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Pressure for ideal gas in terms of stat.

  1. Jan 8, 2009 #1


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    I am trying to deduce the expression for pressure of perfect gas when the momentum distribution [tex]n(p)[/tex] is given.

    Here is how I did. First we assume a box with side length [tex]L_x, L_y, L_z[/tex], when a particle , say moving a long x direction, collide with one side of the wall, the total change of momentum would be

    [tex]\Delta p_x = -2m v_x[/tex]

    Assume it takes time t for one round-trip (from one wall to the oppsite and come back), hence

    [tex]t = \frac{2L_x}{v_x}[/tex]

    and 1/t is the rate of colliding.

    Now consider the average impact per unit time,

    [tex]\overline{f} = \Delta p_x \times (\textnormal{rate of colliding}) = 2mv_x \frac{v_x}{2L_x} = \frac{mv_x^2}{L_x}[/tex]

    For N particles, the total average impact per unit time would be

    [tex]\overline{F} = \sum_i^Nf_i = \sum_i^N \frac{mv_{ix}^2}{L_x}[/tex]

    Hence, the pressure on the side [tex]A=L_yL_z[/tex] woule be

    [tex]P = \frac{\overline{F}}{A} = \frac{\overline{F}}{L_yL_z}[/tex]

    For continuous case, the average impact becomes

    [tex]\overline{F} = \int \frac{pvn(p)}{L_x}dp[/tex]

    So, the pressure becomes

    [tex]P = \frac{\overline{F}}{A} = \int \frac{pvn(p)}{L_xL_yL_z}dp[/tex]

    In unit volume, [tex]L_xL_yL_z=1[/tex], wehave

    [tex]P = \frac{\overline{F}}{A} = \int pvn(p)dp[/tex]

    I know there is something wrong here. The correct answer should be

    [tex]P = \frac{1}{3}\int pvn(p)dp[/tex]

    Well, I don't know where my reasoning is going wrong. From [tex]\overline{F} = \sum_i^N \frac{mv_{ix}^2}{L_x}[/tex] to [tex]\overline{F} = \int \frac{pvn(p)}{L_x}dp[/tex], I feel that there is something missing?
  2. jcsd
  3. Jan 13, 2009 #2
    Because when you say N particles with the side L_yL_z, you say that all of the particles move in the x direction. Whereas in average only one third of the particles move in the x direction.
    So you should divide the average rate of impact by three for all three directions...
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