[Araconan]

Hi!

I am currently reading a section that proves the equality of pressure as described between the Kinetic Molecular Theory and the Ideal Gas Law. (Pages 157-161 of Zumdahl's Chemical Principles 6th Ed. to be more specific). There are a few sections that I don't understand. First, the explanation starts as follows:

Suppose there are n moles of an ideal gas in a cubical container with sides each of length L in meters. Assume each gas particle has a mass m and that it is in rapid, random, straight-line motion colliding with the walls. The collisions will be assumed to be elastic - no loss of kinetic energy occurs. We want to compute the force on the walls from the colliding gas particles and then, since pressure is force per unit area, to obtain an expression for the pressure of the gas. Before we can derive the expression for the pressure of a gas, we must first discuss some characteristics of velocity. Each particle in the gas has a particular velocity u, which can be broken into components u_x, u_y and u_z. First, using u_x and u_y and the Pythagorean theorem, we can obtain u_xy.
u_xy² = u_x² + u_y²

Then, constructing another triangle we find

u² = u_xy² + u_z² or
u² = u_x² + u_y² + u_z²

Now let's consider how an individual gas particle moves. For example, how often does this particle strike the two walls of the box that are perpendicular to the x axis? Note that only the x component of the velocity affects the particle's impacts on these two walls. The larger the x component of the velocity, the faster the particle travels between these two walls, thus producing more impacts per unit of time on these walls.

What I don't understand, is why the collision frequency on the two walls is only affected by the x component of the velocity. Wouldn't this only be true, if the particle was moving horizontally, with zero values for the y and z components of the velocity? So then wouldn't the y and z components also affect the collision frequency on the two walls (the two that are perpendicular to the x axis) since any value for y and z would cause the particle to move diagonally, and so would decrease the frequency of collisions on the two walls? (Since the particle is travelling more distance diagonally, than horizontally)

Thank you in advance!

[Jorriss]

Let me ask you, if you are only concerned with movement in the x-direction, why do you feel movement in other directions matters?

[Araconan]

Because I think that if there are movement in the other directions, will slow down the movement in the x direction. So like if there's movement in the y and z direction, then the particle will be travelling diagonally, and will be travelling more distance between the two walls, and hence colliding less.

[Jorriss]

Because I think that if there are movement in the other directions, will slow down the movement in the x direction. So like if there's movement in the y and z direction, then the particle will be travelling diagonally, and will be travelling more distance between the two walls, and hence colliding less.

Ahh, but in this case they are only considering the component of velocity in the x direction. Raising or lowering the velocity in the y and z directions won't affect our velocity in the x-direction.

Now, if on the other hand, we are told that u² = u_x² + u_y² + u_z² is a constant and the velocity in the y and z directions have changed THEN the x component may change, but this is not what we are considering here.

[Araconan]

Ohh so essentially the y and z values are held constant here, for the purpose of calculating the force exerted only by the x velocity. I see it now, thank you!