Answer 1:
It's hard to have a definite answer because of the
unclear definition of "a square inch of the
world". Nevertheless, the exact number is going
to be a really really big one.
To talk about the number of atoms, it is
necessary to introduce the Avogadro's
number, which is defined as the number of
units (atoms, molecules, etc..) in one mole of any
substance.
Here the Avogadro's number is measured to be
around 6.02 times 10^{23}, and mole is
related to Avogadro's number and the mass or
density of the units.
To give an example, one mole of water is
referred as one Avogadro number of water
molecules, or 18.015 grams of water (the atomic
weight of a water molecule is 18.015 gram per
mole). Here the message is 18.015 grams of
water or 18.015 cm^{3} of water (water
density is 1 gram per cm^{3}) contains one
Avogadro number of water molecules (Or three
times of the Avogadro number of atoms since each
water molecule is formed by two H and one O
atoms).
The density of any substance on Earth can be
quite different. The density of water is 1000
kg/m^{3}, while the density of air is only
1.2 kg/m^{3}. The density of gas hydrogen
is even less, 0.082 kg/m^{3}.
Nevertheless, one m^{3} of any
substance on Earth will contain a lot of atoms,
since the Avogadro's number is huge.
OK, back to the question now. One square inch
of the world, it is not sure how deep or how broad
in space. To talk about of the total number of
atoms, the volume should be correctly specified.
As an example, I can give a rough estimate of a
square inch of water (one square inch = 6.4516
cm^{2}) with different depths:
1) If it is one meter deep, the volume should be
6.4516 cm^{2} times one meter = 645.16
cm^{3}.
This is equivalent to 645.16 grams of water or
645.16/18.015 = 35.81 mole of water.
In terms of atom number, it should be
35.81 * 3 * 6.02*10^{23} =
6.47*10^{25}
Here we multiply by 3 because each water molecule
contains two atoms of hydrogen H and one atom of
oxygen O. The number 6.02 times 10^{23} is
Avogadro's number.
2) If it is only one inch deep, the total number
should be less because of the lesser volume. The
total number should be scaled by the ratio of one
inch to one meter:
6.47*10^{25} * (one inch/one meter)=
6.47*10^{25} * 0.0254 = 1.64*10^{24}
3) If the depth is equal to the deepest ocean
depth, 10994 meters, then the total number of
atoms is about
6.47*10^{25} * 10994 = 7.11 * 10^{29}
To conclude, the atom number is a huge number
in any reasonable amount of substance on
Earth. The Avogadro number is the
order of atom number that you should refer to. The
density can vary from one substance to
another, but such difference is small
compare to the scale of one Avogadro number.
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