Maxwell-Boltzmann speed distribution derivatives

In summary, the conversation discusses the movement of molecules through a vacuum chamber and slits, towards two rotating discs before reaching a detector. The question at hand is determining the speed at which a molecule will reach the detector, given a constant temperature and distance between the discs. The most probable speed is found to be equal to ωL/θ, as determined through a simple dynamics problem.
  • #1
devious b
4
0
Hi everyone,

Molecules move into a vacuum chamber from an oven at constant T. The molecules then pass through a slit. They reach two rotating discs before finally reaching a detector.

Show that a molecule that passes through the first slit will reach the detector if it has speed, u = ωL/θ.

After it passes through the slit, the beam of particles is directed towards 2 discs that are a fixed distance apart (L). They both have a notch in them, the 2nd disc's notch is offset from the 1st by 1/6∏. T is constant, common angular speed is ω & L = 0.262m.

My question is, do I simply say that the most probable speed is equal to u?

√(2KT/m) = u = ωL/θ

Any clues would be appreciated.

Thanks:)
 
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  • #2
I think you are looking too deep into the question. It's not about velocity distribution but they have given you a 'scenario' for a fairly straightforward dynamics problem. They just want to find which speed will get "a" molecule through the shutter which the two spinning slits present.
 
  • #3
You're absolutely right, I completely over-thought the problem. So,

t1 = Time particle takes between the 2 discs = d/u = L/u

t2 = Time taken for disc 2 to reach correct position = θ/ω

For a particle to pass through both notches & hit detector, t1 must equal t2

So,

L/u = θ/ω → Lω/u = θ → u = Lω/θ

Thanks
 

1. What is the Maxwell-Boltzmann speed distribution?

The Maxwell-Boltzmann speed distribution is a probability distribution that describes the distribution of speeds for a gas at a given temperature. It is based on the kinetic theory of gases and assumes that gas molecules follow a Gaussian or normal distribution in terms of their speed.

2. What do the derivatives of the Maxwell-Boltzmann speed distribution represent?

The derivatives of the Maxwell-Boltzmann speed distribution represent the rate of change or slope of the distribution curve at a particular point. They can provide information about the average speed, most probable speed, and root mean square speed of the gas molecules in a given sample.

3. How are the derivatives of the Maxwell-Boltzmann speed distribution calculated?

The derivatives of the Maxwell-Boltzmann speed distribution can be calculated using calculus. The first derivative, known as the first moment, represents the average speed of the molecules and can be found by taking the integral of the distribution curve. The second derivative, known as the second moment, represents the most probable speed and can be found by setting the first derivative equal to zero and solving for the value of the speed. The third derivative, known as the third moment, represents the root mean square speed and can be found by taking the square root of the second derivative.

4. What is the significance of the Maxwell-Boltzmann speed distribution derivatives?

The derivatives of the Maxwell-Boltzmann speed distribution provide important information about the behavior and properties of gases. For example, the first derivative gives the average speed, which is directly related to the temperature of the gas. The second derivative gives the most probable speed, which is the speed at which the majority of molecules in the gas are moving. The third derivative gives the root mean square speed, which is related to the kinetic energy of the gas molecules.

5. How does temperature affect the derivatives of the Maxwell-Boltzmann speed distribution?

As temperature increases, the average speed, most probable speed, and root mean square speed of gas molecules also increase. This is reflected in the derivatives of the Maxwell-Boltzmann speed distribution, as the first derivative (average speed) increases, the second derivative (most probable speed) shifts to the right, and the third derivative (root mean square speed) increases in magnitude.

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