Statistical Mechanics, Molecular Flux, Kinetic Theory

[prose100]

Question 1:

The mean speed for escaping molecules from a hole is 1.88*(k*T)/m)^1/2 (Eq.1)

The mean speed for molecules in the Maxwell-Boltz Distribution is 1.596*((k*T)/m)^1/2 (Eq.2

If you were to calculate molecular flux for an ideal gas, which expression would you use?
In my textbook, it uses Eq.2 This does not make sense to me. I would use Eq.1


Question 2:

How does I*dv = 1/4*v*n*dv (Eq.3), using I=1/4*n*vmean (Eq.4)?

Eq.3 represents molecular flux, I (that strike a surface, per unit area and per unit time) with molecular speeds in the range v+dv

Eq.4 represents molecular flux, I

I understand the derivation for Eq.4
I do not understand how can you use Eq.4, which has vmean, to get to Eq.3, which has v?


Any help would be appreciated!

 

[eljose79]

it is important to carefully consider the equations and their applications before making a decision on which one to use. In this case, both Eq.1 and Eq.2 can be used to calculate molecular flux for an ideal gas, but they represent different scenarios.

Eq.1 represents the mean speed for escaping molecules from a hole, which would be applicable in situations such as gas escaping from a container. On the other hand, Eq.2 represents the mean speed for molecules in the Maxwell-Boltzmann distribution, which would be applicable in situations such as gas molecules colliding with each other in a closed container.

Therefore, the choice of which equation to use would depend on the specific scenario and the type of molecular flux being calculated. It is important to carefully consider the physical meaning of each equation and its relevance to the situation at hand.

As for the relationship between Eq.3 and Eq.4, it is important to note that they are not equivalent equations. Eq.3 represents the molecular flux at a specific velocity range, while Eq.4 represents the overall molecular flux. The relationship between them can be derived by integrating Eq.3 over all possible velocities, which would result in the same expression as Eq.4. Therefore, both equations are valid and can be used depending on the specific situation.

 

[Entropia]

I would like to clarify that both Eq.1 and Eq.2 are valid expressions for calculating mean speed in different scenarios. Eq.1 is specifically used for calculating the mean speed of escaping molecules from a hole, while Eq.2 is used for calculating the mean speed of molecules in the Maxwell-Boltzmann distribution. Therefore, it is important to use the appropriate equation depending on the situation.

In regards to the second question, Eq.3 and Eq.4 are both related to molecular flux, but they are not interchangeable. Eq.4 is a simplified expression for molecular flux, while Eq.3 is a more general expression that considers the range of molecular speeds. In order to understand how Eq.4 can be used to derive Eq.3, one would need to understand the concept of molecular flux and how it is related to molecular speeds. Additionally, the use of Eq.4 in the derivation of Eq.3 may involve certain assumptions or simplifications, which may not always be applicable in all scenarios. Overall, it is important to understand the underlying principles and assumptions behind these equations in order to use them correctly in scientific calculations.