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Kinetic Theory of Gases

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In the laboratory, we can use experimental observations to macroscopically give equations, such as PV = nRT. While this formula here gives us details of what's happening to gases as we change the pressure, volume, number of moles, or temperature, it doesn't give us a view of what's happening at the molecular level. So what we have to do is take this equation and go little bit farther and look at what's happening at the molecular level, and the best way to do this is to use computer simulations. So in this demonstration, we're going to take a look at a computer simulation, which is going to allow us to view the Maxwell Boltzmann distribution. Shown here is a molecular simulation program developed by Professor Shirts at Brigham Young University. The red dots represent atoms in the gas state moving around inside a two dimensional container. What is the first thing we notice? The atoms are continually colliding with the walls of the container and with each other. We also notice that the atoms are moving at different velocities. The velocities of the atoms change as they hit other atoms and transfer energy between each other. The model of a gas is moving particles is called the kinetic molecular theory. This idea of atoms in motion can also be applied to liquids and solids. In the upper left-hand corner of the simulation window, we can see that the simulation graphs the distribution of velocities for the atoms. The graph shows the number of atoms moving at a certain velocity as a function of all possible velocities. The bars represent an instantaneous calculation of the number of atoms with different velocities at any given moment in time. The smooth line represents what is called a Maxwell Boltzmann distribution, which is a theoretical prediction of what the distribution should look like. Since this simulation only has 100 atoms, the calculated velocity distribution fluctuates and does not match up exactly to the Maxwell Boltzmann distribution. However, we can still make some interesting observations. First, we note that most atoms have velocities grouped in the middle around the maximum in the Maxwell Boltzmann distribution. The average velocity of all the particles would be near this maximum. Second, we note that there are very few atoms moving very slowly or very quickly. Lastly, we note that the Maxwell Boltzmann distribution function is not symmetrical. That is, the distribution function has a tail that extends to high velocities, which suggests that in a gas, a few particles can be moving at very high velocities, but the bulk of the particles have velocities around the maximum in the Maxwell Boltzmann curve. As we go back to watching the particles move around in the box, we observed that we can now create a microscopic interpretation of pressure. We can see that the atoms or particles often strike the walls of the container. When a particle hits the wall, it transfers momentum or exerts a force on the wall. Thus, the pressure of the gas is a result of the cumulative impacts of all the gas particles striking the walls of the container. Now let's see what happens when we increase the particles in the container. As we increase the particles in the container, what do you predict will happen? If we change the number of particles from 100 to 200, we then have more particles hitting the wall, and thus, have an increase in pressure. If we reduce the number of particles back down to 100, we will see a decrease in pressure. This is just another manifestation of Avogadro's law. Now let's see what happens and we change the temperature. What do you predict will happen to the particles if we lower the temperature? If we lower the temperature of the gas from T equals 358 down to T equals 10 K, we note that the pressure of the gas has decreased, but the change isn't a result of changing the number of particles, but because the overall velocity of the particles has decreased. We note that the range of speeds has not changed, but the average velocity has decreased significantly. The slower particles strike the walls with less force, and so the pressure is lower. Based on what we just saw in the simulation, we verified the Maxwell Boltzmann distribution. This can be summarized using the Maxwell Boltzmann equation, which tells us that the average velocity is equal to the square root of 3RT divided by M. So the velocity of the particles is then proportional to the square root of the temperature and inversely proportional to that of the molar mass, which was demonstrated in the simulation.