The Maxwell Distribution of Speeds

Substitution into eqn 7.39 of the molar mass of Oj (32.0 gmoL ) and a temperature corresponding to 25°C (that is, 298 K) gives an r.m.s. speed for these molecules of 482 m s. The same calculation for nitrogen molecules gives 515 m s.  

Although eqn 7.40 looks complicated, its features can be picked out quite readily  

Because/is proportional to the range of speeds As, we see that the fraction in the range As increases in proportion to the width of the range. If at a given speed we double the range of interest (but still ensure that it is narrow), then the fraction of molecules in that range doubles too. 

The factor M/2RT multiplying s in the exponent is large when the molar mass, M, is large, so the exponential factor goes most rapidly towards zero when M is large. That teUs us that heavy molecules are unlikely to be found with very high speeds. 

The opposite is true when the temperature, T, is high then the factor M/2RT in the exponent is small, so the etqjonential factor falls towards zero relatively slowly as s increases. This tells us that at high temperatures, 

---

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... 

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. 

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.

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. 

---