Velocity of molecules

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

The atoms are of a complex structure made up of positive nuclei and electrons, of which the latter, at least, are in very rapid motion. If we assume that the electrons are 1/1800 the mass of hydrogen atoms and that they obey the same laws of motion as other atoms, their average velocity would be about sixty times the velocity of molecules of hydrogen (H2). I will not attempt to discuss here the question whether the law of equipartition of energy actually holds for electrons. 

However, in above derivation it has been assumed that molecules of B are stationary, which is not correct. Because the molecules of gases are moving with different velocities and cannot be assumed to be stationary. Thus, the average velocity of molecules uA may be replaced by /8kb T/n i, where kh is Boltzmann s constant and p the reduced mass given by... 

Under ordinary conditions, the molecules of a gas have speed which are not uniform but distributed over a range. In general, the distribution of velocities of molecules A and B, by Maxwell law is given as... 

When other kinds of work are involved, it is necessary to specify more variables, but the point is that when a small number of properties are specified, all the other properties of the system are fixed. This is in contrast with the very large number of properties that have to be specified to describe the microscopic state of a macroscopic system. In classical physics the complete description of a mole of an ideal gas would require the specification of 3NA components in the three directions of spatial coordinates and 3NA components of velocities of molecules, where NA is the Avogadro constant. 

Problem 6 Define the terms average velocity, root mean square velocity and most probable velocity of molecules and how are they related to each Other (Meerut 2006)... 

Equation 3.13 fulfills our basic objective in describing average molecular motion. However, inasmuch as we cannot ordinarily measure directly the average velocity of molecules or ions undergoing transport, it is advantageous to transform 0 into the flux density /, a parameter that is more directly observable. The flux density of a component is the number of moles carried through a unit areain unit time. It is related to that component s mean molecular velocity 0 and concentration c by... 

When we substitute equations (3.255) and (3.256) in the remaining terms of (3.251) we find that we retain a number of awkward cross terms involving P0 whereas symmetry considerations suggest that translational terms should be completely separable for the field free case. The explanation seems to be that we have made a coordinate transformation to the centre of rest mass of all particles rather than to the centre of relativistic mass. Since the translational velocities of molecules are very much less than the speed of light, the contributions of these cross terms in P0 are expected to be very small and we ignore them in further discussion. 

That Eqs. (32) and (34) differ in the dependence of Z i2 on the molecular weights should not be altogether surprising, since in our simple treatment we have assumed that the velocities of molecules after collisions are uncorrelated with their velocities before collisions, whereas in fact such correlations in general exist and are functions of the ratio of the masses of the colliding particles. To take this and other remaining factors properly into account would require a treatment that is beyond the scope of this book. It may suffice to point out here that the square-root quantity in Eq. (34) is equivalent to y/lfpizNo, where... 

Since each molecule thus striking has come from the opposite plate, it carries an excess of tangential component of momentum on striking the fixed plate equal to mV, where V is the difference in velocity. If on the average a fraction a of the molecules achieves the speed of the plate they collide with, it can be shown that, at a steady state, the excess velocity of molecules after striking the fixed plate is (1 — a)V/(2 — a), while the excess velocity of molecules after striking the moving plate is K/(2 — a)."... 

From the classical chemical aspect, isotopes of an element are expected to have same chemical properties. Therefore, it has been believed that the behaviors of isotopes of an element are essentially the same. However, a change in the mass of an atom in a molecule produces changes in vibrational, rotational, and translational motions of the molecule. In the diffusion of gases it has long been known that the velocity of molecules changes with the mass of the molecule, thus producing a physical isotope effect. 

Here the factor ip accounts for effects of composition (cf. Sect. 8.23.6), tortuousity and psd and the square root is the mean thermal velocity of molecules of mass M,-. The value D serves as a reasonable estimate of the diffusion coefficient. 

The above three velocities have been defined to measure the velocity of molecules from different viewpoints. Three kinds of velocities are used for different applications the root mean square velocity (urm ) is used for the calculation of the average dynamic energy, the mean velocity uav for the mean distance between molecules, and the most probable velocity (up) for the distribution of velocity. 

Here, V is the interaction of the electric field with die molecule, N is the concentration of the absorbing molecules, v is tire tlrermal velocity of molecules, and the term v V is responsible for the Doppler effect. The relaxation matrix, I, contains the rates of various radiative and non-radiatlve transitions. 

V local instantaneous mixture velocity, mass average velocity (m/s) modified mass diffusion velocity for species s (m/s) species velocity of molecules of species s generated by chemical reaction (m/s)... 

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