lays a penny, how long the line will be?

 

At $14,000,000,000,000, which will be 1 400 000 000 000 000 pennies. At a penny a pop, let's make it easy and say a penny takes up 1cm. A bit more, but the line will wiggle a bit, okay, so let's stick to a cm penny. There are 100cms to a meter. And there are 1000 meters to a kilometer. Which means a 100 000cms/km. So we need to chop five zeros off - giving us a line 14 000 000 000kms long if we were lay pennies to clear the US debt. That's a line of pennies 14 billion kilometers long.

 

Alternatively, we can stack pennies. Let's say a penny is a millimeter thick. Covering the US debt will then require a stack of pennies 1 400 000 000kms high. That's a stack 1.4 billion kilometers in hight. Or, put another way, 3856 individual stacks of pennies from the Earth to the moon at perigee...

 

...which can buy a hell of a lot of apples.

 

hahaha this is great! :-))))

 

So, are we now saying there are roughly 17 trillion water molecules in a tiny floating steam droplet?

 

I mean the kind of steam that floats in your bathroom after a hot shower.

 

Come on guys! This is bending my brain.

 

A stupid question maybe but how is it possible to know even roughly how many water molecules there are if they can't even be seen??

 

And how do we even know they exist if they can't be seen?

 

Are we right to imagine " things", like tiny ' spheres'?

 

Is it possible we've got this all wrong?

[Mintaka]

hahaha this is great! :-))))

 

So, are we now saying there are roughly 17 trillion water molecules in a tiny floating steam droplet?

 

I mean the kind of steam that floats in your bathroom after a hot shower.

 

Come on guys! This is bending my brain.

 

A stupid question maybe but how is it possible to know even roughly how many water molecules there are if they can't even be seen??

 

And how do we even know they exist if they can't be seen?

 

Are we right to imagine " things", like tiny ' spheres'?

 

Is it possible we've got this all wrong?

 

 

 

My head is still reeling from your confirmation that "there are in fact more water molecules in a glass of water than grains of sand on the earth"..........This seems INCREDIBLE. i live near the sea, i walk on the beach every day, i see how many grains there are, and you add to thaht all the thousands of other beaches in the world,and all the deserts and all the sand lying under the oceans, and ......well I really need to go and lie down in a dark room.

[modest]

As a comparison, the US national debt is about 14 trillion dollars.

...which is, of course, a hell of a lot of apples.

 

:hihi:

 

~modest

[modest]

...

I trust Craig's calculation of 170 trillion in the other thread which is to say that there are 1.7 x 10¹³ water molecules per 0.00001 meter diameter water droplet or 1.7 x 10¹³ apples per 2.77 kilometer diameter apple droplet)

 

The volume of the apple droplet would be about 1.1139 x 10¹⁰ meters cubed making the density about 1572 apples per cubic meter which seems to indicate that there would be very little space between apples (or, very little space between water molecules in water).

 

~modest

 

actually, there would be quite a bit of space for the apples. not so sure about water molecules though. anyway, i thought right off of fuller - as in bucky, not more full - but he proved himself a little daft to me and as he is still dead & not up on the latest, i'll take a bite from a wiki.

 

Sphere packing @wikipedia

...

Random close pack

If we attempt to build a densely packed collection of spheres we will be tempted to always place the next sphere in a hollow between three packed spheres. If five spheres are assembled in this way, they will be consistent with one of the regularly packed arrangements described above. However, the sixth sphere placed in this way will render the structure inconsistent with any regular arrangement. (Chaikin, 2007). This results in the possibility of a random close packing of spheres which is stable against compression.

 

When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%. Recent research predicts analytically that it cannot exceed a density limit of 63.4%[2] This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres (i.e. line segments or disks) will yield a regular packing. ...

 

Yeah, I was assuming a uniform distribution. I think the distance between the center of one apple and the center of another in that case (where the density and size of apples are a larger scaled version of the density and size of water molecules) is the inverse of the density to the power of 1/3... so,

[math]r = \left( \frac{1 \ m^3}{1572 \ \mbox{apples}} \right)^{1/3} = 0.086 \ \mbox{meters}[/math]

 

so, 3.38 inches between the center of one apple and the center of an adjacent apple. I assumed the diameter of the apples to be 3 inches at the start, so there would be only .38 inches of 'empty space' between them (assuming they are packed uniformly and orthogonally like the atoms in a salt crystal). I hadn't realized there was so little space between molecules in liquid water.

 

But, I agree that they wouldn't be packed uniformly like that. If they were arranged randomly then the average distance between apple centers would be:

 

[math]r = \frac{1}{3} \Gamma ( 1/3 ) \left( \frac{4}{3} \pi \right) ^{-1/3} \rho ^{-1/3}[/math]

 

where [math]\Gamma(1/3)[/math] = 2.678938 and [math]\rho[/math] = 1572 is the density, so...

 

[math]r = .55396 \cdot 1572^{-1/3} = 0.0476 \ \mbox{meters}[/math]

 

making the average distance between centers when placed randomly at around 1.874 inches—less than the size of an apple. Clearly not a good approximation as the structure of the apple prevents random placement like wickylicious says.

 

In any case, before responding to Mintaka, I was expecting most of the space in a liquid to be intermolecular rather than molecular—naively assuming that the distance between apples would be at least a few times the diameter of the apple like in a gas. Clearly a bad intuition.

 

Here is an model of water and ice molecules:

 

http://esciencenews.com/articles/2009/08/11/physicists.make.crystalliquid.interface.visible.first.time

 

where the blue molecules are ice and the red are liquid. I would have thought they would be more spread out. I guess I should have realized that liquid water is more dense than ice and therefore pretty tightly packed.

 

~modest

[modest]

So, are we now saying there are roughly 17 trillion water molecules in a tiny floating steam droplet?

 

Yep. Craig solved in the other thread, and I rechecked the calculation:

 

Taking the 0.00001 m diameter above and the molecular mass of water (18 AMU = 3.0e-26 kg), we can calculate that the droplet has a volume of about 5.2e-16 m³, a mass of about 5.2e-13 kg, so consists of about 1.7e13 (170 trillion, as we say in the US) molecules.

 

He made a typo—it should be 17 trillion—but, yeah, it's true at least as an approximation.

 

~modest

[Mintaka]

Yep. Craig solved in the other thread, and I rechecked the calculation:

 

 

 

He made a typo—it should be 17 trillion—but, yeah, it's true at least as an approximation.

 

~modest

 

utterly mind-blowing. Who needs fantasy and fiction? This stuff is amazing.