Solution thermodynamics: summary Definitions: Chemical potential of component i: # "U & # "H & # "F & µi = % ( =% ( =% ( $ "n i ' S,V ,n $ "n i ' S,P,n $ "n i 'T ,V ,n j )i
!
j )i
j )i
# "G & =% ( $ "n i 'T ,P ,n j )i
Partial molar properties (using volume as an example): # "V & and therefore dV = "Vi dn i and V = "Vi n i Vk = % ( $ "n k ' P,T ,n i i i )k
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The chemical potential of a component is equal to its partial molar Gibbs free-energy: µk = Gk = H k " TSk ! ! Upon mixing A and B with molar fractions xA and xB , the Gibbs free-energy change of the system is Δgmix: "gmix x ,x = x A # (µA $ µA0 ) + x B # (µB $ µB0 ) = x A "µA + x B "µB A
B
Remember that Δgmix, µA and µB are all functions of composition…
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The activity of a component in solution is by definition: µA " µA0 = #µA ,mix = RT ln( aA ) Solution models 1-Ideal solutions (think ideal gas mixture or atoms that don’t have any preference for specific neighbors) n
"µk,mix = RT ln( x k ) # "gmix = RT $ x i ln( x i ) i=1 n
"hmix = # x i"H i,mix = 0 ! !
i=1 n
"v mix = # x i"Vi,mix = 0 i=1 n
n
"smix = # x i"Si,mix = $R# x i ln( x i ) i=1
i=1
! !
2-Non-ideal solutions Everything else! We define:
The activity coefficient of each component γi such that "µk,mix = ak = # k x k XS ID And the excess quantities: "µk,mix = RT ln(# k ) + RT ln( x k ) = "µk,mix + "µk,mix n
= # x i"H i,mix (any partial molar enthalpy of mixing is an excess quantity ! i=1 by definition because for ideal solutions, "H ! i,mix = 0 . XS And "Si,mix = "Si,mix # [#Rln( x i )] # ln($ k ) ! The temperature dependence of the activity coefficients is given by "H k,mix = R % 1( ! #' * &T) Note that "h
!
XS mix
2-1 Dilute solutions All solutions obey the dilute solution model provided they are dilute enough. ! 1 as the solvent molar Raoult’s law: the activity coefficient of the solvent tends to # d" & fraction tends to 1. Moreover, lim% S ( = 0 . $ dx S ' x S )1
Henry’s law: the activity coefficient of the solute tends to a constant finite value (≠1) – called Henry’s coefficient- as the molar fraction of the solute tends to 0. # d" & Moreover, lim% i ( ! = 0. $ dx i ' x i )0
2-2 Regular solutions The entropy of mixing of regular solution is equal to the ideal entropy of mixing: ! ID "smix = "smix = #R$ x i ln( x i ) i
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XS As a consequence, "µk,mix = RT ln(# k ) = "H k,mix and XS $ "#H k,mix ' $ "#µk,mix (T ) ' d (#hmix ) XS = 0. ) = *#Sk,mix = 0 + & )=& "T dT % "T ( % ( Note that!the regular solution model does not make any assumption on the form of Δhmix.
2-2-1 A particularly simple regular solution model The simplest form of Δhmix for a regular solution is: "hmix = ax A x B where a is a Tindependent constant (due to the regular solution requirement). In this model, "H A ,mix = ax # $ A = e 2 B
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ax B2 RT
and "H B ,mix = ax # $ B = e ! 2 A
ax A2 RT