Chemical_Equilibrium

Chemical_Equilibrium - Chemical Equilibrium The Gibbs...

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1 Chemical Equilibrium The Gibbs energy of a mixture For a pure substance or for a mixture of fixed composition the fundamental equation of Gibbs free energy is dG = - SdT + VdP If the mole numbers n 1 , n 2 , n 3 of the substances present vary, then G = G (T, P , n 1 , n 2 , n 3 ) The total differential dG is: ... 1 , , , 1 , , , ... , , 3 2 3 2 , 1 3 , 2 1 + + + = dn n G dP P G dT T G dG n n P T n n n T n n n P If the system doesn’t suffer any change in composition, dn 1 = 0, dn 2 = 0, etc ... , , , + + + = i P T i n T n P dn n G dP P G dT T G dG i i and consequently we get, S T G i n P = , V P G i n T = , j n P T i i n G , , = µ chemical potential And i i i n T n P dn dP P G dT T G dG i i + + = µ , , or i i i dn VdP SdT dG + + = µ fundamental equation of chemical thermodynamics Because the composition is constant, µ I is constant = i n i G i i dn dG 0 ' 0 " µ and i i i n G = µ G is an extensive property A m T P A A m A P T Ai Ai G n G n n G , , , , ] [ = = = µ
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2 For a given species, transport will occur spontaneously from a region of high chemical potential to one of low chemical potential. 0 ) ( < = + = i II i I i i II i i I i dn dn dn dG µ µ µ µ - spontaneous The flow of material will continue until the chemical potential has the same value in all regions of the mixture. At equilibrium the chemical potential of each individual species is the same throughout a mixture. Gibbs energy of a gas in a mixture mixture H pure H 2 2 µ µ = condition for equilibrium ) , ( ln ) ( ) , ( 2 2 2 2 2 2 H mixture H o H o H H pure H P T P P RT T P T µ µ µ = + = The chemical potential of a gas in a mixture depends logarithmically on its partial pressure, P . For a gas A: P A = x A P and the above equation becomes: A pure A A o o A mixture A x RT P T x RT P P RT T P T ln ) , ( ln ln ) ( ) , ( + = + + = µ µ µ Is the chemical potential of a gas in a mixture greater or smaller than that of a pure gas? Gibbs energy of mixing for ideal gases G mixing = G f - G i o Xe Xe o Ar Ar o Ne Ne o He He Xe Ar Ne He i G n G n G n G n G G G G G + + + = + + + = ) ln ( ) ln ( ) ln ( ) ln ( Xe o Xe Xe Ar o Ar Ar Ne o Ne Ne He o He He f x RT G n x RT G n x RT G n x RT G n G + + + + + + + + == i i i i i i i f mixing x x nRT x n RT G G G ln ln = = = G mixing < 0 because every term is negative, so mixing at constant T and P is spontaneous i i i P mixing mixing x x nR T G S ln = =
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3 S mixing arises purely from the dependence of S on V at constant T .
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