since you cannot count individual molecules how is the root mean square energy defined? How is is derived?

Question

or proven to be true?

Answer 1

It's a statistical quantity of microscopic particles, which is effectively what you measure when you measure macroscopic temperature.

If you want to understand all the details of how macroscopic temperature and pressure and chemical potential and all that are derived from microscopic QM properties, major in physics and you will have to take statistical thermodynamics and you'll learn all about it.

Answer 2

You can do an excellent estimation of the number of Molecules, or atoms for that matter, using Avogadro's Number and applying the formula for it based upon the weight and chemical composition of any sample.

Answer 3

RMS is an average of the energy of molecules. It's measure by temperature or pressure.

Answer 4

The root-mean-square energy Erms
is given by:

K2rms=1N/V∞0K2n(K)

dK=2π(πkBT)2/3∞0K5/2e−K/kB