Probability distribution mean-square speed

The root mean square speed rrrm of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are i/mean = (8RT/-nM),a and... 

FIG U R E 9.14 The Maxwell-Boltzmann distribution of molecular speeds in nitrogen at three temperatures. The peak in each curve gives the most probable speed, u p, which is slightly smaller than the root-mean-square speed, Urms The average speed Uav (obtained simply by adding the speeds and dividing by the number of molecules in the sample) lies in between. All three measures give comparable estimates of typical molecular speeds and show how these speeds increase with temperature. 

Like any nonsymmetrical probability distribution, the MB speed distribution can be characterized by several different single numbers average speed u, most probable speed and root-mean-square speed ttotal probability to the high side of the most probable speed, these three numbers have this order [Pg.399]

A root-mean-square speed for gas particles is easy to define but should not obscure a key point Gas particles do not all move at the same speed. Also, as implied at the beginning of this section, not all possible speeds are equally probable. Rather, there is a particular distribution of different gas speeds in any sample. What is the mathematical expression that gives us the distribution of gas speeds ... 

Figure 9.12 shows this probability distribution of speeds for oxygen molecules at 298 K. The most probable speed, the mean speed, and the root-mean-square speed are labeled on the speed axis. Compare this figure with Figure 9.7. The most probable value of a velocity component is zero, while the most probable speed is nonzero and the probability of zero speed is zero. This difference is due to the fact that the speed probability density is equal to the area of the spherical shell in velocity space (equal to 4nxP-) times the probability density of the velocities lying in the spherical shell. Zero speed is improbable not because the velocity probability density is zero (it is at its maximum value), but because the area of the spherical shell vanishes at n = 0.

The functional form of the distribution of speeds, s (horizontal axis), of ideal gas particles. This is simply a plot of the function/(s) = exp(-as ) (/on the vertical axis) versus s with an arbitrarily chosen value of a = m/2kT. The three vertical lines mark, from left to right, the s-values of the most probable speed, the average speed, and the root-mean-squared speed. Notice that this function is not symmetrical about its maximum, and as a consequence, the average speed is somewhat greater than the most probable speed. 

There are three characteristic speeds of this distribution the most probable speed Up the average molecule s speed (v) and the root mean square speed. The value of the most probable speed can be found by calculating an extremum of the distribution function ... 

Use the Maxwell-Boltzmann distribution of molecular speeds to calculate root-mean-square, most probable, and average speeds of molecules in a gas (Section 9.5, Problems 41-44). 

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...

The individual molecules of a gas do not all have the same kinetic energy at a given instant Their speeds arc distributed over a wide range the distribution varies with the molar mass of the gas and with temperature. The root-mean-square m speed, r es in proportion to the square root of the absolute temperature and inversely with the square root of the molar mass = V3RT/A1. The most probable speed of... 

A Figure 10.13 Distribution of molecular speeds for nitrogen gas. (a) The effect of temperature on molecular speed. The relative area under the curve for a range of speeds gives the relative fraction of molecules that have those speeds, (b) Position of most probable (Ump), average (Uav), and root-mean-square (u,rns) speeds of gas molecules. The data shown here are for nitrogen gas at 0°C. 

Figure 3.10 Maxwell distribution and characteristic speeds the most probable Dp, the averaged (n) and root mean square velocities.

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