Week 03 - Mean free path, Molecular number density, Impingement rate, Continuum & Molecular state o gases, Exercise problems

2.3 Molecular Impingement Rate, J

It is defined as “Total number of molecules striking the surface from all directions and with all speeds” and given by

                                                         or        

Where ‘n’ is molecular number density.

It is extremely important and can be used to calculate numerous other quantities such as condensation and evaporation rates, monolayer adsorption times, and flow properties of gases.         

2.4 Pressure and Molecular Number Density,  n

The pressure exerted on the surface by the impact of molecules having number density ‘n’at temperature T is given by 

                                                                p = nkT

where k is Boltzmann constant.

It shows that the pressure exerted by an ideal gas is directly proportional to its absolute temperature and the number density of molecules. This is very reasonable because we have seen that n directly determines the rate of molecular impacts, and that  T determines the energy.

 

2.5 Molecular Collisions and Mean Free Path, λ

The mean free path  λ  is defined as “The average distance traveled by a molecule between successive collisions”. The mean free path  is thus the distance traveled in one second divided by the number of collisions per second.

                                                    

Νotice that λ  depends on  n and σ,  the collisional cross section σ = πd2

The probability that a molecule will have a free path equal to or greater than  x is given by

                                                       

The total number of molecules which will have free paths of x or longer is given by

                                                      

The familiar exponential curve of above equation is plotted below showing fraction of molecules having free path equal or greater than х.

Remembering that exp (-1) = 0.37, we may deduce that 63% of free paths are shorter than  λ . Since e3= 20 to a very good approximation, 95% are shorter than 3 λ ; only 1 in 22,000 is greater than 10 λ

                  

                               Fig. Fraction of molecules having free paths longer than x

  

                                                                     

 

2.6 Knudsen Number: Continuum and Molecular States of Gas

The molecule–molecule collisions is the characteristic process that determines gas behavior. In other words, the mean free path  λ  in the gas at atmospheric pressure is very much less than the characteristic dimension of its container. For a cubical box, that would be the length of a side, or for a pipe, its diameter. But from above Table it is evident that at low pressures, mean free path will become comparable with or greater than a container’s characteristic dimension. Molecule–surface collisions will now dominate gas behavior and molecule–molecule collisions become quite rare.

It is useful to have a criterion for distinguishing between the  fluid (or what is better called the continuum  state) and the molecular state. This is provided by the Knudsen Number Kn defined as Kn =  λ /D, where  λ  is the mean free path and D is a characteristic dimension. There is, of course, in the nature of things, no sharp change between the continuum and molecular states, but rather a gradual transition between them. These states, or  regimes as they are sometimes called, are conventionally taken to be

                                            Kn < 0.01: CONTINUUM

                                           Kn > 1: MOLECULAR

 and in between,                    0.01 < Kn < 1: TRANSITIONAL

Kn is a useful dimensionless number because it automatically incorporates information about the gas condition in relation to the container size.

For example, at a pressure such that  λ  is 5 cm, the gas in a 1-cm cube would be in a molecular state, but in a large vacuum tank of dimension 3 m at the same pressure, conditions would be more nearly continuum.

 

 

 

 

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