Statistical_Mechanics_2_Chap._8-10

# Statistical_Mechanics_2_Chap._8-10 - Chapter 8 Chapter 8...

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Unformatted text preview: Chapter 8. Chapter 8. Ideal Polyatomic Gas Ideal Polyatomic Gas H trans int Η = Η + 3 2 2 2 trans MkT q V h π = trans int ε ε ε = + trans int q q q = M : the total mass of the molecule ‒ n atoms in a polyatomic molecule → require 3 n coordinates to specify the polyatomic molecule ‒ 3 for translation ‒ 2 (or 3) for orientation of linear or non-linear molecule ‒ 3 n − 5 (or 3 n − 6) for relative location of n molecules 3 n − 5 relative coordinates for linear molecule 3 n − 6 relative coordinates for non-linear molecule Selection rule for rotational level For diatomic molecules, 2 1 ( ) ( ) ( ) 2 e e u r u r k r r = +- + L (1) 2 2 2 1 Η 2 2 kx m x ∂ = + ∂ h (2) 1 1 ( ) , 2 2 n k n h m ε ν ν π = + = (3) For polyatomic molecules, the potential energy will be a quadratic function of 3 n- 5 or 3 n- 6 relative coordinates such as x 2- x 1 , x 3- x 1 , y 2- y 1 , and so on. Terms such as ( x 2- x 1 ) 2 has cross terms ( x 1 x 2 , x 1 x 3 ,···), which makes the potential energy complicated . Introduce a new coordinate system, Q 1 , Q 2 ,···, which eliminates the cross terms. 2 2 2 2 1 1 Η 2 2 j j j j j j k Q Q α α μ = = ∂ = - + ∂ ∑ ∑ h (4) Since eq (4) is the Hamiltonian of a sum of independent harmonic oscillators, where 3 n- 5 or 3 n- 6, and are effective reduced masses and force constants. α = j μ j k 1 1 2 1 2 j j j j j j n h k α ε ν ν π μ = = + = ∑ (5) 0, 1, 2, j n = L Q j : normal coordinates The normal coordinates for a linear triatomic molecule (CO 2 ) (a) a symmetric stretch, (b) a bending mode. (b) an asymmetric stretch, (a) (b) (c) The normal coordinates for a non-linear triatomic molecule (H 2 O) (a) (b) (c) where Since each normal mode of vibration makes an independent contribution to thermodynamic functions, (6) ( 29 1 1 j j T vib T j e q e ν ν α-Θ-Θ = =- ∏ 1 E 2 1 j j T j j vib T j e Nk e ν ν α ν ν-Θ-Θ = Θ Θ = + - ∑ 2 , 1 1 j j T j V vib T j e C Nk T e ν ν α ν-Θ-Θ = Θ = + - ∑ j j h k ν ν Θ = If the molecule is linear, such as CO 2 and C 2 H 2 , the problem is exactly the same as for a diatomic molecule. The Rotational Partition Function 2 1 n j j j I m d = = ∑ (8) where the moment of inertia I is given by ( 29 2 2 1 8 J J J h I ε π + = (7) 0, 1, 2, J = L where d j is the distance of the j th nucleus from the center of mass of the molecule. 2 1 J J ϖ = + 1 1 1 1 1 1 n cm j j j n cm j j j n cm j j j x m x M y m y M z m z M = = = = = = ∑ ∑ ∑ (9) The center of mass The rotational partition function of a linear polyatomic molecule 2 2 8 rot r IkT T q h π σ σ = = Θ The principal moments It is always possible to choose X, Y, and Z axes such that: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 1 2 2 1 2 2 1 1 n xx j j cm j cm j n yy j j cm j cm j n zz j j cm j cm j n xy j j cm j cm j I m y y z z I m...
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Statistical_Mechanics_2_Chap._8-10 - Chapter 8 Chapter 8...

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