# Advanced Thermo chap6 - 1 Chapter 6 The Thermodynamics of...

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Unformatted text preview: 1 Chapter 6 The Thermodynamics of Multi-component Mixtures 6.1 The thermodynamic description of mixtures for a C component mixture, the equations of state of the forms: ) ,....... , , , ( ) ,....... , , , ( 1 2 1 2 1- = = C C x x x P T U U N N N P T U U or ) ,....... , , , ( ) ,....... , , , ( 1 2 1 2 1- = = C C x x x P T V V N N N P T V V or The simplest description: linear combination description ) , ( ) ,....... , , , ( 1 1 2 1 ∑ = =- C i i i C P T U x x x x P T U ) , ( ˆ ) ,....... , , , ( ˆ 1 1 2 1 ∑ = =- C i i i C P T U x x x x P T U or the descriptions above imply that the internal energy of mixing is zero, which is not true for most of the mixtures. 2 Definition of a Partial molar property ( 29 i j N P T i i i N N x P T ≠ ∂ ∂ = = , , ) , , ( θ θ θ always satisfies i θ ) , , ( 1 x P T x C i i i ∑ = = θ θ for example, the partial molar Gibbs free energy is defined as ( 29 j i N P T i i N G N G ≠ ∂ ∂ = , , and ) , , ( 1 x P T G x G C i i i ∑ = = proof of the linear combination equality C N N N N + + + = ....... 2 1 N N x i i / = ( 29 ∑ ∂ ∂ ⋅ ∂ ∂ + = ∂ ∂ + ∂ ∂ = ∂ ∂ ⋅ = ⋅ = = ≠ ≠ ≠ ≠ C j j x P T j N P T N P T N P T C C N x x N N N N N N N N N N N N N P T N x x x P T N N j i j j j 2 1 , , , , 1 , , 1 , , 1 3 2 3 2 1 1 1 ) ,....., , , , ( ) ,..... , , , ( θ θ θ θ θ θ θ θ N x N N N N N N N N N x j j j j j- =- = ∂ ∂ = ∂ ∂ = ∂ ∂ 2 1 1 1 1 ( 29 ∑ ∂ ∂- = ∂ ∂ = ≠ C j x P T j j j i x x N N 2 , , 1 θ θ θ also true for V , H , S , A and U 3 ( 29 ( 29 ( 29 ∑ ∂ ∂- ∂ ∂ + = ∑ ∂ ∂- +- ∂ ∂ + = ∑ ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ∂ ∂ + = ∑ ∂ ∂ ⋅ ∂ ∂ + = ∂ ∂ + ∂ ∂ = ∂ ∂ = ≠ = ≠ = = ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ C k x P T k k x P T j C j k k x P T k k j x P T j C j k k j k x P T k j j x P T j C k j j x P T j N P T j N P T j N P T j k i j i k i j i k i j i j i j i j i j i x x x x x x x N x x N N x x N N N N x N N N N N N N 2 , , , , , 2 , , , , , 2 , , , , 2 , , , , , , , , 1 θ θ θ θ θ θ θ θ θ θ θ θ θ θ for 1 ≠ j ∑ ∑ ∂ ∂- ∂ ∂ + + ∑ ∂ ∂- = ∑ = = = = C l C k k k l l l l C j j j k C k k x x x x x x x x x x x 2 2 2 1 1 1 θ θ θ θ θ θ ( 29 θ θ θ θ θ θ θ θ = ∑ ∑ ∂ ∂- ∑ ∑ ∂ ∂ + = ∑ ∑ ∂ ∂- ∑ ∂ ∂- + ∑ = ∑ = = = = = = = = = 1 2 2 2 2 2 2 2 1 1 1 C l C k k k l C l C k k k l C l C k k k l C k k k C l l k C k k x x x x x x x x x x x x x x 4 isothermal volume change on mixing [ ] ∑- = ∑- ∑ = ∑- = ∆ = = = = C i i i i C i i i C i i i C i i i C C mix P T V x P T V N P T V N x P T V x N P T V N N N N P T V N N N P T V 1 1 1 1 2 1 2 1 ) , ( ) , , ( ) , ( ) , , ( ) , ( ) ,...., , , , ( ) ,...., , , , ( [ ] ∑- = ∑- ∑ = ∑- = ∆ = = = = C i i i i C i i i C i i i C i i i...
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Advanced Thermo chap6 - 1 Chapter 6 The Thermodynamics of...

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