Statistical Mechanics: Calculating Pressure on a 3D Box Wall

F: In summary, the conversation discusses calculating the force per unit area on a wall of a particle confined within a cube-shaped box. The mean pressure on the wall is then found by averaging over all possible states, using the property that the average values of the quantum numbers are equal by symmetry. This mean pressure can be expressed in terms of the mean energy of the particle and the volume of the box. The process is done through a quasistatic process, and guidance is needed on how to solve it using the method of pressure.
  • #1
vladittude0583
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Homework Statement



Consider a particle confined within a box in the shape of a cube of edges Lx=Ly=Lz. The possible energy levels of this particle are then given by the quantized energy for a particle in a 3D box.

Calculate explicitly the force per unit area (or pressure) on this wall. By averaging over all possible states, find an expression for the mean pressure on this wall. (Exploit the property that the average values of the quantum numbers must all be equal by symmetry). Show that this mean pressure can be very simply expressed in terms of the mean energy of the particle and the volume of the box.

Homework Equations



In part (a), it was said that the particle exerts a force on a wall perpendicular to the x-axis such that Fx= - dE/dLx (these are partial derivatives).

The Attempt at a Solution



I don't want the final answer, however, I just need guidance on how to get there

1) This is through a quasistatic process such that dW=(mean pressurve)*dV
2) What does it mean when "exploit the property that the average values of the quantum numbers are all equa by symmetry?"
3) I know that the length along the x-axis changing by an amount dLx whereas the area of the wall is A=LyLz=constant. How do I related these values together?
4) I understand how to solve it using Kinetic Theory of Gases, however, my professr prefers using the method of pressure, etc.
5) Just some guidance is greatly appreciated. Thanks!
 
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  • #2
Regarding your question 3, it could mean that you will be able to determine a quantity that is given by the sum of the three average values, but not the average values individually. In that case you could exploit a symmetry to say that each of the average values would be 1/3 of the total.

Torquil
 

1. How is pressure calculated in a 3D box using statistical mechanics?

Pressure in a 3D box can be calculated using the ideal gas law, which relates pressure, volume, temperature, and the number of molecules. In statistical mechanics, this is represented by the equation P = (N/V)kT, where N is the number of molecules, V is the volume, k is the Boltzmann constant, and T is the temperature.

2. What is the significance of the molecular velocity distribution in calculating pressure?

The molecular velocity distribution is important in calculating pressure because it determines the frequency and magnitude of collisions between molecules and the walls of the box. The higher the velocity, the more frequent and forceful the collisions, resulting in higher pressure.

3. How does the size of the box affect the calculated pressure?

The size of the box directly affects the calculated pressure. As the volume of the box increases, the pressure decreases because there is more space for the molecules to move around and less frequent collisions with the walls. Similarly, a smaller volume results in a higher pressure.

4. Can temperature affect the calculated pressure in a 3D box?

Yes, temperature has a direct impact on the calculated pressure. As temperature increases, the velocity and kinetic energy of the molecules also increase, resulting in higher pressure. Conversely, a decrease in temperature leads to lower pressure.

5. How does the number of molecules in the box affect the calculated pressure?

The number of molecules in the box also has a significant impact on the calculated pressure. As the number of molecules increases, there are more collisions with the walls, resulting in higher pressure. Similarly, a decrease in the number of molecules leads to a decrease in pressure.

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