# Lect08 - Applying Boltzmann Statistics Atmospheres Heat...

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Physics 213: Lecture 8, Pg 1 Applying Boltzmann Statistics: Applying Boltzmann Statistics: Atmospheres, Heat Capacities, EM Radiation Atmospheres, Heat Capacities, EM Radiation Lecture 8 Lecture 8 Re fe re nc e  fo r this   Le c ture : Ele m e nts   C h 9 Re fe re nc e  fo r Le c ture  9:         Ele m e nts   C h 10 Law of atmospheres Simple Harmonic Oscillators: C V of molecules – for real! Planck Distribution of Electromagnetic Radiation Global Warming

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Physics 213: Lecture 8, Pg 2 Last time: Boltzmann Distribution Last time: Boltzmann Distribution If we have a system that is coupled to a heat reservoir at temperature T, we saw last time that the entropy of the reservoir decreases when the small system extracts energy E n from it. Therefore, this will be less likely (fewer microstates). The probability for the small system to be in a particular state with energy E n is given by the Boltzmann factor: / / n E kT n P e Z - = with n / 1 so Z n n E kT n P e - = = Z is known as the “partition function”.
Physics 213: Lecture 8, Pg 3 The Boltzmann atmosphere The Boltzmann atmosphere How does atmospheric pressure vary with height? 0 Pressure p(h) p(0) h Earth’s surface p=(N/V)kT in equilibrium, how would T vary with height? a) increase b) decrease c) constant The ratio of probabilities gives the ratio of densities, which gives the ratio of pressures (for fixed T), i.e., ( 29 ( 29 kT mgh e h kT p V N / ; - = = 0 ρ ( 29 ( 29 kT mgh e p h p / 0 - = kT mgh e P h P / ) ( ) ( - = 0 For every state of motion for a molecule at sea level, there’s one up at height h that’s identical except for position. Their energies are the same except for mgh. The chance of finding a molecule in those states is just proportional to the Boltzmann factor. (We can ignore the state with no molecules, and for sparse atmospheres we can ignore all states with more than one molecule present.)

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Physics 213: Lecture 8, Pg 4 Boltzmann atmosphere Boltzmann atmosphere T = 227K 0 10 20 30 40 50 h (km ) 1.0 0.1 0.01 0.001 p(h)/p(0) Actual data from rocket flights (from Kittel, Thermal Physics) We can define a characteristic height by mg kT h where e e p h p c h h kT mgh c / ) ( ) ( / / = - - 0 From this log graph h c 7 km is the height at which the atmospheric pressure drops by a factor of 1/e 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 <h>=kT/mg mgh/kT p(h)/p(0)
Physics 213: Lecture 8, Pg 5 Act 1 Act 1 What is the ratio of atmospheric pressure in Denver (elevation 1 mi) to that at sea level? In other words, why is it wrong to play baseball in Coors Stadium? (Assume an atmosphere of N 2 .) K T K J x k molecule g x mol molecules x mol g N weight molecular m A 273 / 10 38 . 1 / 10 7 . 4 / 10 022 . 6 / 28 23 23 23 = = = = = - - a) 1.00 b) 1.22 c) 0.82

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Physics 213: Lecture 8, Pg 6 Law of Atmospheres- Reprise Law of Atmospheres- Reprise We have now QUANTITATIVELY answered many of the questions that arose earlier in the course:
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Lect08 - Applying Boltzmann Statistics Atmospheres Heat...

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