KMT Distribution of Speed in Gases

KMT: Distribution of Speed in Gases The average kinetic energy of the particles in a gas sample can be calculated from the mass of the gas and the average speed of the gas particles: Mass, Speed, and Average Kinetic Energy of Some Gases at 273 K: H2 He NH3 N2 Ar F2 Gas CO2 Cl2 38 44.01 70.9 423 393 310 3.40 3.41 MW(g/mol) 2.02 4 17.03 28.02 29.95 V (m/s) 1836 1305 632 493 477 3.40 3.41 3.40 3.41 3.41 3.40 KE (kJ/mol) = 1/2 mV2 The “theoretical math part” of KMT states that the average kinetic energy of any gas at some temperature can be calculated by: (KE)avg = (3/2) RT At 273 K (0 C) KE = 3.4 kJ/mol; R is the Universal Gas Constant and = 8.3014 J/(molK) Clearly, there is good agreement between the average values from the table and the theoretical value. What this equation tells us is that the average kinetic energy of gas particles is directly proportional to the absolute temperature. What does “average” mean here? As particles collide, energy can be transferred. In any collection of particles, some will be moving slowly and others quickly. Most, of course, will be moving at some intermediate speed. This situation is shown in the box at left below where the “tail” on each gas particle is proportional to the speed of the gas particle. Even in this “snapshot” mode where the particles aren’t moving, it is hard to see what’s going on. Suppose that we took these particles and made a chart where we lined up the gas particles according to their speed as shown on the right. If we connect the height of each column of particles we can get a better sense of how the different speeds are distributed in the collection. For a gas sample, KMT tells us that the particle speeds in a typical gas sample are distributed like the graph at left below. The graph on the right shows finish times for a typical large marathon te reinforce that these curves are just a different way to keep track of the individual speeds. The distribution graph for a gas sample looks similar to the “Bell” curve (dotted lines) that you might be familiar with, but is asymmetric with fewer items at low values and a trailing off at higher values. The asymmetry of the distribution graph means the average speed of a particle in the collection is slightly higher than the most probable speed that the top of the curve represents. Distribution of Speeds in a Gas Sample Distribution of Finish Times in a Marathon Fraction of Molecules vs Speed 7.00E-03 average speed Fraction of Molecules 6.00E-03 5.00E-03 4.00E-03 3.00E-03 2.00E-03 1.00E-03 0.00E+00 0 100 200 300 400 500 Speed (m/s) What happens to this distribution when we change the temperature? As you increase the temperature, the curve shifts to higher speeds just like you might expect (below left). What also happens is a broadening of the curve. At lower temperatures, collisions between the particles are weaker and tend not to transfer energy around at much. Consequently, a narrow distribution of speeds results. At higher temperature, a larger fraction of the particles in the collection is at a higher energy. Another way to think about why the curves behave this way is to really think about what’s being shown here. Since the number of particles in the collection hasn’t changed, the area under each curve must be the same. Imagine grabbing the top of the lower energy curve and moving it to higher speeds while keeping the start point anchored at zero. Can you see that this would necessarily produce the higher energy curve? Although this point is just a curiosity at this stage, try to remember how the distribution of speeds (and therefore energies) in a collection of molecules changes with temperature. Although the results here are for gases, the ideas are universal and apply to liquids, solids, and solutions. We know that different molecules with different masses have the same average kinetic energy (KE) at the same temperature. But, how does the mass affect the distribution of speeds? As shown below at right, a lighter gas will have broader and flatter distribution than a heavier gas. A small energy transfer for a lighter gas means a proportionately larger change in velocity compared to a heavier gas and the speeds of the lighter gas will be more spread out as a result. Another curiosity, but one that may help shape your thoughts moving forward! Effect of Temperature on Distribution Effect of Mass on Distribution Fraction of Molecules vs Speed Fraction of M ole cules vs Spe ed 7.00E-03 7.00E-03 6.00E-03 lower temperature 5.00E-03 Fraction of Molecules Fraction of Molecules 6.00E-03 4.00E-03 higher temperature 3.00E-03 2.00E-03 1.00E-03 0.00E+00 0 200 400 600 Spe e d (m/s) 800 1000 heavier gas 5.00E-03 4.00E-03 3.00E-03 lighter gas 2.00E-03 1.00E-03 0.00E+00 0 200 400 600 800 1000 Speed (m/s) Final Note: Since the KE of a molecule is proportional to the speed, the distributions of KE’s of particles in the collection are qualitatively similar. The graphs of KE vs. number (or fraction) of particles looks the same and behaves the same way towards changing temperatures or masses.