Help Needed: Finding Mean Free Path of Cs+ Ions

[lara-quark]

I was hoping somebody could help me out with this problem I'm stuck on:

So in a mass spectrometer, an ionizing current of 2 pA causes ions of Cs+ to be produced. these ions have an energy of about 10 kev. I need to find the average distance (mean free path) between any two Cs+ ions at this current.

Any help will be more than appreciated

Here's what I'm thinking:
dq/dt = i
so i = dq/dx*dx/dt
i can get velocity from the energy: 1/2mv^2 ( i use mass of Cs atom)
if i use dq = charge of an electron: since i have a singly ionized Cs atom
i get a distance of about 1.7 cm. now i know that at ultra high vacuum the distance between these ions is of the order of 10^5 m . that's all i have to go on. i cannot check if my answer is correct. hence this posting.
alternatively i can use the mean free path equation but i don't know if
can i use the same equation for mean free path of atoms as given my the kinetic theory of gases for this problem?? is there some factor that you need to put into account for them being ions ?

 

[Aaron Barnard]

The mean free path equation is:
l = (1/nσ)^(1/2)
where n is the number density of the particles, and σ is their collision cross section.

In this case, since you are dealing with ions, you need to take into account the fact that they will be attracted or repelled by other ions in the vicinity. This will affect their mean free path, so you should factor in the electrostatic forces between them.
You can calculate the effective number density of ions using the Boltzmann equation and the ionization potential of Cs:
n = N_0 exp(-I/kT)
where N_0 is the number density of neutral atoms, I is the ionization potential, k is the Boltzmann constant, and T is the temperature.
Once you have the number density, you can use the mean free path equation to calculate the distance between two ions.

 

[Graeme Robinson]

First of all, it's great that you are actively seeking help and trying to solve this problem on your own. It shows determination and a willingness to learn. Let's break down the problem and see if we can find a solution.

To find the mean free path of Cs+ ions, we need to use the kinetic theory of gases. This theory states that the average distance traveled by a particle between collisions is equal to its average speed multiplied by the mean time between collisions. In this case, the particles are Cs+ ions and the collisions are with other ions.

To find the average speed of the ions, we can use the kinetic energy equation you mentioned: KE = 1/2mv^2. We know the mass of a Cs+ ion and the kinetic energy, so we can solve for the speed.

Next, we need to find the mean time between collisions. This can be calculated using the collision frequency, which is the number of collisions per unit time. The collision frequency can be found using the ionizing current of 2 pA and the charge of a single ion.

Once we have the average speed and mean time between collisions, we can plug them into the mean free path equation: λ = v/ν.

Note that this equation is valid for neutral particles, so we need to take into account the fact that Cs+ ions are charged. This means they will experience some repulsion from other ions, which may affect their average speed and collision frequency. However, for a rough estimate, we can use this equation.

Finally, we can compare our result to the distance of 10^5 m at ultra high vacuum. If our result is in the same order of magnitude, then we can be confident in our answer.

I hope this helps and good luck with your problem! Remember, always double check your calculations and assumptions. If you're still unsure, don't hesitate to ask for clarification or assistance.