Gibbs-Duhem/Partial Molar Properties

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Presentation transcript:

Gibbs-Duhem/Partial Molar Properties

Gibbs-Duhem Equation Useful for defining equilibrium concepts Integrate from state 1 to state 2 State 1: no mass (U,S,V,N)=0 State 2: some mass (U,S,V,N)<>0

Define U

Gibbs-Duhem Plug it in! When we are at equilbrium

Partial Molar Properties Consider equilibrium M(T,P,ni) M is any property (V, S, G, etc..) Partial molar property

Notation and Definition*** Look at derivative for definition M – extensive property m – intensive property To evaluate, just take the derivative

Terminology**** Example: mix 50 mL of water with 50 mL of ethanol, what happens? - Vi – V –

Derive contribution of partial molar properties

Derive contribution of partial molar properties

Simplifications System size can be anything System size can change by any amount Independent and arbitrary Both “brackets” are 0

Derive contribution of partial molar properties Gibbs-Duhem Eq

Derive for special case of binary system (from SVNA) Gibbs-Duhem Eq

Derive for special case of binary system (from SVNA)

Derive for special case of binary system (from SVNA)

Derive for special case of binary system (from Koretsky) Gibbs-Duhem Eq

Analytical Analysis From virial EOS, determine v1, v2, v, vIS, Δvmix, , and T=333 K P = 10 bar B11=-910 cc/mol B22=-1330 cc/mol B12=-2005 cc/mol y1=0.2

Analytical Analysis For v, just plug it in For v1 and v2, what should you do?

Analytical Analysis For v, just plug it in

Analytical analysis Ideal solution -

Analytical Analysis Binary system

Alternative Analytical Analysis If this WASN’T a binary system

Summary Gibbs-Duhem Defined partial molar properties Developed formula for binary mixtures Analytical solution Other than partial molar properties, plug and/or chug Remember formula for ideal solution

Summary For binary partial molar properties Convert all y2y1 (y2=1-y1) Take derivative Plug derivate and mixture property into formula simplify

Summary Partial molar properties (general) Convert from intensive to extensive Multiply by total moles Convert mole fractions to ni Take derivative Remember nT is a function of n1 n2 is NOT a function of n1 Simplify