The Maxwell-Boltzmann Distribution

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1 The Maxwell-Boltzmann Distribution Gases are composed of atoms or molecules. These atoms or molecules do not really interact with each other except through collisions. In many cases, we may think of a gas as a collection of tiny billiard balls flying through space, hitting one another again and again. Even if we were to think that all atoms or molecules had the same speed to begin with (although we do not), the constant collisions would result in a spread of many speeds. Some atoms could have very high speeds, others low ones. In real gases at equilibrium there is a distribution of speeds (Fig. E07.4.1). This distribution is called the Maxwell-Boltzmann distribution, and it depends on temperature, as shown. The high temperature curve has proportionally many more fast molecules or atoms than the low temperature curves. As the temperature rises, the highest point on the curve is pushed out to higher v, and the maximum is pushed down toward the axis. All curves shown in Fig. E have a similarity in their shape. Fig. E The number of particles at speed v varies with the absolute temperature, as the shape of the curve changes. However, the distribution really has the same profile; the distribution is pushed to the left and upward as temperature decreases, and is pushed to the right and down as the temperature increases.

2 Energy, Ch. 7, extension 4 The Maxwell-Boltzmann Distribution 2 It is also possible to show that the average kinetic energy of an atom or molecule, < 1 2 mv2 >, is directly proportional to the absolute temperature T. For example, for a gas of individual atoms such as neon, < 1 2 mv2 > = 3 2 kt, where m is the atomic mass and k is a constant known as Boltzmann s constant. The average of the square of the speed increases with temperature, just as is seen in the curves. We can actually experience the effects of the Maxwell-Boltzmann distribution by performing a few evaporation experiments. What is evaporation, and why does it require energy? Recall that the atoms or molecules of all materials are in constant motion. Materials of course differ. We are generally familiar with three states, or phases, of material: solid, liquid, and gas. To understand evaporation, we must look at the differences among the three phases. In solids, the molecules are cemented into their relative positions despite their incessant motion. There is a structure that is maintained. We are familiar with many solids that have crystalline structure. The symmetry of the crystal reflects the symmetry of the bonds between the atoms or molecules composing the crystal. Most solids do not exhibit such a clearly ordered structure, but they are structured nevertheless. In liquids, the interatomic or intermolecular forces attracting the atoms or molecules to each other are weaker than those in solids. This allows the atoms or molecules to move somewhat in relation to one another, while still keeping the liquid together. This weakened interatomic force is the reason that liquids can take on the shapes of their containers. The attractive force between liquid molecules is evidenced by the phenomenon of surface tension. The beading of water, for example, is due to surface tension. Inside a

3 Energy, Ch. 7, extension 4 The Maxwell-Boltzmann Distribution 3 liquid, and away from the boundary, molecules are surrounded by other molecules of the same type and are thus attracted in all directions equally. This is another way of saying that there is no net force at all on molecules inside the liquid; those molecules are in equilibrium. For the molecules on or very near the liquid surface, however, attractive forces come only from inside the liquid to act on them. Thus, they are attracted back to the liquid surface should they try to depart. In gases, such as water vapor, the molecules are essentially free to move wherever they are going. That is, the attractive forces are essentially negligible. There is no surface tension. In all these phases, molecules are dancing about in some way. In the solid, the molecules vibrate in various ways about their fixed positions in the solid lattice. In liquids, this dance of the molecules also occurs. Even in transparent liquids, we are not able to see the motion of the liquid molecules. In some liquids that contain very large molecules, the large molecules can be kicked about randomly under the impact of the smaller, invisible liquid molecules. If the molecules are large enough to be seen, we can then see the effect of the random motion on the larger molecules. This visible motion is termed Brownian motion. The properties of materials at temperatures above absolute zero that is, those materials exhibiting the molecular motion are described by distributions giving the relative numbers of molecules at the different speeds of motion possible for the material. The Maxwell-Boltzmann distribution of Fig. E07.1 a illustrates the properties of the distribution of speeds in typical gases or liquids: there is a range of speeds of molecules, and there is a most probable speed at a given temperature. Thus, in any gas or liquid, there are many molecules going fast and many going slow. In fact, the distribution shows

