# Relation between energy annd pressure

Me and and my friend were having discussion about the motion of molecules of gas.We talked about their velocity ,kinetic energy and much more. He asked me to derive a relation between pressure and energy. I was unable to explain him that... Can anyone explain the relation?

Bystander
Homework Helper
Gold Member
He asked me to derive a relation between pressure and energy.
How would you begin? What information do you have?

How would you begin? What information do you have?
P=density*Vx*2

T= 2/3k 1/2mv*2

BvU
Homework Helper
You could read up on the gas law, e.g. here
Your equations feature in there, but one appears molecular to me and the other macroscopic:

##{1\over 2} mv^2 = {3\over 2} kT## average kinetic energy for a molecule.

##P = \rho v_x^2 = {mass\over V} v_x^2 ## may be ok if in the right context.

I can wrangle a bit with formulas: $${1\over 2} mv^2 = {3\over 2} kT\ \Rightarrow\ {1\over 2} mv_x^2 = {1\over 2} kT$$ (the energy is equally distributed over the three degrees of freedom),

##\displaystyle \ \Rightarrow\ v_x^2 = {kT\over m} ##

mass = number of molecules * mass of a molecule = number of moles * ##N_A## * mass of a molecule (Avogadro number); write mass = ##n\; N_A\; m##

Leaves

##pV = n\;N_A\;m \ {kT\over m} = n\; N_A \;kT##;

then use

##R_G = N_A \; k## (gas constant) to get the ideal gas law

##pV = nR_GT##.

pressure times volume has the dimension of energy.

It's not the whole story, but quite a big part of it.