What's keeping us from simply redefining Avogadro's Number / the Mole as a definite integer? This might be a question to ask in a Chemistry site, but because there is a lot of talk about redefining many units of measurements in terms of Avogadro's Number / the Mole, I was wondering why we don't just redefine the Mole to infinite precision, since it is basically inherently an integer.
This might be the only unit/physical constant that can be defined to infinite precision.  The only unit that is an "integer".
I guess it's not really a physical constant, per se, in that it is not a property of nature.  But then we can easily create a definition of the Kilogram that doesn't change over time as exactly something-something moles worth of Carbon-12.
 A: Basically, you are proposing to redefine the kilogram, and your approach has been proposed and recently (in october 2010) abandoned ( http://en.wikipedia.org/wiki/Kilogram#Carbon-12 ). 
I think the reason why the Watt-balance approach has been preferred for the future definition of the kilogramme was mainly technological : it is more precise and would allow more practical realization of the kilogramme.
A: Basically, there's no reason why we couldn't redefine the mole as as simple integer number of atoms or molecules. In fact, as other users have mentioned, there's a lot of people who'd like to do that.
On the other hand, that's not how chemists actually use the mole in practice. You simply can't count 6×10^23 atoms or molecules, nor do you need to. What is important for chemists is to know (for example) that there are the same number of atoms in 58 grams of iron as in 12 grams of carbon, and so on for all the other elements. It's not important to know exactly what that number is, just that it's the same number, and, for much of the history of chemistry, we had absolutely no idea what the number was.
I should point out as well that the mole is not the only unit which is an "integer". If you take the coulomb as a unit of electric charge, that should be equal to an integer number of elementary charges, shouldn't it? Actually, it's not, for historical reasons, and there doesn't seem to be much enthusiasm for making it an integer number of elementary charges either.
You can find my paper discussing this in more technical terms at http://precedings.nature.com/documents/5138/version/1
A: There was a proposal in 2006 trying to define NA as an exact number[1,2]:
$$ N_A^* = 84\;446\;888^3 = 6.022\;141\;410\;704\;090\;840\;990\;72 \times 10^{23} $$
the problem? This value is incorrect, as the currently most accurate result is[3] 
$$ N_A = 6.022\;140\;84(18) \times 10^{23} $$
i.e. $N_A^*$ is now 3 s.d. away from $N_A$. As I have commented, if we randomly pick a number within the current error bound and call it $N_A$, we risk the problem that a better experiment for the old definition will invalidate the proposed value. To be safe about the validity of that number, you need to produce an equally accurate experiment to show that it is actually valid (like the 299792458 m/s in the definition of meter, and 9192631770 Hz in the definition of second.)
Also, the rationale for redefinition of SI base unit always involve that the current one isn't accurate enough or hard to realize: 


*

*second (1967): 


*

*the definition ... is inadequate for the present needs of metrology


*meter (1960): 


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*the international Prototype does not define the metre with an accuracy adequate for the present needs of metrology, 

*it is moreover desirable to adopt a natural and indestructible standard,


*meter (1983):


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*the present definition does not allow a sufficiently precise realization of the metre for all requirements


*candela (1979):


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*the time has come to give the candela a definition that will allow an improvement in both the ease of realization and the precision of photometric standards, ...



Do the current experiments which reduces to NA = 12 gram of carbon-12 atoms not accurate enough or hard to realize? I don't think so; 9 significant figures are already very accurate. However, the redefinition of mole would be put on board on 2011 (24th CGPM). One proposal is to define[4]
$$ N_A \overset{\underset{\mathrm{def}}{}}{=} 6.022\;141\;5 \times 10^{23} \mathrm{mol}^{-1}, $$ to decouple kilogram from the definition of mole. So if this path is taken, the only thing that keep us from defining it as a definite number to 10 significant figures is because "the conference haven't started yet".
But infinite precision? That would be a long way before we can reach and need that.
Ref:


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*Ronald Fox and Theodore Hill, A Proposed Exact Integer Value for Avogadro's Number. http://arxiv.org/abs/physics/0612087

*Ronald Fox and Theodore Hill, An Exact Value for Avogadro's Number. http://www.americanscientist.org/issues/pub/2007/2/an-exact-value-for-avogadros-number/3

*B. Andreas, Y. Azuma, G. Bartl, et. al., An accurate determination of the Avogadro constant by counting the atoms in a 28Si crystal http://arxiv.org/abs/1010.2317

*Ian M Mills, Peter J Mohr, Terry J Quinn, et. al., Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI-2005). http://iopscience.iop.org/0026-1394/43/3/006
A: The problem is that you want your unit definitions to be realizable - so specifying "1 mol is long number molecules, 1 gram is 1/12 of the mass of one mol of $C_{12}$" is nice for your thought process, but as long as there is no practical way to count molecules at such scales to a precision of better than $10^{-9}$ (which I think is the precision of the kilogram standard), there is no operational advantage to it.
