[Chondrally (OP)]

The old accepted number for Avogadros theory is 6.023x10(23) particles per Mol.
Possible more accurate number:
the length of a water molecule is 0.27 x10(-9) metres
The diameter of a water molecule is 0.275x10^(-9) metres
10^(-9) is almost at the limit of space dimensions in spacetime because of the uncertainty principle and was set at CERN in particle physics experiments.
1 Mol of water is 18ml or 18 grams
18x10^(-6) cubic metres/(length of water molecule)/(area of water molecule) cylinder approximation
18x10^(-6) m^3/(.27x10(-9)m)/(.275x10(-9)m/2)^2/pi=1.122x10(24)  particles/Mol .  I guess Avogadros number of 6.023x10^23 still stands if it were different it would affect pharmaceutical calculations everywehere chemical reactions and thermodynamic calculations..
I realize that modelling of water at the atomic level is a super complicated method and may involve quantum mechanics and statistical mechanics and hydrogen bonded networks,
so what would be a good approximation of the volume of a water molecule on average?  Does anybody know at Cambridge?
I would like to apologize for my error earlier and appreciate the post following for making the correction. 
What is the paper where Avogadros number is first calculated and when was it done?

[Kryptid]

There is so much wrong with this...

10^(-9) is almost at the limit of space dimensions in spacetime...

It's nowhere close, actually. The proton charge radius is about 8.77 x 10^-16 meters, for example. The upper limit on the electron's radius is on the order of 10^-22 meters.

One of the problems with your calculations is that you seem to assume that a particular volume of water is completely filled with water molecules. You need to make some allowance for the empty space between those molecules as well. There is also the complicating factor that water is not composed completely of H₂O molecules, but also contains hydronium, hydroxide and more complicated ions (even free protons may have some fleeting existence as they move between molecules). Another problem is that your equation for figuring out the volume of a cylinder is wrong. It's not length divided by area. It's the length of the cylinder multiplied by the radius of the cylinder squared multiplied by pi.

Ignoring the fact that water molecules aren't cylinders (or even an approximation of a cylinder), that would yield a volume of pi x (1.375 x 10^-10 meters)² x (2.5 x 10^-10 meters) = 1.48 x 10^-29 cubic meters. Then (1.8 x 10^-5 cubic meters)/(1.48 x 10^-29 cubic meters) = 1.216 x 10²⁴ molecules. This is significantly closer to the standard value of a mole than yours. However, as I said before, you also need to consider that there is some amount of empty space between the water molecules. For that reason, you cannot use this method to calculate the value of a mole.

[Bored chemist]

1 Mol of water is 18ml or 18 grams

At what temperature?

the length of a water molecule is 0.27 x10(-9) metres
The diameter of a water molecule is 0.275x10^(-9) metres

At what temperature?


All of this misses the point.
Avogadro's constant is DEFINED as  N_A = 6.02214076×10²³ mol⁻¹

You can't measure it meaningfully.
If you get a different answer, you are wrong.

It's also instructive to learn that you can't "create" accuracy by doing arithmetic.If you start with measurements of the size or molar mass of a molecule that are only accurate to 2 significant figures, you can't calculate a value that is accurate to better than 2 significant figures.