Speed distribution, Maxwell

Maxwell distribution A relation describing the way in which molecular speeds or energies are shared among gas molecules, 121... 

S FIGURE 4.27 The Maxwell distribution t again, but now the curves correspond to the speeds of a single substance inf molar fyg mass 50 g-mol 1) at different... 

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

J I I Describe the effect of molar mass and temperature on the Maxwell distribution of molecular speeds (Section 4.11). 

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 distribution of molecular speeds The formula for calculating the percentage of molecules that move at any given speed in a gas at a specified temperature. 

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

Figure 1. The Maxwell-Boltzmctnn speed distribution for N2 at 300 K illustrating a calculation described in the text.

Exercise. Compute from (4.14) the jump moments, taking for F the Maxwell distribution. Show that (4.1) and (4.2) hold when V is small compared to the average speed of the gas molecules, and can therefore be used to describe equilibrium fluctuations if Mpm. 

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). 

FIGURE 4.27 The range of molecular speeds for several gases, as given by the Maxwell distribution. All the curves correspond to the same temperature. The greater the molar mass, the lower the average speed and the narrower the spread of speeds. 

The molecules of all gases have a wide range of speeds. As the temperature increases, the root mean square speed and the range of speeds both increase. The details of the range of speeds are described by the Maxwell distribution, Eq. 28. 

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 function f M) was predicted theoretically by Maxwell and Boltzmann about 60 years before it was first measured. It is called the Maxwell-Boltzmann speed distribution for a gas of molecules of mass m at temperature T and it has the following form ... 

An alternative interpretation of the Maxwell-Boltzmann speed distribution is helpful in statistical analysis of the experiment. Experimentally, the probability that a molecule selected from the gas will have speed in the range Au is defined as the fraction AN/N discussed earlier. Because AN/N is equal to f u) Au, we interpret this product as the probability predicted from theory that any molecule selected from the gas will have speed between u and u + Au. In this way we think of the Maxwell-Boltzmann speed distribntion f(u) as a probability distribution. It is necessary to restrict Au to very small ranges compared with u to make sure the probability distribution is a continuous function of u. An elementary introdnction to probability distributions and their applications is given in Appendix C.6. We suggest you review that material now. 

FIGURE 9.15 Mathematical form of the Maxwell-Boltzmann speed distribution. The factor cuts off the distribution at small values of u, whereas the exponential factor causes it to die off at large values of u. The competition between these effects causes the distribution to achieve its maximum value at intermediate values of u. 

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

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