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Lect03 - Le cture3 Exam s and Proble s ple m Mechanic...

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Lecture 3, p 1 Mechanic & thermodynamics Equipartition First Law of Thermodynamics Ideal gases Isothermal and adiabatic processes Lecture3 Examples and Problems
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Lecture 3, p 2 Consider a fixed volume of an ideal gas. Because pV = NkT, if you double either T or N, p goes up by a factor of 2. If you double N, how many times as often will a particular molecule hit the container walls? A) x1 B) x1.4 C) x2 D) x4 ACT 1: Ideal gas behavior
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Lecture 3, p 3 Consider a fixed volume of an ideal gas. Because pV = NkT, if you double either T or N, p goes up by a factor of 2. If you double N, how many times as often will a particular molecule hit the container walls? A) x1 B) x1.4 C) x2 D) x4 In an ideal gas, the molecules are non-interacting except for occasional elastic collisions, so the motion of an individual molecule does not depend on the others. The total collision rate is proportional to N, but the rate per molecule is independent of N. Solution
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Lecture 3, p 4 Dalton’s Law of Partial Pressures for Ideal Gases In the derivation of pV=NkT, we never assumed that the molecules were all the same. All we used was equipartition and that they don’t interact much. So, N is the total number of molecules in the gas, independent of type. Because p is proportional to N, if a gas has multiple components, the total pressure is the sum of the individual partial pressures: p total = p 1 + p 2 + p 3 +… , where p i = N i kT / V Example: Air is 78% N 2 (by number of molecules, not by mass), so the partial Pressure of the N is 0.78 atmospheres.
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Lecture 3, p 5 Internal Energy of Ideal Gases Dalton’s law does not hold for internal energy, because the energy per molecule does depend on the type. The energy per molecule is proportional to the number of energy modes. Example: Consider a 10 liter gas bottle that has one mole each of helium and nitrogen gas, at room temperature, T = 295 K. What is the thermal energy of each gas? (R = 8.314 J/mol . K = 0.08206 l.atm/mol . K)
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Lecture 3, p 6 Solution Dalton’s law does not hold for internal energy, because the energy per molecule does depend on the type. The energy per molecule is proportional to the number of energy modes. Example: Consider a 10 liter gas bottle that has one mole each of helium and nitrogen gas, at room temperature, T = 295 K. What is the thermal energy of each gas? (R = 8.314 J/mol . K = 0.08206 l.atm/mol . K) U He = (3/2)nRT = 3679 J U N2 = (5/2)nRT = 6132 J Note: p = nRT / V = 2.42 atm for both gases.
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Lecture 3, p 7 A Quick Probability Problem We’ll spend a lot of time calculating probabilities. Here’s a quick introduction. This lecture room is approximately a cube 15 m on a side. Calculate the probability that all the air molecules will be found in the left half of the room.
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