§8 The mean free path of the molecules.

Effective diameter

 Gas molecules are in a state of chaotic motion continuously bump into each other. Between two successive collisions of molecules move uniformly in straight lines, passing with a path, which is called the mean free path. In general, the length of the path between successive collisions is different but since we are dealing with a large number of molecules and they are in random motion, we can speak of the mean free path:

The minimum distance that converge in a collision centers of two molecules, called the effective diameter of the molecule.

It depends on the speed of the colliding molecules, that is, the temperature (the effective diameter decreases with increasing. For a second (t = 1 s) molecule transits on the average parth equal in magnitude to the average velocity.

If for one second, she undergoes an average z collision,
, then


To determine ν believe that the molecule has the shape of a ball, and moves among other fixed molecules. This molecule is confronted only with those molecules whose centers are at a distance d, ie, lie within the "broken" a cylinder of radius d.

The average number of collisions per second is equal to one the number of molecules in the volume of "broken" cylinder.

where n - the concentration of the molecules, and

­- average speed of the molecules, or the path traveled by it in 1 second

 - the average number of collisions

Taking into account a motion of other molecules:



§9 Transport Phenomena

Transport phenomena combine a group of processes associated with the irregularities of density, temperature and velocity of the orderly movement of individual layers of material. Alignment leads to inhomogeneities in transport phenomena.

Transport phenomena in gases and liquids consist in the fact that these substances an ordered, directed mass transport (diffusion), momentum (internal energy) and internal energy (thermal conductivity). In the gas breaks complete randomness of the molecules and the distribution of molecular velocities. Deviations from the law of Maxwell explained directional transfer of physical characteristics of the material in the transport phenomena.

We only consider the one-dimensional phenomena, in which the physical quantities determining these phenomena depend only on one coordinate

   1. Thermal conductivity.

The phenomenon of thermal conductivity observed in the different parts of the considered gas temperature are different. Consideration of the effects of heat conduction from the microscopic point of view, shows that the amount of heat transported through the area ΔS, perpendicular to the direction of transfer is directly proportional to the thermal conductivity χ, which depends on the type of substance or gas, the temperature gradient , the value area ΔS and observation time Δt

Fourier law:

The minus sign in the Fourier law shows that heat is transferred in the direction of decreasing temperature T.

With molecular-kinetic phenomena in terms of thermal conductivity is explained as follows. In the area of ??gas, where the temperature is higher, the kinetic energy of the random thermal motion of the molecules is greater than the area where the temperature is lower. As a result of random thermal motion of the molecules move from the area where the region above T, where T is less. However, they suffer from a kinetic energy greater of the average kinetic energy possessed by the molecules in the field of lower energy. Due to continuous collisions of molecules over time the process of alignment of the mean kinetic energy, that is, temperature equalization.    The thermal conductivity χ is equal

Where cV - specific heat capacity of gas at constant volume (a quantity of heat necessary for heating 1 kg of gas on 1 K at constant volume).

      -  density of gas   -  average thermal velocity of the molecules

 -  medial free length.

     Physical sense χ: the thermal conductivity χ is numerically equal to density of a thermal stream  at  the temperature gradientequal 1

      2. Diffusion
The phenomenon of diffusion is the spontaneous mixing of molecules of different gases or liquids. Diffusion phenomenon is observed in solids. In those cases where a chemically pure homogeneous gas concentration of molecules will be different, there is a transfer of molecules, leading to equalize (or concentration) of molecules. This phenomenon is of self-diffusion. For simplicity we assume that the density is inhomogeneous along the x axis.
Of the phenomenon of self-diffusion from the macroscopic point of view was Fick, who established the following law: the mass of the gas transported through the area
ΔS, perpendicular to the direction of the transfer during Δt is proportional to the self-diffusion coefficient D, which depends on the type of gas, the density gradient  the value of the site ΔS and the observation time Δt. 

 - Fick’s law

     The minus sign indicates that the mass of the gas transported in the direction of decreasing density. Self-diffusion coefficient D is numerically equal to the mass of gas transferred per unit time through a unit area perpendicular to the direction of transport, with a gradient of density equal to one


 - fluence

According to the kinetic theory of gases

3. The internal friction (viscosity)

The phenomenon of internal friction is observed in the case where the different layers of gas are moving at different speeds. In this case, the layers are decelerated more rapidly moving slowly. At the macroscopic motion of the gas layers (ie, wall motion as a whole) has an impact microscopic thermal motion of the molecules.

Consider one layer of gas moving at a speed v1 and gas layer 2, moving at a speed v2 v1 > v2. As a result of the thermal random motion of molecule A from layer 1 to layer 2 switch and change its momentum from the value mv to any value mv ' (v2 < v' < v1).

The molecules B in a layer 2 as a result of the heat goes into the random motion of the layer 1 and change its momentum from the value mv2 to the value of mv'' (v2 < v'' < v1), that is, the molecules in the layer above the former two, once in the layer 1, collisions with molecules it accelerates its orderly movement, and ordered the moving molecules of the layer 1 is slowing. On the contrary, the transition of molecules from a fast-moving layer 1 to layer 2, they carry large momenta and intermolecular collisions at layer 2 speed motion of the molecules of this layer.
The phenomenon of internal friction is described by Newton's viscous force F, acting between two layers of gas is directly proportional to the internal friction coefficient
η, the velocity gradient and the size of square of ΔS.

(Impulse dp, carried through the area dS in the time Δt, is directly proportional to the internal friction coefficient η, the velocity gradient , the value of the area dS and observation time dt).    

 - Newton's law

The minus sign indicates that the viscous force is opposite to the velocity gradient, that is, the momentum transferred in the direction of decreasing velocity. Coefficient of internal friction is given by

Relation between the coefficients for the transport phenomena


To the list of lectures