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bjnartowt
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Homework Statement
If you poke a hole in a container full of gas: the gas will start leaking out. In this problem, you will make a rough estimate of the rate at which gas escapes through a hole: effusion. (This assumes the hole is sufficiently small).
Consider such a hole of area "A". The molecules that would have collided with it will instead escape through the hole.
Assume that nothing enters through the hole. Then: show that the number of molecules “N”, is governed by:
\frac{{dN}}{{dt}} = - \frac{A}{{2V}}\sqrt {\frac{{kT}}{m}} N
Homework Equations
2L = \Delta t \cdot \overline {{v_x}} (round trip time for collision, but the factor of 2 coming from considering the walls of collision in just one dimension)
{V = L \cdot A} (volume = length times area)
{PV = N{k_B}T} (ideal gas law)
\overline {{v_x}} \approx \sqrt {\overline {{v_x}^2} } (root-mean-square/mean-velocity approximation)
\sqrt {\frac{{kT}}{m}} = \sqrt {\overline {{v_x}^2} } (I derived this result and know it to be true: it's from assuming (1/2)*mv^2 = (1/2)*kT: that is, thermal/kinetic energy equality in one dimension)
The Attempt at a Solution
work backwards: start reading from bottom up...sorry:
\begin{array}{l}<br /> \frac{{dN}}{{dt}} = - \frac{{{V^2}}}{{N \cdot \Delta t}} \\ <br /> = {\left. { - \frac{V}{{\Delta t \cdot \overline {{v_x}} }}\overline {{v_x}} \frac{{{k_B}T}}{P}} \right|_{2L = \Delta t \cdot \overline {{v_x}} }} \\ <br /> = {\left. { - \frac{{AL}}{{2L}}\overline {{v_x}} \frac{{{k_B}T}}{P}} \right|_{V = L \cdot A}} \\ <br /> = {\left. { - \frac{A}{2}\overline {{v_x}} \frac{{{k_B}T}}{P}} \right|_{PV = N{k_B}T}} \\ <br /> = {\left. { - \frac{A}{{2V}}\overline {{v_x}} N} \right|_{\overline {{v_x}} \approx \sqrt {\overline {{v_x}^2} } }} \\ <br /> = {\left. { - \frac{A}{{2V}}\sqrt {\overline {{v_x}^2} } N} \right|_{\sqrt {\frac{{kT}}{m}} = \sqrt {\overline {{v_x}^2} } }} \\ <br /> \frac{{dN}}{{dt}} = - \frac{A}{{2V}}\sqrt {\frac{{kT}}{m}} N \\ <br /> \end{array}
Well…hard to say what I wanted d/dt to look like, so it's no wonder this just looks like algebraic junk. Well … I know A/V has units of inverse-length, and should be propotional to the volume of escaping air…
Ansatz: the numer of particles striking the area, “A”, is a fraction of the total area, which is V/L, where “L” is some length of “gas” perpendicular to the area “A” related to the velocity.
Is that a good ansatz? I'm looking for a "given" to start the derivation of this differential equation with. I'm sure I could make the quantities I wanted appear with the (2) Relevant Equations...