Equation for pressure of a 2D gas

[Chasing_Time]

Homework Statement

Hi all. This isn't quite a homework problem but I figured this is the best place to post it. I was working through the derivation of Pressure (P = (1/3)*n*m*v^2 bar) via kinetic theory in Sears + Salinger's Thermodynamics / Kinetic Theory / Statistical Thermodynamics textbook in chapter 9. I was curious to derive an equation of Pressure as force / length in a plane.

I am getting confused in the use of angles, since in two dimensions there is not a polar angle + azimuthal angle. While I want to integrate over 2 pi to account for the entire circle in the plane, it seems logical to integrate over 0 to pi/2 in consideration of both a surface containing all molecules of a given speed + direction and the trajectories in elastic collisions. I am sure some of my expressions are incorrect and maybe pointing these out or guiding their correction will be helpful.

Homework Equations

The basic idea is to derive an expression for flux, or the number of molecules with velocities in some direction determined by phi and with some speed v per unit length per unit time, then derive an expression for momentum, and then to combine these for an expression of pressure and integrate properly.

The Attempt at a Solution

Let N be the total number of molecules in a surface of area A. Imagining that there is a vector representing the magnitude and direction of velocity for each molecule and transfer all these vectors to a common origin so that they form a circle of arbitary radius with center at the origin. The average number of points per unit length is N/(2*pi*r). The average number of points in some element of length delta_L is delta_N = (N/ (2*pi*r))*delta_L. Delta_L can be expressed as r*delta_phi, so that N_phi = (N/(2*pi))*delta_phi is the number of molecules having velocities in a direction between phi and phi + delta_phi. Dividing this by area, we get delta n_phi = (n/(2*pi))*delta_phi. The area of some slant surface containing all molecules with direction phi and speed v is delta_A = delta_L*cos(phi)*v*delta_t. Then, using the expression for n_phi, we can define an N_phiv = (1/2pi)*v*delta_n_v*cos(phi)*delta_phi*delta_L*delta_t, which is the number of molecules with speed between v and v + delta_v, direction between phi and phi + delta_phi.

With this, we can define a flux as PHI = (delta_N_phiv)/(delta_L*delta_t) = (1/2pi)*v*delta_n_v*cos(phi)*delta_phi.

We can also define a delta_P_phiv = (1/2pi)*v*delta_n_v*cos(phi)*delta_phi*2mvcos(phi), where 2mvcos(phi) comes from conservation of momentum assuming elastic collisions with surface only. To integrate this expression and get pressure P, the integral over n and v will yield the square of the root mean square velocity, but I'm not sure of the integral over phi (what bounds? is this even the correct expression?).

I think my biggest uncertainty is in my definition of delta_A, or the area of a slant surface containing all molecules with speed v and direction phi, and subsequently the expression for momentum. I don't seem to be processing the geometry right. I'm sure this is something well known. Thanks for your time.

 

[anathema]

Thank you for sharing your thoughts and attempts to derive an equation for pressure as force/length in a plane. It is always exciting to see someone actively engaged in understanding and applying scientific concepts.

First, I would like to clarify that the equation for pressure that you have mentioned (P = (1/3)*n*m*v^2 bar) is the expression for pressure in a gas, specifically an ideal gas, as derived from the kinetic theory of gases. This equation is valid for a gas in a three-dimensional space, where the molecules have velocities in three dimensions. In your attempt, you are trying to derive an equation for pressure in a two-dimensional plane, which is not applicable to a gas.

In a two-dimensional plane, the concept of pressure as force/length does not apply because there is no volume for the gas to occupy and exert pressure on. In this case, the concept of pressure as force/area would be more appropriate. However, in a two-dimensional plane, the kinetic theory of gases does not apply, as the molecules do not have velocities in three dimensions. Instead, we would need to use the concept of surface tension to understand the behavior of the molecules in a two-dimensional plane.

Regarding your attempt to derive an equation for pressure in a two-dimensional plane, I would like to point out a few things. Firstly, your expression for delta_A, the area of a slant surface containing all molecules with speed v and direction phi, is incorrect. The correct expression would be delta_A = r*delta_phi*v*delta_t, where r is the radius of the circle formed by the molecules, and delta_t is the time interval over which the molecules are observed. This expression takes into account the fact that the molecules are moving along a circular path with a constant speed v.

Secondly, the expression for delta_n_phi, the number of molecules having velocities in a direction between phi and phi + delta_phi, is also incorrect. It should be delta_n_phi = (n/(2*pi*r))*delta_phi*delta_t, where r is the radius of the circle formed by the molecules, and delta_t is the time interval over which the molecules are observed. This expression takes into account the fact that the molecules are moving along a circular path with a constant speed v.

Lastly, the expression for delta_P_phiv, the change in momentum of the molecules with speed between v and v + delta_v and direction between phi and phi

 

[kerimek]

Hi there, thank you for sharing your work with us. It looks like you are on the right track in deriving an equation for pressure in a 2D gas using kinetic theory. However, there are a few things that need to be clarified and corrected in your attempt.

Firstly, the equation you have provided (P = (1/3)*n*m*v^2 bar) is actually for the pressure of a 3D gas, not a 2D gas. In 2D, the pressure equation is P = (1/2)*n*m*v^2.

Next, your approach for deriving an expression for flux is correct, but there are a few errors in your calculations. The number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N = (N/(2*pi*r))*delta_L, as you have correctly stated. However, the number of molecules in some element of length delta_L should be delta_N