- #1

rogdal

- 14

- 2

- Homework Statement
- Given a box of dimensions ##V=L \times L \times L## in the ##xyz## space which has ##N \gg1## particles of a certain gas, the velocity of a random particle is given by the Maxwell-Boltzmann distribution:

$$\rho_v(\vec{v})=\frac{1}{\sqrt{(2\pi \beta)^3}}e^{-\frac{v^2}{2\beta}},$$

where ##\beta = k_BT/m##

With this data, prove that the probability distribution function with which a particle chosen at random collides against the wall located in the plane ##z = L## is:

$$\rho_{collision}(\vec{v})=2\rho_v(\vec{v})\theta(v_z),$$

where ##\theta(v_z)## is the Heaviside theta function.

- Relevant Equations
- $$\rho_v(\vec{v})=\frac{1}{\sqrt{(2\pi \beta)^3}}e^{-\frac{v^2}{2\beta}}$$

I have considered two scenarios:

1) A particle that has just collided with the wall at ##z=L## is moving with a velocity ##v_z<0## moving away from the wall. Hence, the probability that this particle has of colliding again is ##0##, so its distribution is also ##0##.

2) A particle moving with positive ##v_z##. Since the particles are constrained to move in that box, it will necessarily collide against the wall, and assuming a completely elastic collision, the velocity distribution in this case would be ##2\rho_v(\vec{v})## because after the collision the particle emerges with the same velocity it had before.

Would these arguments be correct?

Thanks in advance!

1) A particle that has just collided with the wall at ##z=L## is moving with a velocity ##v_z<0## moving away from the wall. Hence, the probability that this particle has of colliding again is ##0##, so its distribution is also ##0##.

2) A particle moving with positive ##v_z##. Since the particles are constrained to move in that box, it will necessarily collide against the wall, and assuming a completely elastic collision, the velocity distribution in this case would be ##2\rho_v(\vec{v})## because after the collision the particle emerges with the same velocity it had before.

Would these arguments be correct?

Thanks in advance!