Lecture 14

# Lecture 14 - Review Ideal Gas Law Chap 19:1-4 The First Law...

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May 22, 2008 Physics 40B Lecture 14 1 For each particle K = 1 2 mv rms 2 = 3 2 k B T For all particles E = K Total = 3 2 nRT E = U " Internal Energy depends only on temperature (no potential energy) Chap. 19:1-4 The First Law of Thermodynamics Review Ideal Gas Law pV = nRT R = 8.31 J/mole K (0.0821 l atm/mole K) Kinetic Theory of Gases: let pV = nRT = Nk B T where N = nN A and k B = R N A k B = 1.38 x 10 " 23 J / K is the Boltzmann's Constant p = nN A m ( ) 3 V v 2 # v rms = v 2 (square root of mean of square) Since pV = nN A m 3 v rms 2 = nRT # v rms = 3 RT N A m = 3 k B T m = 3 RT M Monatomic ideal gas: E = 3/2 nRT, (internal energy of ideal gas only depends on T ) C V = dE/(ndT) = 3/2 R C P = 5/2R γ = C P /C V = 1.67 For diatomic gases , E = 5/2nRT, C V = 5/2 R, C P = 7/2 R, = 7/5 = 1.4

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May 22, 2008 Physics 40B Lecture 14 2 Example: Calculate Speed T = 300 K Hydrogen: v rms = 1920 m/s Nitrogen: v rms = 517 m/s Example: RMS Calculation N = 5, 11, 32, 67, 89 N = 5 + 11 + 32 + 67 + 89 5 = 40.8 N rms = 5 2 + 11 2 + 32 2 + 67 2 + 89 2 5 = 52.1 Oxygen f ( v ) = 4 " m 2 k B T # \$ % ( 3 2 v 2 e ) mv 2 /2 k B T v most probable = 2 k b T m v AVG = 8 k b T m v rms = 3 k b T m Maxwell-Boltzmann Distribution
May 22, 2008 Physics 40B Lecture 14 3 Kinetic Energy as Internal Energy of Ideal Gas The kinetic energy for a single particle is K = 1 2 mv 2 . The average kinetic energy for each particle is K = 1 2 mv 2 = 1 2 mv rms 2 = 1 2 m 3 k B T m " # \$ % = 3 2 k B T Since there is no potential energy for an ideal gas, the total energy is E = K Total = ( nN A ) 3 2 k B T ( ) = 3 2 nRT E = U is the Internal Energy of the gas. It depends only on temperature.

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Lecture 14 - Review Ideal Gas Law Chap 19:1-4 The First Law...

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