Mean free path and collision cross section

In summary, the task at hand is to calculate the mean free path for an ion with a radius of 9 x 10‐8 cm in various pressures. The most commonly used equation for this is λ= 1/Nσ, where N is the gas number density and σ is the collision cross section. However, the gas number density is unknown and can be calculated using Avogadro's number if the moles per unit volume are known. The second equation given, λ= 7×10−6/ r coll2 ×P(torr), may also be used but its validity is uncertain without further research. The identity of the gas is not specified, but it is assumed to be nitrogen.
  • #1
Etox
2
0

Homework Statement



I do need to calculate the mean free path for an ion with a radius of 9 x 10‐8 cm in pressures of 10‐8 Torr, 10‐6 Torr, 10‐4 Torr and 10‐2 Torr

Homework Equations



There are different equations I found but the most one is λ= 1/Nσ
N gas number density which I do not know how to calculate based on radius and pressure.
And σ which is collision cross section σ =πd2
Another equation I found = 7×10−6/ r coll2 ×P(torr)

The Attempt at a Solution


I did convert radius to nm (0.9 nm) and applied the second equation
mean free path= 7×10−6/ 0.81 ×P(10-8) and so on for the different pressure
but I am not sure if this attempt is wright

These calculations are to use in the ion free path (ion mobility) in mass spectrometer instruments.
 
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  • #2
If you can calculate the moles per unit volume, then use Avogadro's number to convert to number density, would that do?
 
  • #3
the identity of gas is not mentioned but I assume nitrogen.
would the second equation be possible in which needs only the pressure and cross section section to calculate the mean free path.
 
  • #4
Etox said:
the identity of gas is not mentioned but I assume nitrogen.
would the second equation be possible in which needs only the pressure and cross section section to calculate the mean free path.

I'm afraid that I'm not familiar with the second equation that you gave, so I can't advise (not without doing a bit of research).
 
  • #5


I can provide some guidance on how to approach this problem. The mean free path is a measure of the average distance an ion travels before colliding with a gas molecule. It can be calculated using the equation λ = 1/(Nσ), where N is the number density of gas molecules and σ is the collision cross section of the ion and gas molecule.

To calculate N, you can use the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. Since we are given pressure and can assume room temperature, we can calculate the number density using the equation N = n/V = (P/RT). The volume can be calculated using the formula for the volume of a sphere, V = (4/3)πr^3, where r is the radius of the ion.

To calculate the collision cross section, we can use the equation σ = πd^2, where d is the diameter of the ion. Since we are given the radius, we can calculate d = 2r. Plugging these values into the equation, we can calculate the collision cross section for the given ion.

Once we have calculated N and σ, we can plug them into the equation for mean free path, λ = 1/(Nσ), to get the mean free path for the ion at each pressure. Remember to convert the pressure from Torr to Pascals before plugging it into the equation.

In summary, to calculate the mean free path for an ion with a given radius at different pressures, we need to use the ideal gas law to calculate the number density of gas molecules, use the formula for the volume of a sphere to calculate the volume, use the equation σ = πd^2 to calculate the collision cross section, and finally plug these values into the equation for mean free path, λ = 1/(Nσ). I hope this helps in your calculations.
 

What is mean free path?

Mean free path is a term used in physics and engineering to describe the average distance a particle travels before undergoing a collision with another particle or object.

How is mean free path calculated?

The mean free path can be calculated by dividing the total distance traveled by the number of collisions that occur during that distance.

What factors affect mean free path?

The mean free path of a particle is affected by the density of the medium it is traveling through, the size and shape of the particles, and the temperature and pressure of the medium.

What is collision cross section?

Collision cross section is a measure of the probability of a collision occurring between two particles or objects. It is dependent on the size and shape of the particles and the angle at which they collide.

How is collision cross section related to mean free path?

The mean free path and collision cross section are inversely related. As the collision cross section increases, the mean free path decreases, meaning there is a higher likelihood of collisions occurring between particles.

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