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# Proof: U = (3/2)PV or U = (3/2)nRT

Conceptual proof that the internal energy of an ideal gas system is 3/2 PV. Created by Sal Khan.

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• Why do we count the number of particles striking a wall as 1/3 instead of 1/6 (the six faces of a cube)? • i'm still confused, why do we have to divide the particles by 3? :s • Daniel, the particles are divided by 3 because you have 1/3 of the particles going in one direction (up and down), 1/3 of the particles going in another direction (side to side), and 1/3 of the particles going in another direction (forward or backward). For example, suppose I have 9 particles in the cube; that means N = 9. That means that 3 are going up and down, 3 are going to the sides, and 3 are going forward and backward. So, if I asked you to talk about the pressure in one direction, you would talk about only 3 of the particles, which is N/3 (=9/3=3).
Hopefully, that helps!
• Even though people have tried explaining the time being 2x / v, it is still confusing for me.

Force is defined as follows: "If a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ P = F * t". So if the time, t, is how long the force was applied for, then why include the time that the particle is not applying a force to the wall? Which is what is done when you include the time taken to travel 2x. It is not applying a force to the wall during that time.

Let's make a macroscopic example of this situation. An elastic ball is rolling from one wall in a buiding to another with mass, m, and velocity, v. It rolls back and forth one time by 'bouncing' off the wall elastically as it reaches the wall on the other side. The time of the bounce is 1 second, in other words the time the ball has to apply a force to the wall and have its momentum changed. It takes 10 second for the ball to reach the wall and come back to its initial position.

Calculating the force applied to the wall as shown in the video means F = m* delta v / delta t, where delta t is 10 seconds (the time it takes the ball to go to and from the wall), and delta v is 2v because the difference is the velocity vector direction has changed so you get delta v = v - (-v) = 2v. So F = m * 2v / 10 = m*v/5

If I calculate it according to how I originally understand how force equals the change in momentum I get the following:

F = m * delta p / delta t, where delta t is the 1 second the ball is in contact with the wall during the 'bounce' and delta p is the same as above: 2v. We get F = m * 2v / 1 = 2*mv

Clearly the method shown in the video gives a much smaller force than when considering time as only the time when the object is applying the force to the wall. What am I misunderstanding here? • In the video, it is safe to assume that the force is applied over a time period of 2x/v, because within that time period the particle hits each wall once. Imagine with your bouncy ball, you had it bouncing between 2 buildings. It would hit both buildings every 10 seconds, so it would apply the force of those two collisions every ten seconds. This may seem confusing on a small scale, but on a large scale with many particles, that averages out to about the same number of collisions per second, which is really what we're trying to calculate with this.
• U=(3/2)PV is ONLY true for an IDEAL gas. Is this correct? • At it's said that all of the internal energy of the system is in kinetic energy because it's an ideal monoatomic gas. Even in a monoatomic gas, such as He, every atom has potential and kinetic energy "stored" within it due to the electrons moving about the nucleus. Why isn't this energy also included as part of the internal energy? • At shouldn't the change in momentum for the molecule that got ricochet off the wall be minus 2mv (because change in momentum for the molecule= final momentum - initial momentum that is equal to minus 2mv). Did Sal considered momentum imparted to the wall that is equal to plus 2mv? • Yes u are right Sal considered that momentum imparted to wall by particle.when we calculate force as rate of change of momentum , we take force exerted by molecule on the wall not the force exerted by wall on the molecule. force exerted by wall on the molecule would be -2mv/t but force exerted by molecule on the wall will be 2mv/t because they are action reaction forces and will be equal in magnitude but opposite in direction.
• I thought Total Thermal Energy---not Internal Energy, U---is the "Kinetic energy of all particles."
Or is the "Total Thermal Energy" the same as "Internal Energy," U, for Ideal gases, since there are no potential energies of attraction between gaseous species? • isn't the change in momentum equal to -2mv? my physics teacher told me that so i'm a little bit confused right now.   