Lecture 33: Kinetic Gas Theory - Ideal Gas Law - Isothermal Atmosphere - Phase Diagrams - Phase Transitions
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1. Intro to Ideal-Gas Law and Avogadro's Number:
Liquids are near incompressible, but gases are not; the density of gases can be increased with relative ease by increasing the pressure, but that is not the case for liquids. The ideal-gas law is introduced and explained; it is a good approximation for the compressibility of most gases. Avogadro's number is the number of molecules/mole, and defined as the number of atoms in 12 grams of carbon12.
2. Ideal-Gas Law Insights:
The ideal-gas law predicts that the volume of a mole of gas for a given temperature and pressure is independent of the molecular mass of the gas. The momentum transfer per second from the gas molecules to the vessel walls is proportional to mv^2 (which is reminiscent of the kinetic energy of the molecules). This is proportional to the gas pressure. If two different gases have the same temperature, the molecules must have the same average translational kinetic energy. Thus the molecules with the lowest mass must have, on average, the highest speed.
3. Ideal-Gas Law Experimentally Applied:
A demonstration of the ideal-gas law uses a pressure gauge that measures overpressure (the pressure in excess of the atmospheric pressure). The temperature of a fixed number of air molecules in a fixed volume is increased from melting ice temperature (273 K, pressure=1atm) to boiling water temperature (373 K). The resulting pressure increase is measured.
4. Phase Diagrams and Phase Transitions:
Introduction to phase diagrams and phase transitions. Taking a gas at a constant temperature, and using a piston to increase its pressure, the gas volume decreases as the pressure increases until you approach the gas->liquid phase transition. At constant pressure of one atmosphere, but with increasing temperature, you start with ice at low temperature, which becomes liquid water at 273 K, and the water will boil at 373 K, and it will become water vapor (gas) above this temperature.
5. A Fire Extinguisher:
A fire extinguisher is filled with CO2. Given the dimensions of the tank (i.e. its volume), room temperature (293 K), the mass of CO2 in the extinguisher (from the label), and the ideal-gas law (this law is only valid if there is ONLY gas inside and NO liquid), the pressure is calculated inside the cylinder. It is concluded that it can't be just gas, there must also be liquid CO2 in the fire extinguisher. The phase diagram for C02 shows a phase transition at 60 atm at 293 K; the CO2 gas and liquid would co-exist in thermal equilibrium at room temperature and 60 atm. If you tried to further compress at room temperature, more gas would turn into liquid but the pressure would remain at 60 atm until all the gas had turned into liquid (after which the pressure can increase).
6. Boiling Water - Part 1:
In Lecture 27 there was discussion of hydrostatic pressure, the overpressure submarines must survive as they go deeper in the water. Conversely the atmospheric (barometric) pressure should decrease with increasing altitude, but with a different height dependence (because air is compressible, its density changes with pressure). A differential equation is solved to determine that in an isothermal atmosphere the pressure decreases exponentially with altitude. Consequently the boiling point of water decreases with altitude. A demo of water boiling at room temperature but low pressure is shown.
7. Boiling Water - Part 2:
While waiting for the pressure in the bell jar to decrease, Professor Lewin starts a second demo, boiling the water (373 K and 1 atm) in a can so the air in the can gets displaced by water vapor (he then seals the can, and lets it cool). Inside the can is liquid water and water vapor in thermal equilibrium, as the can cools the vapor condenses into liquid, and the pressure in the can decreases; the pressure in the can should drop to about 17 mm Hg (about 0.02 atm) as the can cools to room temperature. The can implodes due to the external force of the atmospheric pressure. Meanwhile, the water in the bell jar boils at room temperature! A third demo involves air-filled balloons that shrink much more than naively predicted using the ideal-gas law. What's going on?
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