Model for Sudden Decompression of Ideal Gas in Box

[intervoxel]

Imagine an infinitely long square box of side L. This box is isolated from the ambient and contains a number of N molecules of an ideal gas in a volume L^3 in thermal equilibrium located at one end of the box at time t=0.

I found that the evolution of this system can be modeled by the decay equation

n(t) = N e^{-At}.

Where n is the number of particles in the volume L^3 and A is a scalling constant.

My question is: Is there a better model for this system? (maybe hopefuly including absolute temperature T, L and Boltzman constant k_B)

 

[intervoxel]

Please, help. I'm stuck a long time in this.

So far I got the following formula which gives the time the i-th particle takes to reach the barrier at x=L:

$$t_i = \frac{2 L - x_i}{\overline{v} \cos(a_i)}$$

where

$$x_i$$ is a random variable between 0 and L
$$a_i$$ is a random variable between 0 and $$\pi /2$$
$$\overline{v}$$ is the average speed of a gas particle

What I need is $$n(t) = f(N, L, \overline{v},t)$$

where

N is the total number of particles
n(t) is the the number of particles in the original volume $$L^3$$ after time t

Any reference book or article?

Thanks

 

[pallidin]

Perhaps a Google search of "sudden decompression equations" with an additional search term of NASA, LANL or something similar might give some insight.

 

[intervoxel]

Thank you for you suggestion, pallidin, but I couldn't find anything.

I'm checking the consistency of the following formula I worked out:

$$\boxed{ \;\;n(t) = N exp\left[-\left(\frac{4\ln{2}\sqrt{\frac{3k_BT}{m}}}{\pi L}\right)t\right].\;\; }$$

where $$k_B$$ is Boltzmann constant, T is the absolute temperature and m is the atomic weight of the monoatomic gas molecule.