Distribution of speeds

For a sample at diennal equilibrium there is a distribution of speeds which depends on the mass of the molecules and on the temperature according to the Boltzmaim distribution. This results in a line shape of the form... 

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... 

When the speed profile of a gas is measured in this way, the results give a distribution like the one shown in Figure 5-lb. If all the molecules traveled at the same speed, they would reach the detector at the same time, in a single clump. Instead, faster molecules move ahead of the main packet, and slower molecules fall behind. This experiment shows that molecules in a gas have a distribution of speeds. 

Distribution of Speeds in Gas— Maxwell-Boltzmann Distribution (Rae D6jur)... 

Equation 10.19 is the distribution of speeds (which are always positive) in one dimension. However, negative and positive x velocities are equally likely, so Eq. 10.19 only considered... 

Figure 1.1a shows the Gaussian function. The Lorentzian shape is similar to the Gaussian, but falls off more slowly. The Doppler shift of radiation from an emitting molecule is proportional to its velocity component in the direction of observation. The one-dimensional distribution of speeds in a gas is a Gaussian function. (See any physical-chemistry text.) Hence when Doppler broadening is dominant, we get a Gaussian-shaped line. 

In section 2 the distribution of speeds among the molecules of a gas was discussed. According to Maxwell s law the fractional number of molecules possessing energies corresponding to velocity components between u and u + du is given by the expression... 

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... 

Just because the average speed of helium atoms at 298 K is 1.36 km/s doesn t mean that all helium atoms are moving at that speed or that a given atom will travel from Maine to California in one hour. As shown in Figure 9.12, there is a broad distribution of speeds among particles in a gas, a distribution that flattens out and moves higher as the temperature increases. Furthermore, an individual gas particle is likely to travel only a very short distance before it collides with another particle and bounces off in a different direction. Thus, the actual path followed by a gas particle is a random zigzag. 

Maxwell-Boltzmann distribution of speeds for H2 at 100, 400, and 900 K. Note that ump increases linearly with Tl/2. 

Gas molecules are constantly moving, with a random distribution of directions and speeds. This is also a very reasonable assumption, unless the molecules in the gas were prepared in some way which (for example) made all of them move at nearly the same speed—and even then, collisions with the walls would randomize the distribution in practice. This distribution of speeds will be found using the Boltzmann distribution we discussed in Chapter 4. 

Often we are not interested in the distribution of speeds of the product molecules, so we may accordingly multiply both sides of the equation by dvc and integrate over the speed ... 

In a gas of atoms at finite temperature, the atoms move according to the Maxwell38-Boltzmann39 distribution of speeds, which collectively cause a Doppler broadening A/., /2,Soppier that is typically two orders of magnitude greater than the natural linewidth A/l1/2 ae... 

Under great volume consumption rates of melts in the same experiments, an interesting effect was observed the extrudate became narrowed at the channel slit size 1.9 x 9.9 mm. The extrudate was 2 mm thick but only 5.5 + 0.5 mm wide. Reduction of the extrudate cross-section means transfer to a multilayer flow. It can be assumed that this phenomenon is caused by the existence of temperature cross-gradients along the channel section. This leads to a non-linear distribution of speed gradients and tensions over the channel section and along the flow axis, that in some places conditions for the melts-cohesion disintegration exist (see Sect. 4.4). 

The gas molecules are constantly moving in random directions with a distribution of speeds. 

Molecules are assumed to be widely separated point masses that interact only during collisions, move randomly with a distribution of speeds, and experience elastic collisions with walls... 

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 graph in Figure 10.12 shows the distribution of speeds of the particles in a container of gas. The most common speed (kinetic energy) of the particles is represented by the peak near the center of the graph. According to the kinetic theory, the temperature of a material is a measure of the average kinetic energy of the particles in the material. For example. 

The graph below shows the distribution of speeds of the particles of a gas at two different temperatures. Which curve, A or B, corresponds to the higher of the two temperatures Explain your reasoning... 

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