# Thermodynamics of separation What is the minimum work to separate a mixture into it’s pure components? Ex. Mining, Desalination, Material Purification,

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Thermodynamics of separation What is the minimum work to separate a mixture into it’s pure components? Ex. Mining, Desalination, Material Purification, Recycling.

Balance Eq’ns for Mass, Energy & Entropy S irr

Minimum Work of Separation

Gibbs Free Energy of Mixing*  g o mix =  h o mix – T 0  s o mix.  g o mix  – T 0  s mix = – T 0 (s 12 – x 1 s 1 – x 2 s 2 ) For non-interacting molecules entropy can dominate often resulting in a negative Gibbs Free Energy and hence spontaneous mixing. I.e.  g o mix < 0 * at standard conditions

S = k ln  Boltzmann’s entropy equation How many ways can “r” atoms be positioned in a lattice with “n” locations?

w min = T 0  s mix = k T 0 (ln  12 ) Ex. 4 atoms in 8 locations

Using Stirling’s Approximation Where x is mol fraction r/n, and R = k N avo ln N! = N ln N - N

Multi-component System

“Separation”

“Extraction”

Separation Examples From the atmosphere From the Ocean Solutions –Polymer –Water based –Liquid metals (activity coef)

The minimum work to separate O 2 from the atmosphere In wet air you get 3.97 kJ/mol : compare with Szargut Table from the EngineeringToolBox.com

energy requirements for mining and milling, possible future trends Chapman and Roberts p 113 & 116 underground ~ 1000/g (MJ/t metal) open pit ~ 400/g (MJ/t metal)

Sherwood plot showing the relationship between the concentration of a target material in a feed stream and the market value of (or cost to remove) the target material [Grübler 1998].

Exergy of a Mixture

CRUST at T o, p o Ore value at mine Pure ore (e.g. Fe 2 O 3 ) Pure metal Metal alloy Mixing in product Mixing in waste stream Further mixing and corrosion Exergy Purification Stages Recycle to pure metal Theoretical Exergy Values for a metal extracted from the earth ’ s crust shown at various stages of a product life cycle (not to scale)

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