Chem 451 Kinetic Theory of Gases PhET Simulation http://phet.colorado.edu/get_phet/simlauncher.php The discussion in the first lectures centered on the ideal gas equation of state and the modifications to it resulting from both attractie and repulsie particle-particle interactions: Ideal Gas: p RT p RT a b an der Waals: ( more on this later) The pressure was treated as a continuous ariable, with no explicit recognition of the molecular origin of pressure. This was justified implicitly by the idea that the number of particles inoled was enormous, and that smooth, continuous behaior was a ery good approximation. In the last class, we examined the kinetic modeling of an ideal gas, and deried the ideal gas equation of state for particles with mass M. Assumptions: No particle particle interactions. Elastic collisions with the container walls We calculated the momentum transfer to the wall from a single collision, and related the momentum to force and then to pressure. By adding up the pressure contributions from all wall collisions in some unit of time, we obtained an expression that looked ery similar to the Ideal Gas equation of state: ( x + y + z ) M M p where refers to the aerage alue. We thus related the temperature to the aerage energy of the particles. In class we deried expressions for the mean speed,, and the mean squared speed, for the Maxwell distribution of speeds. 8kT π m and 3 kt m. Notice that for this distribution. We should keep this in mind as we utilize the PhEt simulation to explore gas behaior. Now the PhEt kinetic theory simulation, aailable by selecting gas properties from simulations listed at 1
PhET Kinetic Theory Simulation http://phet.colorado.edu/get_phet/simlauncher.php enables us to examine and extend these concepts. W e will use it to make predictions and to see how the graininess of collisions smoothes out with increasing numbers of particles. We will also see the effect of particle particle elastic collisions. The assumptions in the simulation are as follows: Hard sphere molecules Elastic collisions with walls Molecule-molecule collisions are also hard sphere elastic, and can be turned on and off. Heating or cooling the particles raises or lowers all speeds proportionately No heat is transmitted through the walls, except as aboe. The simulation is two-dimensional; ie, z z 0. Extra Problem on First Problem set You should work together in groups on this problem. You may turn in one solution with all three names on it. You can access the PhET website at the address gien aboe. You might, howeer, hae to install the free JAA webstart on your computer. Alternatiely, the computers just outside of the Math Physics Library are aailable and set up to handle PhET. If you are haing difficulty, contact either Carl Lineberger or Jesse Marcum. 1. There are no units gien in the simulation. Howeer, the temperature and speed scales are about correct for the heay particle haing mass 8 amu, with the temperature in K and the speed in meters/sec. Assuming that the display is running in real time, what is the approximate width of the box?. Place 1 and 100 heay particles in the box with interactions turned off. Deduce the following for each case. How do the mean speed and temperature ary with time? How does pressure ary with time? Why? 3. Place 1 and 100 heay particles in the box with interactions turned on. Deduce the following for each case. How do the mean speed and temperature ary with time? How does pressure ary with time? Why? 4. Obtain 100 particles in the box with a temperature near room temperature and randomized trajectories. Turn off the interactions. Using this ensemble, inestigate the dependence of p, and P with T. What is an appropriate equation of state for this gas? 5. What is the approximate mass of the light particle, assuming that the heay one has mass 8 amu?
6. Obtain 00 particles in the box with a temperature near room temperature and randomized trajectories. Leae the interactions on. Using this ensemble, inestigate the dependence of p, and P with T. Plot your results and see what conclusions you can obtain. In this case, is really the area of the box for the two-dimensional gas. Use the ruler to obtain. 7. We see that the pressure and aerage speed bounce around a lot. At fixed temperature, how does the amplitude of the fluctuations scale with the number of particles in the box? 3
Group 8. Place a small number of particles in the box. Moe the left wall of the box in and out. Do so with and without allowing particles to collide with the moing wall. What is the difference in the gas properties in these two circumstances? 9. In the real world, the speed of the gas particles is much greater than that of the piston, and the direct kinetic energy transfer from the piston to the gas is not significant. Under this ery slow motion condition, the there is no heat transfer into or out of the gas, and the process is referred to as being adiabatic. The motion of much of our atmosphere is quasi adiabatic, with indiidual parcels of the atmosphere behaing nearly adiabatically. In a Colorado windstorm, parcels of air from the continental diide (p 0.6 atm) descend rapidly into Boulder (p0.8 atm). Assuming that the air on the diide is pretty cold (-0º), what would be the thermal effect of a windstorm in Boulder? Group 3 10. Turn on the center of mass markers. Inject some 100 particles without molecule interactions. Allow time for the distribution to appear relatiely uniform spatially. At this point, turn on molecular interactions. Estimate how long it takes for the energy and speed distribution to become stable. How long does it take for the spatial distribution to become uniform? Are the times different? Why? Do any of the following make a difference in the times? a) particle particle interactions b) number of particles c) temperature d) mass of the particles You may want to use a watch to estimate the equilibration times. Plot the results. Can we draw any general conclusions? 11. Let us look at the same set up to study particle diffusion. Equilibrate 300 heay particles at, say 300ºK. Inject about 10 light particles. Using the cm markers, how long does it take the light particles to diffuse and become uniformly mixed? Use a watch to estimate the equilibration time. How does this time scale with the initial number of particles (aried from 10 to 300) and initial temperature (aried from 1 to 500 K)? Plot these results and see if you can draw any conclusions/ 1. We want to try to look at the approach to equilibrium in another manner. With interactions turned off, inject approximately equal numbers (say 75) of heay and light particles. Note the energy and speed distributions after each injection. Turn on interactions. What happens to the distributions? Are they different in shape? Mean? Which one(s) are fastest to approach their equilibrium? What is driing the system to equilibrium? 4
Group 4 13. Let s look at the process of eaporation. Place 00 particles in the box and equilibrate them at 300 K. Open the coer and record the number of particles remaining and the temperature as a function of time. Plot the results and deduce a scaling law. Does it make sense? What is the effect of initial temperature? What is the importance of moleculemolecule interactions? 14. Let s now do this with graity turned on. What conclusions do you reach about the cooling process here? Group 5 15. Let s look at diffusion in another fashion. Equilibrate 400 heay molecules at 300 K with collisions on. Inject 10 light particles and estimate the time required for the light particles to reach an equilibrium spatial distribution and equilibrium speed distribution? Are they the same? Why or why not? How do these times scale with the number of initial heay particles? 16. Finally equilibrate 500 heay particles in the box, and then cool them to near absolute zero. (Easy in a simulation, but hard in the real world!) Now add perhaps 5 light particles and again estimate the time required to diffuse halfway across the box. What if we added 10 particles? 300 particles? 5