Ideal Gas Equation: Deriving pV=Nm<c>^2/3

[PFuser1232]

I need a detailed derivation of why pV = $\frac{Nm<c>^{2}}{3}$for an ideal gas using the example of a gas molecule placed in a cube-shaped container, the derivation in my book isn't that clear.$^{}$

 

[Borek]

Have you tried to google for "derivation of ideal gas equation"?

 

[jtbell]

Or if that doesn't help, maybe you could tell us about the derivation in your book (or better yet, give us a link to it if you find it somewhere online) and describe to us what you find unclear about it. Then maybe someone can get you un-stuck.

 

[klimatos]

Try:

en.wikipedia.org/wiki/Kinetic_theory

The shape of the container is irrelevant, but I think you need more than one molecule.

 

[PFuser1232]

Try:

en.wikipedia.org/wiki/Kinetic_theory

The shape of the container is irrelevant, but I think you need more than one molecule.

So, is the equation valid for any container of any shape?

 

[Borek]

Yes. It is easier to derive for a simple container, but turns out it works for every container.

Note that you can always select an imaginary cube INSIDE gas, and assume - as everything around is identical to the gas inside the selected volume - that for every molecule leaving the imaginary cube, one identical molecule enters the cube. This is equivalent to the molecules bouncing on the cube walls, which is part of assumptions required when deriving the equation. And as every dV inside of the volume of gas behaves the same way, equation holds also for whole V.

 

[klimatos]

So, is the equation valid for any container of any shape?

Yes, although your original post uses V which I assume is the volume of one mole at the specified pressuire and N which I assume is Avogadro's number. To apply to any size container, that specification has to be changed.

Since N/V = n, where n is the number density of molecules per unit volume, the equation becomes
P = nm<c>²/3. This would apply to any shape and size of macroscopic container.

It would help a lot if you would define the terms used in the OP. Not all authors use the same common terms in the same way. For instance, the term c is often used to designate the speed of light in a vacuum, and p is often used to designate momentum.

 

[PFuser1232]

Yes, although your original post uses V which I assume is the volume of one mole at the specified pressuire and N which I assume is Avogadro's number. To apply to any size container, that specification has to be changed.

Since N/V = n, where n is the number density of molecules per unit volume, the equation becomes
P = nm<c>²/3. This would apply to any shape and size of macroscopic container.

It would help a lot if you would define the terms used in the OP. Not all authors use the same common terms in the same way. For instance, the term c is often used to designate the speed of light in a vacuum, and p is often used to designate momentum.

Actually N is the number of molecules, N_A is avogadro's constant, p is pressure, n is the amount in moles, and V is the total volume of the container.