4 Energy, Ch. 7, extension 4 The Maxwell-Boltzmann Distribution 4 that we can measure the temperature by determining the average molecular speed (see Fig. E07.1 b). Molecules approaching the boundary of a liquid generally are trapped back into the liquid by the surface tension. The very fast molecules may be able to penetrate the surface barrier and escape the liquid altogether. The more surface available, of course, the faster the rate of escape. The rate also increases as the temperature increases because of the boosting of molecules to higher speeds. Fig. E This illustrates what happens to a distribution when almost all the faster particles are suddenly removed. The effect is a cooling, since temperature is a measure of average thermal energy. We may distinguish two sorts of evaporation. In liquids with a relatively small surface area, the liquid absorbs heat from the surroundings to keep the temperature of the liquid fixed as the faster molecules escape. The absorbed heat speeds up some molecules, restoring the original Maxwell-Boltzmann distribution. The total number of molecules in the liquid decreases, but the distribution which depends only on temperature remains the same. In the second sort of evaporation, the surface area is relatively large. Thus, the faster molecules leave the surface rapidly enough that the liquid cannot absorb enough heat to restore the original distribution. In this case of non-equilibrium evaporation, the distribution quickly becomes truncated. (It looks as shown in Fig. E07.4.2). Since the absolute temperature measures the average kinetic energy of a gas, a departure of high-

5 Energy, Ch. 7, extension 4 The Maxwell-Boltzmann Distribution 5 kinetic-energy molecules, shown in Fig. E07.4.2, will cause a thermometer to record a lower temperature. This effect is enhanced when the material in question has a smaller surface tension. For example, the effect is much greater for alcohol than for water. Conversely, the temperature drop is much smaller for a liquid with a large surface tension. Lubricating oil exhibits a tiny temperature drop compared to water. Gasoline spilled on our hands makes the skin feel cool because the gasoline molecules evaporate rapidly. This effect can easily be seen with tap water, a paper towel, and a thermometer able to register temperatures around 10 C to 20 C. Wet the towel and wrap it around the bulb of the thermometer. You should be able to observe a decrease in temperature of a few degrees. The Maxwell-Boltzmann distribution can also be used to explain how the hydrogen and helium originally in Earth s atmosphere disappeared, while the nitrogen and oxygen show little sign of imminent departure. Recall that for a gas of individual atoms < 1 2 mv2 > = 3 2 kt. For a gas of molecules such as oxygen and nitrogen, which have two atoms in a molecule, a similar relation, < 1 2 mv2 > = 5 2 kt, holds. The atmosphere s temperature changes little on average from year to year; for a fixed T, the smaller m is, the larger < v 2 > will be. The square root of < v 2 >, called the root-mean-square speed (v rms ) is a measure of how fast the atoms or molecules are whizzing around. The root-mean-square speed for hydrogen is 14 times as great as that for nitrogen because the mass of the nitrogen molecule is fourteen times that of hydrogen. Similarly, the root-mean-square speed of helium is 7 times as great as that for nitrogen because the mass of a nitrogen molecule is seven times that of a helium atom. In fact, the

6 Energy, Ch. 7, extension 4 The Maxwell-Boltzmann Distribution 6 root-mean-square speeds in Earth s atmosphere of hydrogen, helium, nitrogen, and oxygen are, respectively, 1.93 km/s, 1.37 km/s, 0.52 km/s, and 0.48 km/s. Escape speed is the speed an object has to go to escape Earth s gravity. Escape speed for Earth is only 11.2 km/s. A rocket ship to Mars must go at a speed greater than 11.2 km/s to escape from Earth. If a gas molecule has a speed in excess of 11.2 km/s, it will escape totally from Earth. Hydrogen and helium have a mean speed that is a significant fraction of the escape speed. For this reason, there is almost no hydrogen or helium in Earth s present atmosphere. Because of the rapid (exponential) decrease in number of particles with increasing speed, the distribution tells us that there are proportionally about a million times more hydrogen molecules with speeds exceeding 11.2 km/s than nitrogen molecules. Because of the large volume of the atmosphere, the loss of hydrogen molecules proceeds in a quasi-equilibrium fashion, with slower molecules gaining energy and the distribution preserving its shape. After a geologically short time, almost all hydrogen will have escaped, while practically no nitrogen will have been lost.

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