Physics 111. Lecture 34 (Walker 17.2,17.4-5) Kinetic Theory of Gases Phases of Matter Latent Heat

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1 Physics 111 Lecture 34 (Walker 17.2,17.4-5) Kinetic Theory of Gases Phases of Matter Latent Heat Dec. 7, 2009 Kinetic Theory Pressure is the result of collisions between gas molecules and walls of container. It depends on the mass and speed of the molecules, and on the container size: Lecture 34 1/22 Lecture 34 2/22 Kinetic Theory* Not all molecules in a gas will have the same speed; their speeds are represented by the Maxwell distribution, and depend on the temperature and mass of the molecules. Kinetic Theory* We replace the speed in the previous expression for pressure with the average speed: Including the other two directions and considering N molecules instead of just one, Therefore, pressure in a gas is proportional to average kinetic energy of its molecules. Lecture 34 3/22 Lecture 34 4/22

2 Kinetic Theory* Comparing this expression with the ideal gas law allows us to relate average internal kinetic energy and temperature: Kinetic Theory* Solving for the rms speed gives: The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas. The square root of is called the root mean square (rms) speed. Lecture 34 5/22 Lecture 34 6/22 Example What is the RMS speed of atoms of helium gas at T = 300K? mass of He atom = (4g/mol)/(6.02x10 23 /mol) = 6.6x10-27 kg Total Internal Kinetic Energy - Ideal Gas The internal kinetic energy of an ideal gas is the sum of the kinetic energies of all its molecules. In the case where each molecule consists of a single atom, this may be written: = (3/2)nRT 23 3(1.38x10 J / K)(300K) = = 1370 m/s 6.6x10 kg v RMS 27 Lecture 34 7/22 Since there is no internal potential energy in an ideal gas, the total internal energy U is just the internal kinetic energy. Lecture 34 8/22

3 Example What is the total internal kinetic energy of a mole of monatomic gas at T=300K? U = (3/2)nRT = (3/2)(8.31J/mol-K)(300K) = 3740 J This much energy would have to be removed from the gas to cool it to absolute zero. Lecture 34 9/22 Internal Energy U: The internal energy U of an object is the total of all the random kinetic energies and potential energies of all the atoms in the object. The amount of internal kinetic energy depends on temperature, the type of material, and how much material there is. has more internal kinetic energy than The internal potential energy depends on the amount of material, the phase (state) of the material, and maybe on volume and pressure. Lecture 34 10/22 The Three Basic Phases of Matter Solid Liquid Gas If a liquid is put into a sealed container so that there is a vacuum above it, some of the molecules in the liquid will vaporize. Once a sufficient number have done so, some will begin to condense back into the liquid. Equilibrium is reached when the numbers remain constant. Sequence of increasing molecule motion (and energy) Lecture 34 11/22 Lecture 34 12/22

4 The pressure of the gas when it is in equilibrium with the liquid is called the equilibrium vapor pressure, and will depend on the temperature. The vaporization curve determines the boiling point of a liquid: A liquid boils at the temperature at which its vapor pressure equals the external pressure. This explains why water boils at a lower temperature at lower pressure and why you should never insist on a 3-minute egg in Denver! Lecture 34 13/22 Lecture 34 14/22 A liquid in a closed container will come to equilibrium with its vapor. However, an open liquid will not, as its vapor keeps escaping it will continue to vaporize without reaching equilibrium. As the molecules that escape from the liquid are the higher-energy ones, this has the effect of cooling the liquid. This is why sweating cools us off. If we look at the Maxwell speed distributions for water at different temperatures, we see that there is not much difference between the 30 C curve and the 100 C curve. This means that, if 100 C water molecules can escape, many 30 C molecules can also. Lecture 34 15/22 Lecture 34 16/22

5 Latent Heat Energy is required for a material to change phase, even though its temperature is not changing. Instead of raising the temperature, the heat goes into changing the phase of the material melting ice, for example. Temp ( C) Latent Heats The heat required to convert from one phase to another is called the latent heat. The latent heat, L, is the heat that must be added to or removed from one kilogram of a substance to convert it from one phase to another. During the conversion process, the temperature of the system remains constant. Heat Added Lecture 34 18/22 Latent Heat Heat of fusion, L F : heat (in J) required to change 1.0 kg of material from solid to liquid Heat of vaporization, L V : heat (in J) required to change 1.0 kg of material from liquid to vapor Latent Heats For melting a mass m of material that it already at the melting temperature, the heat required is Q = ml F For vaporizing a mass m of material that it already at the boiling point, the heat required is Q = ml V Lecture 34 20/22

6 Example How much heat must be removed from 2 kg of water at 0 C in order to freeze it? Q = ml F = (2 kg)(334 kj/kg) = 668 kj End of Lecture 34 For Wed., Dec. 9, read Walker , Homework Assignment 17a is due at 11:00 PM on Thursday, Dec. 10. During the time the water is freezing, the temperature will stay at 0 C. Lecture 34 21/22 Lecture 34 22/22