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Equation for pressure of a 2D gas

  1. Nov 4, 2009 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant 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.

    3. 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.
  2. jcsd
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