This post was written without looking up anything.
I am a physicist, and this makes me think about the world we live in in those terms. For example, I think about the weather, and the role moisture plays in it (it’s a big one). So, how much moisture do you have when the relative humidity is 100%? “100%” does not mean you are under water. It means that the air is saturated with moisture, and any additional moisture will condensate and form clouds or fog.
Long ago, I looked up and remembered, that at 10, 20, and 30°C, one cubic metre can contain at most about 10, 20, and 30g of water (at the usual atmospheric pressures on the earth’s surface.) How much is that compared to the mass of the air in that volume? Air is light, but it’s not nothing. I have not memorised the densities of all the major gases. I do have memorised a very useful general property: The volume of one mole of an ideal gas at room temperature is about 23 litres. How does this help? And why do I know this? Second, the concept of moles and ideal gases is fundamental, which is why I remember it, and the number 23 is easy enough, too. First, we can calculate the mass of the air in a certain volume.
The way the mole is defined, and knowing the atomic weight numbers of oxygen and nitrogen (16 and 14, which are things a physicist could know), we find that 23 litres of air should be about 29 grammes. To get from 23 litres to 1 cubic metre, we need about a factor of 44, so 29g*44 = 1276g, so about 1.3 kilogrammes of air in a cubic metre. Which is a lot more than 30g of moisture.
This means that on a humid day, the air only SEEMS thicker than usual.
Alternatively: 44 mol per cubic metre means that each cubic metre of air contains about 44*600000000000000000000000 = 26400000000000000000000000 air molecules. The large number is the approximate value of Avogadro’s constant, which gives the number of atoms or molecules in a mole. If you knew the masses of oxygen and nitrogen in grammes per atom, you could also get the total mass, but it would be less elegant and involve a very large and a very small number. If you do not know them, like I don’t, you could remember the electron mass (about 1/1000000000000000000000000000000g, easy number), the fact that nucleons are about 1800 times heavier, and arrive at about 1.4kg per cubic metre. This is similar enough to the above estimate.
And, now that we know this, how large a helium ballon do you need to float? Helium is 4⁄29 of the density of air, so you get about 1.3*25⁄29 kilogrammes or about 1.1kg of “lift”. To be precise, the lift would be the force to lift that kind of mass against the force of gravity. Assuming 100kg for an adult plus clothes plus balloon, 100/(1.1) is about 100*0.9 or 90 cubic metres of helium to achieve liftoff. The volume of a sphere is about 4 times the radius cubed; 3*3*3 is 27, so you need a balloon with a diameter of 6 metres.
What about hydrogen? Its atomic weight is only 1⁄4 of that of helium, so you need only one quarter the volume? No. The relative lift compared to helium is only (29-1)/29 instead of 25⁄29. An extra 12% that is not usually worth the extra fire risk.
I hope you liked this tour of one physicists mind.