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```"Bradd W. Szonye" <bradd+srfi@xxxxxxxxxx> writes:

>> What makes you think Avogadro's number is an integer?
>
> Because there's no such thing as half an atom of carbon. Furthermore, as
> you mentioned yourself, the error in measuring the canonical gram is
> much larger than the mass of an atom, so why /wouldn't/ it be an
> integer? Indeed, if we were to switch the gram from a sample to a more
> fundamental definition, I would expect something like "1/12 the mass of
> one mole of carbon-12," with the mole fixed at an integral number of
> atoms.

You have to pick one of the following:

1) There *is no* precise definition of the kilogram, since it's
defined by an artifact which changes over time.

2) There will some day be a precise definition of the kilogram, and
that precise definition will assign Avogradro's number an integral
value.

3) There isn't a commonly agreed-upon precise definition of the
kilogram, but we can invent one by either of the following two
means:
3a) The exact value is the time average of the mass of the standard
kilogram.
3b) The exact value is the mass of the standard kilogram at a
uniquely defined moment in time.

If (1), then the statement that Avogadro's number is an integer makes
no sense.

Either variation of (3) clearly shows that there is no reason to
expect an integral number of atoms in a kilogram.

Number (2) is more interesting.

To make this work, we would need to standardize the kilogram as being
N atoms of some substance.  We would need a technological means of
counting atoms, which would yield measurements more precise than
gravitational balances using the standard kilogram (the current
method).  We would fix N by reference to the existing standard
kilogram, and we would choose N to be an integer.

This presumed atom-counting technology presumably works for some
substances better than others.  At the time that the mole was defined,
carbon-12 was picked because counting atoms roughly was best done with
carbon-12.  But the technology necessary to count atoms in this
ultra-precise scenario would surely need to be very different from the
methods currently used to measure the atomic mass unit.

So there is no reason to think that carbon-12 will be the substance
used.  (For example, it seems very likely that we would need to use a
substance which is very nonreactive.)  It is therefore extremely
likely that some *other* substance will be used, not carbon-12.  So
the kilogram might then be defined as N atoms of, say, xenon-132.  But
the mass of xenon-132 and the mass of carbon-12 are not integral with
respect to each other (xenon-132 weighs about 131.9041535 u, and
carbon-12 weighs exactly 12.

Thomas

```