Mean relative speed

When the temperature is T and the molar masses are MA and MK, this mean relative speed is... 

Although the mean relative speed of the molecules increases with temperature, and the collision frequency therefore increases as well, Eq. 16 shows that the mean relative speed increases only as the square root of the temperature. This dependence is far too weak to account for observation. If we used Eq. 16 to predict the temperature dependence of reaction rates, we would conclude that an increase in temperature of 10°C at about room temperature (from 273 K to 283 K) increases the collision frequency by a factor of only 1.02, whereas experiments show that many reaction rates double over that range. Another factor must be affecting the rate. 

Maxwell, J. C., 164 Maxwell distribution of speeds, 164, 560 mean bond enthalpy, 254 mean relative speed, 559 mechanical equilibrium, 290 mechanics, 1... 

It is of interest to compare the half-widths at half-intensity of the spectral functions of the three systems shown in Fig. 3.2. These amount to roughly 140, 80 and 50 cm-1 for He-Ar, Ne-Ar and Ar-Kr, respectively, which are enormous widths if compared to the widths of common Doppler profiles, etc. The observed widths reflect the short lifetimes of collisional complexes. From the theory of Fourier transforms we know that the product of lifetime, At, and bandwidth, A/, is of the order of unity, Eq. 1.5. The duration of the fly-by interaction is given roughly by the range of the induced dipole function, Eq. 4.30 (1/a = 0.73 a.u. for He-Ar), divided by the mean relative speed, Eq. 2.12. We obtain readily ... 

Massieu function 48 mathematical constants 83, 90 mathematical functions 83 mathematical operators 84 mathematical symbols 81-86 matrices 83, 85 matrix element of operator 16 maxwell 115 Maxwell equations 123 mean free path 56 mean international ohm 114 mean international volt 114 mean ionic activity 58 mean ionic activity coefficient 58 mean ionic molality 58 mean life 22, 93 mean relative speed 56 mechanics classical 12 quantum 16 mega 74 melting 51 metre 70,71,110 micro 74 micron 110 mile 110 Miller indices 38 milli 74... 

In the first approximation the collision phenomena are described in terms of hard sphere molecular diameters, which are independent of temperature. Actually, the diameters decrease with higher temperature, approaching individual limits [1]. Let us consider a single molecular entity with the mass m and the diameter afmj, which diffuses through a gas consisting mainly of more abundant dissimilar molecules of the mass m2, the diameter dm.2 and the concentration no. If the collision diameter is mi.2 = (dm, 1 + <7m.2)/2, the tracer molecule must collide each second with the host molecules contained in a volume of about nm 2um. Because the host molecules also move, the mean relative speed u 1,2 is... 

Fig. 10 Plots of (7) and (8) for RT within v = 0 from N2(0 10) and VRT N2 (1 10) —> (0 Aj). Filled squares represent the A-plot (7), circles the E-plot (8) for RT and triangles that for VRT. The vertical arrow indicates the mean relative speed at 300 K. From this it is evident that only velocities in the high-energy region of the MB distribution may open the VRT channels are hence the process is of low inherent probability. The shaded region indicates those channels and velocities for which energy and AM conservation are simultaneously conserved...

One of the things we need to know is the collision number between molecule A, which is representative of a Maxwellian distribution of speeds of A molecules, and B molecules themselves possessing a Maxwellian distribution of speed. This can be done by defining a collision volume determined not by ca but by a mean relative speed, c, between the Maxwellian populations of A and B. This approach has been discussed by Benson [S.W. Benson, The Foundations of Chemical Kinetics, McGraw-Hill Book Co.,... 

To understand heat conduction, diffusion, viscosity and chemical kinetics the mechanistic view of molecule motion is of fundamental importance. The fundamental quantity is the mean-free path, i. e. the distance of a molecule between two collisions with any other molecule. The number of collisions between a molecule and a wall was shown in Chapter 4.1.1.2 to be z = CNQvdtl6. Similarly, we can calculate the number of collisions between molecules from a geometric view. We denote that all molecules have the mean speed v and their mean relative speed with respect to the colliding molecule is g. When two molecules collide, the distance between their centers is d in the case of identical molecules, d corresponds to the effective diameter of the molecule. Hence, this molecule will collide in the time dt with any molecule centre that lies in a cylinder of a diameter 2d with the area Jid and length gdt (it follows that the volume is Jtd gdt). The area where d is the molecule (particle) diameter is also called collisional cross section a. This is a measure of the area (centered on the centre of the mass of one of the particles) through which the particles cannot pass each other without colliding. Hence, the number of collisions is z = c n gdt. A more correct derivation, taking into account the motion of all other molecules with a Maxwell distribution (see below), leads to the same expression for z but with a factor of V2. We have to consider the relative speed, which is the vector difference between the velocities of two objects A and B (here for A relative to B) ... 

The integral in (2.14), which is the mean relative speed is most easily evaluated by transforming to center-of-mass and relative coordinates,... 

For the example of a dilute mixture of in Ng gas at 1 atm and 300 K considered in Section 2.2, we find that the mean relative speed is 516 m sec . Assuming, as before, that Jab 0-4 nm, we obtain a collision frequency 6.4 X 10 sec and a mean free path Slnm. It is worth noting that Asig is much larger than the mean interparticle distance, which is only 3.4 nm at this density. 

If both particles are moving at the mean speed, we identify their relative speed as the mean relative speed, denoted by (i>rei) ... 

Our derivation is crude, but Eq. (9.8-17) is the correct formula for the mean relative speed. 

Note that the mean speed enters in this formula, not the mean relative speed. However, incorporation of the /2 factor in the formula for the mean free path allows us to write... 

Notice how reasonable this equation is. The rate of collisions of a molecule is proportional to the collision cross section, to the number density of molecules, and to the mean relative speed. Under ordinary conditions, a gas molecule undergoes billions of collisions per second. 

Our derivation is crude, but Eq. (9.8-26) is the correct expression for the mean relative speed. For a pair of identical particles, jx is equal to m/2, so that Eq. (9.8-26) is valid for that case as well as for two different substances. 

Calculate the mean relative speed of nitrogen and oxygen molecules at 298 K. Solution... 

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