Notes for this calculation
1 The calculation is made under "Prove it" of 1024.
2 The exact composition of much of the earth is not known with certainty. So it is impossible to determine the value of the average atomic weight of the chemicals found in the earth. A reasonable estimate of this value however can be made.
Most scientists believe that the central core of the earth, 4300 miles thick, is composed mainly of iron. The atomic weight of iron is about 56.
The upper portion of the earth's solid crust, to a depth of a few miles, is composed mainly of a few elements. About 47% of this crust is oxygen, 27% silicon, 8% aluminum, and 5% iron. Four more elements account for the next 11.3% of the crust. They are calcium, magnesium, potassium and sodium. The remaining portion of the crust, i.e. about 1.7% is comprised of 80 other elements.
This does not include the stuff of the oceans, which consists mainly of one part oxygen to every two parts hydrogen, and also many other elements in much smaller amounts found in solution.
It also does not take into account the atmosphere, which is mainly composed of nitrogen (close to 80%) and oxygen (much of the remaining 20%), plus a few other elements in minute amounts. Thus, at the earth's surface, the element that is most prevalent is oxygen, whose atomic weight is 16.
For purposes of the above calculation, I have supposed that the average atomic weight of all earth stuff is somewhere between the atomic weight of oxygen (16) and that of iron (56). I set it at 40.
If the figure is actually higher, there would be less atoms than what we is computed above. If the actual figure is lower, there will be more atoms than what is computed above.
It is extremely unlikely that the total number of atoms in the earth is higher than 1051 or lower than 1048.
If the average atomic weight is 5 times higher, 200, much higher than the atomic weight of any element found in significant measure at the earth's surface, the total number of atoms would still be one fifth the number given above, or about 1.7 x 1049.
Conversely, if the average atomic number is much lower, e.g. 4, the atomic weight of helium, the total atoms would only be ten times as great, still not quite up to 1051.
5 Note that the number we obtain for the total atoms on earth is only about 2000 times the number of molecules of water on earth. This is not at all farfetched.
The atomic weight we arbitrarily chose for the average atomic weight of all earth stuff is twice as much as the molecular weight of water. Remember also that the volume of the entire earth is only about 760 times the volume of all the water. Indeed, if our calculation of atoms would be based purely on volume, the total number of atoms on earth would be only about 375 times the number of water molecules.
However, here we have done the calculation the proper way, by using weight as the measure. Although the planet as a whole occupies only about 760 times as much space as its water, the weight of the earth is a much greater multiple than the weight of the water. The stuff within the earth is extremely compressed. Hence, the total weight of the earth is in fact about 4000 times the total weight of the water.
When we use an atomic weight of 40, which is more than double the molecular weight of water, the total number of atoms in the earth turn out to be about 2000 times the number of molecules in the water.
6 For an explanation of how to multiply big numbers written in scientific notation, see this topic under "About Big Numbers."