Kinetic Energy of Molecules Escaping Through Hole

In summary, the conversation is about a perfect gas in a container at equilibrium that escapes into a vacuum through a small circular hole in the container wall. The number of molecules escaping per unit area per unit time is given by nC/4 and the rate of mass outflow can be expressed using the flux equation. The question at hand is to show that the mean kinetic energy of the escaping molecules is greater than that of the molecules inside the container in the ratio of 4/3. The solution involves calculating the kinetic energy flux of the escaping molecules and dividing it by the number of molecules escaping, resulting in an answer of 2kT, which is 4/3 of the average molecular kinetic energy.
  • #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
Here is a way to do it: calculate the kinetic energy flux of the escaping molecules using equation (3.1) of the book as a basis (assuming, for example, that the wall with the hole is perperdicular to the x1-axis). Then divide it by the result you found in (a). This should yield 2kT, which is 4/3 of the average molecular kinetic energy 3/2 kT. Hope this helps.
 

FAQ: Kinetic Energy of Molecules Escaping Through Hole

1. What is the kinetic energy of molecules escaping through a hole?

The kinetic energy of molecules escaping through a hole refers to the energy possessed by the molecules as they move through the hole. This energy is a result of the molecules' random motion and is directly related to their velocity.

2. How does the size of the hole affect the kinetic energy of escaping molecules?

The size of the hole has a direct impact on the kinetic energy of escaping molecules. As the size of the hole decreases, the molecules have less space to move through and therefore, their velocity increases, resulting in a higher kinetic energy.

3. What factors contribute to the kinetic energy of molecules escaping through a hole?

The kinetic energy of molecules escaping through a hole is influenced by several factors including the temperature of the gas, the size of the hole, and the number of molecules present. Higher temperatures and smaller hole sizes lead to an increase in the kinetic energy of escaping molecules.

4. How is the kinetic energy of escaping molecules related to the rate of diffusion?

The kinetic energy of escaping molecules is directly proportional to the rate of diffusion. This means that as the kinetic energy increases, the rate of diffusion also increases. This is because molecules with higher kinetic energy have a greater tendency to move and spread out, leading to a faster rate of diffusion.

5. Can the kinetic energy of escaping molecules be measured?

Yes, the kinetic energy of escaping molecules can be measured using various techniques such as spectroscopy or mass spectrometry. These methods involve measuring the velocity of the molecules and using it to calculate their kinetic energy.

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