Momentum transfer average

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

The pressure can be computed by calculating the average momentum transfer across a fixed plane per unit area and time. The result is P =nkgT + Pn, where Pn is the nonideal contribution to the pressure which, for small A, takes the form... 

CCD data by circular averaging and expressed as a function of scattering angle in terms of the wave vector of momentum transfer q, with q = 4n sin (())//-, where 2d is the total scattering angle and 2 the X-ray wavelength. 

Now we will switch over to a macroscopic view, looking at the average force and pressure, rather than the instantaneous values. Collisions with the right wall (and thus force on the wall) happen every time the molecule makes a round trip between the left and right walls, thus traveling a total distance 2L the time between collisions is then 2L/Vy. So the average force on the right wall from these collisions is the momentum transferred to the wall per unit time ... 

The kinetic theory relied on converting the momentum transfer from individual collisions (which are very abrupt) into an average pressure. This will only be valid if the pressure we observe is the average of many events on an everyday timescale—in which case the fluctuations are small. This is a reasonable approximation, as we can illustrate by an example which might reflect an attempt to measure these fluctuations. 

The average collision generates a momentum transfer of 2 m ( vy ) (Equation 7.7) which for N2 is roughly... 

The rate at which the momentum transfer takes place is dependent on the rate at which the molecules move across the fluid layers. In a gas, the molecules would move about with some average speed proportional to the square root of the absolute temperature since, in the kinetic theory of gases, we identify temperature with the mean kinetic energy of a molecule. The faster the molecules move, the more momentum they will transport. Hence we should expect the viscosity of a gas to be approximately proportional to the square root of temperature, and this expectation is corroborated fairly well by experiment. The viscosities of some typical fluids are given in Appendix A. 

Equations 3.13 are known as the Stefan-Maxwell equations and are valid when the total pressure and temperature gradients as well as external forces can be neglected. They have the physical meaning that the rate of momentum transfer between two species is proportional to their concentrations and to the difference in their velocities. The molar average velocities of the species v, and v are defined in a such way that the molar fluxes of the various species are... 

One can similarly define average integral, momentum-transfer and total cross sections. 

Consider turbulent flow in a horizontal pipe, and the upward eddy motion of fluid particles in a layer of lower velocity to an adjacent layer of higher velocity through a differential area ri4 as a result of the velocity fluctuation v, as shown in Fig. 6-21. The mass flow rate of the fluid panicles rising through dA is pu dA, arid its net effect on the layer above dA is a reduction in its average flow velocity because of nioraentum transfer to the fluid particles with lower average flow velocity. This momentum transfer causes the horizontal velocity of the fluid particles to increase by and thus its momentum in the horizontal direction to increase at a rate of pv dA)u, which must be equal to the decrease in the momentum of the upper fluid layer. 

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