This is a juicy question that needs science, mathematics and understanding of the world around us.
You may also be interested in reading about the first half of this question in ''How many atoms are there in our world?''.
Molecules are different from atoms in that they are made up of atoms like tiny building blocks put together in precise orders and shapes. Most molecules are still extremely small.
Some molecules are made up of 2 or 3 atoms like oxygen that we breath and water that we drink. Some molecules can be thousands of atoms long like rubber in elastic bands and others can be trillions of atoms long like your DNA.
For this exploration of this question we will assume that our world consists of the solid planet we stand on and all the gaseous atmosphere that we breath.
It is understood from studying volcanoes that once rock melts from the heat of earth's core that molecules break down as the the order and shape of the rock crystals melts to form a liquid we know as lava. We also know from geologists that on average the earth is molten 60km below the surface and the top solid layer, know as the crust is mainly made from 2 molecules. These 2 molecules make up Silica and Alumina which are both types of sand and rock that you can find on beaches, deserts, mountains and at the bottom of the oceans. Maps show us that the surface area of the earth's crust is about half a billion square kilometres (500,000,000 sqkm). If it has a thickness of 60km then the total volume is thickness x area = 30 billion cubic kilometres (30,000,000,000cukm).
Geologists studying rocks from very deep mines and volcanoes erupting lava from inside the earth show that each cubic kilometre weighs about 2.7 million million tons (2,700,000,000,000). We call this its density and is also expressed at 3 grams per cubic centimeter.
volume x density = total weight of the earth's crust made of molecules
30,000,000,000cukm x 2,700,000,000,000tons/cukm = 81,000,000,000,000,000,000,000tons (eighty one billion billion tons)
Now we need to find out how many molecules are in eighty one billion billion tons of crust made up of silica and alumina.
First we need to know how much one molecule of silica and alumina weighs.
Scientific data tells us one Silica molecule weighs 0.0000000000000000000000000001ton and one Alumina molecule weighs 0.00000000000000000000000000017ton. Just to make our sums a little bit easier to work with we will make an educated guess that the average molecule weight found in the earth's crust is somewhere between the weight of silica and alumina. We will pick a figure of 0.000000000000000000000012ton.
We now see how many molecules would weigh the same as the total weight of earth's crust with a sum.
Total weight of earth's crust / average weight of molecule in crust = total number of molecules in crust
81,000,000,000,000,000,000tons / 0.00000000000000000000012tons = 675,000,000,000,000,000,000,000,000,000,000,000,000,000 molecules
The earth's crust contains six hundred and seventy five thousand, trillion, trillion, trillion molecules.
Now for the atmosphere. We know from astronauts and space probes that our atmosphere is about 100 kilometres thick to the point of where it meets space and it is 78% nitrogen and 21% oxygen gas molecules that make up the majority. It weighs about 5,000,000,000,000tons (five billion tons).
The average weight of nitrogen and oxygen molecules is 0.00000000000000000000005ton
Total weight of earth's atmosphere / average weight of atmospeheric gas molecule = total number of molecules in atmosphere
5,000,000,000,000tons /0.00000000000000000000005ton = 100,000,000,000,000,000,000,000,000,000,000,000
The earth's atmosphere contains one hundred billion, trillion, trillion gas molecules.
To find out how many molecules there are in the world we add the number found in the earth's crust to the number found in the earth's atmosphere.
675,000,000,000,000,000,000,000,000,000,000,000,000,000 + 100,000,000,000,000,000,000,000,000,000,000,000 = 675,000,100,000,000,000,000,000,000,000,000,000,000,000 molecules
Six hundred and seventy five duodecillion molecules.
You can see that whilst the atmosphere is very large and very very heavy with so many molecules it makes only a tiny difference when added to the number of molecules is added to the number of molecules in the earth's crust.
Thanks for your fantastic question!