Kinetic Theory: How many molecules enter pinhole in vacuum?

In summary, the problem involves a vacuum with a pressure of 10^-3 Torr and an external pressure of 1atm and 300K. A pinhole with an area of 10^-10cm^2 is made in the vacuum and it is assumed that all molecules striking the hole pass through. The question asks for the number of molecules that leak into the vacuum in 1 hour. Using the formula Flux = <v>*N/4V and Rate = Flux times A, an arbitrary volume can be used to calculate the number of molecules per volume, N/V. Assuming an order-of-magnitude answer is sufficient, any reasonable value for the mass of a molecule can be used to solve the problem. The final answer
  • #1
Lagraaaange
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Homework Statement


Pressure in vacuum is 10^-3 Torr. External Pressure is 1atm and 300K. Pinhole of 10^-10cm^2 made in vacuum. Assuming all molecules striking hole pass through, how many molecules leak into vacuum in 1hour?
Answer: 10^17 molecules

Homework Equations


Flux = <v>*N/4V
Rate = Flux times A

The Attempt at a Solution


I can make an DE: dN/dt = <v>N/4V*A and solve this but I have no info on N0, initial molecules. I also don't have volume. So I don't see how to obtain a numerical result.
 
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  • #2
Ask yourself what pressure is.
 
  • #3
You only need N/V, molecules per volume, which you can get with the given information.
Alternatively, take an arbitrary volume, it will cancel out in the calculations anyway.
 
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  • #4
I don't see how I can get this with the information
mfb said:
You only need N/V, molecules per volume, which you can get with the given information.
Alternatively, take an arbitrary volume, it will cancel out in the calculations anyway.
P/kT = N/V = 2.45molecules/m^3 = ρ
Also <KE> = 3/2KT = 6.21E-21J = m/2 *<v^2>.
Now particles within vΔt make it to hole; also those within AvΔtρ make it.
Now I'm stuck. I can't get v since I don't have m. Also I'm unsure whether to invoke a Maxwell Distribution to get all velocities.
 
  • #5
Looks like they are only asking for an order-of-magnitude answer. So, you can use any reasonable value for the mass of a molecule.
 
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1. How does the kinetic theory explain the movement of molecules entering a pinhole in a vacuum?

The kinetic theory states that all particles, including molecules, are in constant motion. This means that even in a vacuum, where there is no air or other particles, molecules will still be moving. When a pinhole is present, some of these molecules will collide with the edges of the pinhole, causing them to change direction and enter the opening.

2. How many molecules can enter a pinhole in a vacuum?

The number of molecules that can enter a pinhole in a vacuum depends on a few factors, including the size of the pinhole and the pressure of the surrounding atmosphere. Generally, the smaller the pinhole and the lower the pressure, the fewer molecules will enter. However, in a true vacuum, there is still a chance that some molecules will enter the pinhole due to their random motion.

3. Can any type of molecule enter a pinhole in a vacuum?

Yes, any type of molecule can potentially enter a pinhole in a vacuum. This includes gases, liquids, and solids, as all particles are constantly in motion according to the kinetic theory. However, the likelihood of different types of molecules entering a pinhole may vary due to their size and other properties.

4. What happens to the molecules that enter a pinhole in a vacuum?

Once molecules enter a pinhole in a vacuum, they will continue to move in a straight line until they collide with another object or the walls of the pinhole. Some may also pass through the pinhole and enter the surrounding vacuum. This process will continue until the pressure inside and outside the pinhole reaches equilibrium.

5. How does the kinetic theory apply to real-world situations involving pinholes?

The kinetic theory is a fundamental concept in understanding the behavior of particles, including molecules, in various situations. In the case of pinholes, the theory explains how and why molecules can enter even in a vacuum. This understanding has many real-world applications, such as in the design of vacuum systems and in the study of gas behavior in different environments.

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