Maxwell distribution of speeds

Consider the Maxwell distribution of speeds found in Fig. 4.27. (a) From the graph, find the location that represents the most probable speed of the molecules at each temperature. 

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... 

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

This expression for f(v) is now called the Maxwell distribution of speeds. AN is the number of molecules in the narrow range of speeds between v and v + Au, and N is the total number of molecules in the sample (Box 4.2). 

Let s consider the fraction of molecules that collide with a kinetic energy equal to or greater than Emm. Because kinetic energy is proportional to the square of the speed, this fraction can be obtained from the Maxwell distribution of speeds (Section 4.13). As indicated for a specific reaction by the shaded area under the blue curve in Fig. 13.17, at room... 

The equation above demonstrates that the kinetic of a gas is proportional to the temperature. 4.11 The Maxwell Distribution of Speeds... 

Note Calculus can be used with the Maxwell distribution of speeds to obtain the following properties that are not mentioned in the text. 

The molecules of each species possess a Maxwell distribution of speeds. For molecules of species A, for example, their mean speed from the Maxwell distribution is vA = (%kBT lnmA)Xl 2, and the relative velocity of the A, B collision partners is... 

FIGURE 5 Maxwell distributions of speeds for molecular nitrogen at 25°C (298 K) and 500°C (773 K). Arrows indicate v and vms for each case. The most probable velocity has been arbitrarily scaled to unity in each case. 

The reaction rate, expressed as the number of reactions per volume and per time, is proportional to the astrophysical S-factor. At the temperatures and densities relevant to the stellar environments the interacting nuclei have a Maxwell distribution of speeds. This distribution has also to be taken into account when determining the reaction rate. An introduction to astrophysical S-factors and reaction rates can be found in many textbooks on nuclear astrophysics, e.g., Arnett (1996), Rolfs and Rodney (1988), Iliadis (2007), Boyd (2008). 

Fig. F.IO The Maxwell distribution of speeds and its variation with the temperature. Note the broadening of the distribution and the shift of the mean speed (denoted by the locations of the vertical dotted lines) to higher values as the temperature is increased.

I I 15. The properties ofthe Maxwell distribution of speeds are summarized in Figs F. 10 and Ell. 

Fig. 7.14 According to the Maxwell distribution of speeds Further information 7.1), as the temperature increases, so does the fraction of gas phase molecules with a speed that exceeds a minimum value. Because the kinetic energy is proportional to the square of the speed, it follows that more molecules can collide with a minimum kinetic energy Emin = Ea (the activation energy) at higher temperatures.

Let V represent a particle s speed, N the total number of particles,/(v) the Maxwell distribution of speeds, A a finite change, and d an infinitesimal change not density) in the following relationships. 

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