Relation between energy annd pressure

[Haseeb Ali]

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]

He asked me to derive a relation between pressure and energy.

How would you begin? What information do you have?

 

[Haseeb Ali]

How would you begin? What information do you have?

P=density*Vx*2

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

 

[BvU]

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.