Kinetic Energy Molecules Escaping Small Hole

In summary, a gas escapes from a hole in a wall of a container at equilibrium. The number of molecules escaping per unit area per unit time is given by nC/4. The mean kinetic energy of the escaping molecules is greater than that of the molecules inside the container in the ratio of 4/3.
  • #1
TheAmorphist
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Homework Statement


"A perfect gas containing a single species of molecular weight M is in a container at equilibrium. Gas escapes into a vacuum through a small circular hole of Area A in the wall of the container. Assume wall container is negligibly thick and planer in vicinity to hole. The diameter of the hale is appreciably smaller than the mean free path, but larger than molecular diameter.
a. Show that the number of molecules escaping from the hole per unit area per unit time is given by nC/4.
b.Obtain an expression for the rate of mass outflow.
(What I actually need help on)c. Show that the mean kinetic energy of the escaping molecules if greater than that of the molecules inside the container in the ratio of 4/3.

Homework Equations


The flux equation of F=(Int)nf(Ci)QCnDVc, where f(Ci) is the Maxwellian Velocity Distribution, Q is some quantity (energy here) and integration is performed over the range of velocity space of interest.

The Attempt at a Solution



I have completed the nitty gritty of parts a and b, but simply cannot make any progress on part C. I'm assuming that the gas "within the container" have a mean kinetic energy of (3/2)KT, as they have 3 degrees of freedom. My attempt at finding the kinetic energy of those escaping the hole has consisted of integrating the flux equation a number of times to find a constant factor that resulted in a ratio of "4/3" but I feel like this is in inappropriate way to go about this. Any help or at least pointing me in the right direction would be really helpful.
 
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  • #2
Any help? Anybody? This question is from "Physical Gas Dynamics" by Vincenti and Krieger
 
  • #3
I'm reposting this in the "Advanced Physics" forum. That is probably where it belonged to begin with. Can a moderator delete this topic?
 

FAQ: Kinetic Energy Molecules Escaping Small Hole

1. What is kinetic energy?

Kinetic energy is the energy that an object possesses due to its motion. It is a result of the object's mass and velocity.

2. How do molecules escape through a small hole?

Molecules escape through a small hole when their kinetic energy is greater than the attractive forces between them. This happens because the molecules near the hole are moving faster and have enough energy to overcome the attractive forces.

3. Why do smaller molecules escape through a small hole more easily?

Smaller molecules have a higher average velocity than larger molecules at the same temperature. This means they have a higher kinetic energy, making it easier for them to escape through a small hole.

4. What is the relationship between temperature and the number of molecules escaping through a small hole?

The number of molecules escaping through a small hole increases with temperature. This is because as temperature increases, the average velocity of the molecules increases, leading to a higher number of molecules with enough kinetic energy to escape.

5. How does the size of the hole affect the rate of molecules escaping?

The size of the hole affects the rate of molecules escaping through it. Smaller holes will have a higher rate of molecules escaping compared to larger holes, as smaller molecules will have a higher chance of having enough kinetic energy to escape through a smaller opening.

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