Let c represent the rms speed of molecules.(Click to see animation)
KE = 1/2 m(square of c)
= 3/2kT
Starting from this single molecule 's kinetic energy relationship with temperature , you can derive the other relationship easily. Note :m is mass of a single molecule, if Nm =M (mass of the gas )
Internal energy U = 3/2 NkT ( which is kinetic energy of all the molecules )
This is only true for ideal gas , as for real gas Internal energy consists of sum of kinetic energy and potential energy of all the molecules.
If you relate to ideal gas law , pV= nRT which can also be written as pV= NkT .
The internal energy of the container of gas is also U=3/2 nRT or U= 3/2 pV.
This formula U=3/2 NkT helps you to understand why first law of thermodynamics uses this fact that increase in internal energy is proportional to the increase in temperature.
KE = 3/2 kT or 1/2 m(square of c)
Apply this formula:
If a container has different types of gas , which also mean they have the same temperture since the molecules exist together. They all have the same KE , therefore a molcules with twice the mass will only have c/(square root 2).
If a container having the same type of molecules , has its temperature raise to twice its initial temperature , then its speed would increase to (square root 2) c .
It is advised that you use the equation and write it twice, one for each case mentioned. Avoid using proportionality and memorising it. This will prevent overlooking some important. information.
