Explaining the size of a water molecule

[Twodogs]

I am a longtime producer and host of a children's radio show on our local public radio station.
It's springtime and the rivers are running and I'm working up a little bit of shtick about water for the show. Water molecules are among the smallest, and one goal of the piece is to relate the size of a human body to that of a water molecule in a tangible and accessible fashion.

So we imagine an ant that is 1/2” (12.7 mm) long crawling on the leg of a person 5’6” (1676 mm) tall. Fortunately it is a friendly, vegetarian ant and doesn’t bite. Then we imagine that this ant has a much smaller ant crawling on its leg and, further, that this tiny critter has an even tinier critter on its leg and so forth until we get down to a extremely small critter that is the size of a water molecule.

The question is how many times we would need to repeat that recursive process until we reach the size of a molecule given that a water molecule is 2.75E-07 mm (0.000000275 mm) in diameter.

I have made numerous attempts at various calculations and come up with some bizarre answers. My latest equation has "x" appear as a power of a constant, but I am very rusty at dealing at solving that sort of thing.

First, I would very much appreciate some guidance in getting to the right answer.
Secondly, if the question as posed is not clear or could be better phrased, I would be glad to hear suggestions.

Thanks very much,
Don in Colorado

 

[mfb]

That is a big ant.
The ant to human ratio is 12.7/1676 = 0.00758. The ant on the ant (second level ant) is 12.7/1676 times the length of the ant, or 12.7/1676 * 12.7mm which is also equal to $\frac{12.7}{1676} \cdot \frac{12.7}{1676} \cdot 1676mm$ starting at the human. And so on. The third level ant then has a size of $\left( \frac{12.7}{1676} \right)^3 \cdot 1676mm$ The nth level ant has $\left( \frac{12.7}{1676} \right)^n \cdot 1676mm$. Setting that equal to the size of the water molecule: $\left( \frac{12.7}{1676} \right)^n \cdot 1676mm = 2.75\cdot10^{-07} mm$. While you can solve it step by step (divide by 1676 mm, take the logarithm on both sides, simplify), the children won't see these steps anyway. You can just plug it into an online calculator, telling you that n=4.6. The fourth level ant is too big, the fifth level ant is too small.

If you replace the 1676 mm human by a child, the fifth level ant becomes a better approximation (with a perfect match somewhere around 1050 mm). If you use a smaller ant, the fourth level ant becomes a better approximation (with a perfect match at 6 mm).

 

[Twodogs]

That is a big ant.
The ant to human ratio is 12.7/1676 = 0.00758. The ant on the ant (second level ant) is 12.7/1676 times the length of the ant, or 12.7/1676 * 12.7mm which is also equal to $\frac{12.7}{1676} \cdot \frac{12.7}{1676} \cdot 1676mm$ starting at the human. And so on. The third level ant then has a size of $\left( \frac{12.7}{1676} \right)^3 \cdot 1676mm$ The nth level ant has $\left( \frac{12.7}{1676} \right)^n \cdot 1676mm$. Setting that equal to the size of the water molecule: $\left( \frac{12.7}{1676} \right)^n \cdot 1676mm = 2.75\cdot10^{-07} mm$. While you can solve it step by step (divide by 1676 mm, take the logarithm on both sides, simplify), the children won't see these steps anyway. You can just plug it into an online calculator, telling you that n=4.6. The fourth level ant is too big, the fifth level ant is too small.

If you replace the 1676 mm human by a child, the fifth level ant becomes a better approximation (with a perfect match somewhere around 1050 mm). If you use a smaller ant, the fourth level ant becomes a better approximation (with a perfect match at 6 mm).

Ah, I very much appreciate your walking me through this. I could not get a handle on it. Regards.

 

[mfb]

By the way, if you want to make intermediate steps, with the original numbers:
The second level ant is as large as a human hair is wide. The third level ant is a bit smaller than a typical bacterium. The fourth level ant is a complex molecule, 20 times the size of the hydrogen molecule.

 

[Twodogs]

By the way, if you want to make intermediate steps, with the original numbers:
The second level ant is as large as a human hair is wide. The third level ant is a bit smaller than a typical bacterium. The fourth level ant is a complex molecule, 20 times the size of the hydrogen molecule.

Great, that is useful. And apparently ants range in size up to 52 mm or about 2 inches. That would be an encounter.