Find Equation for Ideal Gas in Cube Container

[jamie_o]

Hello, I am having some difficulty following the method for finding an equation for an ideal gas. There are a few different forms, but I'm proving 1. For an indiviual particle of a gas in a cube container side length L, it is traveling with a velocity of u1 on the x-axis (its x component of velocity is u1). This collides with one surface and rebounds elastically. This would have a velocity -u1 as no kinetic energy was lost. I want to find the force this particle exerts on the wall by its x component. So to find Force i would use rate of change of momentum. the distance before it collides with the surface is 2L. so time = 2L/u1.

so for my equation I now have force = change of momentum
2L/u1

change in momentum equals mv - mu. according to the principal of conservation of momentum would i be correct in saying mv - mu = 0 ?? (v = final velocity, u = initial)

if so, -mu1 - mu1 is the change in momentum -2mu1. Now i was told the change in momentum should be equal to 2mu1. Which leaves me with a problem. Does -2mu1 = 0 ? if so can i easily get 2mu1 by adding it to both sides? which gives me a positive change in momentum. Seems odd to me, can someone please explain that and if that would be the correct way of doing it? I have no problems getting the P = 1/3 rho (mean velocity squared) equation from that, but the change in momentum is bothering me. I could just remember 2mu1, but I would rather know how to work it out properly. Any help is very much appreciated, thanks

 

[Diane_]

Another term for "change in momentum" is "impulse". You can get the impulse by taking the product of the force exerted and the time it is exerted. Now you need to find a way to get rid of the time: consider that the pressure is the force exerted over an area, and this will depend on the force exerted by each particle (which you're working on), but also on the total number of particles that strike that area in a given time.

Is that enough?

 

[wais]

Hi there,

Finding equations for ideal gas behavior can be tricky, so I completely understand your difficulty. Let's break down the steps to finding the equation for an ideal gas in a cube container:

1. Start with the definition of force: force = rate of change of momentum. In your case, the force is the force exerted by the particle on the wall, and the rate of change of momentum is the change in momentum divided by the time it takes for the particle to collide with the wall.

2. Next, we need to find the change in momentum. Remember, momentum is defined as mass times velocity. In this case, the mass of the particle remains constant, so we just need to focus on the change in velocity. The initial velocity is u1, and the final velocity is -u1 (since the particle rebounds with the same velocity in the opposite direction). So the change in velocity is -u1 - u1 = -2u1.

3. Now we need to find the time it takes for the particle to collide with the wall. You correctly stated that the distance before collision is 2L, and the velocity is u1. So we can use the formula time = distance/velocity, which gives us a time of 2L/u1.

4. Putting it all together, we have force = (-2u1)/ (2L/u1) = -2mu1^2/L. This is the force exerted by one particle on the wall.

5. To find the total force exerted by all the particles in the cube container, we need to sum up the forces exerted by each individual particle. Since there are N particles in the container, the total force would be N times the force exerted by one particle. So the final equation would be F = -2Nmu1^2/L.

I hope this explanation helps clarify the process for finding the equation for an ideal gas in a cube container. Remember, the change in momentum is negative because the particle is rebounding in the opposite direction, and we need to consider the total number of particles in the container to find the total force. Let me know if you have any further questions. Good luck!