Lecture_9_092308

Lecture_9_092308 - Lecture 9 September 23, 2008 Chapter 2 -...

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Page 1 of 10 Lecture 9 – September 23, 2008 Chapter 2 - Supplemental Handout on Equipartition Internal Energy Internal Energy, U, is the sum of the potential and kinetic energy of the system. It is a state function, i.e. it only depends on its current state, not how it got there. Δ U = U f - U i For any gas, the translational kinetic energy can be written as: 2 z 2 y 2 x K mv 2 1 mv 2 1 mv 2 1 E + + = The equipartition theorem says that all degrees of freedom for molecular motion have equal energy. Quantitatively, it says that each degree of freedom contributes an average energy of 1/2kT to the total energy where k = Boltzmann's constant = R/N A = 1.38x10 -23 J/entity. For a perfect monatomic gas (He, Ne, Ar, Kr, Xe, . ..) the only degrees of freedom are translational so: U = U t = U (0) + 3/2 kT For n moles of perfect monatomic gasses: U = U t = U (0) + 3/2 nN A kT = U (0) + 3/2 nRT and U = RT 2 / 3 ) 0 ( U U t + = Where U(0) is the energy at T = 0 and can be regarded as 0. For monatomic gases, this is essentially true, and RT 2 3 U U t = = can be regarded as fact. For Linear and Diatomic molecules (H 2 , N 2 , O 2 , F 2 , Cl 2 , Br 2 , I 2 , . ..), there are two rotational degrees of freedom, while non-linear molecules have 3 rotational degrees of freedom, hence:
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Page 2 of 10 Diatomic or linear polyatomic: RT RT 2 2 RT 2 1 RT 2 1 U r = = + = Polyatomic (non-linear): RT 2 3 RT 2 1 RT 2 1 RT 2 1 U r = + + = The equipartition theorem also works well here.
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Page 3 of 10
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Page 4 of 10 Finally, one needs to consider the vibrational motion in polyatomic molecules: Linear Polyatomics: () RT 5 N 3 U v = Non-linear Polyatomics: () RT 6 N 3 U v = Hence, Equipartition predicts : Monatomics: RT 2 3 0 0 RT 2 3 U U U U v r t = + + = + + = Diatomics: RT 2 7 RT RT RT 2 3 U U U U v r t = + + = + + = Linear Polyatomics: () RT 5 N 3 RT RT 2 3 U U U U v r t + + = + + =
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Page 5 of 10 Non-linear Polyatomics: () RT 6 N 3 RT 2 3 RT 2 3 U U U U v r t + + = + + = Reality: Monatomic: RT 2 3 U = Diatomic: RT 2 7 to RT 2 5 U = Polyatomics: Forget it, as it is too complex to rationalize. Why the problem, the vibrational part is not correct.
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Lecture_9_092308 - Lecture 9 September 23, 2008 Chapter 2 -...

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