BEH. 462/3.962J Molecular Principles of Biomaterials Spring 2003 Rcoc IRCOOH [RCOOH]+Rcoo RCOO ]+K10m+k。10m+10 IRCOOHI o Outside gel C+=V+Cs Eqn 25 o Our relationship for the ionic chemical potentials is now qn (AH4Dion-(4u)on =V m RT 2(, -C)=mRT(++c-c:-c") o Using Eqn 20, Eqn 21, Eqn 24, and Eqn 25, Eqn 26 becomes Eqn 27 (AWon -(Alion =Vm, RT vc,+vC+2-ic:=vm.RTV vn1R722-i(c1-c) How can we relate cs and cs*? o We can make simplifications for a 1: 1 cation: anion electrolyte o The chemical potentials of the mobile ions must also be equilibrated inside/outside the gel Eqn 28 o Add eqn 29 to Eqn 28 Eqn 31 RTina++ rtina-= RTina" ++ rina v. o Therefore we can write Assuming dilute solutions where the activities are approximately equal to the concentrations Lecture 9-polyelectrolyte hydrogels 14of17BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Eqn 23 [RCOO− ] Ka H+ i = [RCOO− ] [RCOOH] [ ] Ka Ka 10− pKa = = = = = H+ [RCOOH]+ [RCOO− ] [RCOO− ] 1+ Ka [ ]+ Ka 10− pH + Ka 10− pH +10− pKa H 1+ + [RCOOH] [ ] o Outside gel: Eqn 24 c+* = ν+cs* Eqn 25 c-* = ν-cs* o Our relationship for the ionic chemical potentials is now: all−ions ( )* ( ) * Eqn 26 ∆µ 1 ion = vm,1RT ∑(c j − c * j)=vm,1RT(c+ + c− − c+ − c− * 1 ion − ∆µ ) j o Using Eqn 20, Eqn 21, Eqn 24, and Eqn 25, Eqn 26 becomes:     * Eqn 27 ( ) ∆µ * ( )1 ion = vm,1RT ν +cs + ν −c− + ic2 − ˆcs   = vm,1RT ν cs + ic2 − ˆcs  1  ion − ∆µ ν ˆ ν * z− z−   ν(cs − c = s vm,1RT  ic2 − ˆ * )   z− o How can we relate cs and cs*? o We can make simplifications for a 1:1 cation:anion electrolyte: o The chemical potentials of the mobile ions must also be equilibrated inside/outside the gel: Eqn 28 µ+ = µ+* Eqn 29 µ- = µ-* o Add Eqn 29 to Eqn 28: Eqn 30 µ+ + µ-= µ+* + µ-* ν + Eqn 31 RT ln a+ + RT ln a− ν − = RT ln a+ *ν + + RT ln a− *ν − o Therefore we can write: ν + a− ν − Eqn 32 a+ = a+ *ν + a− *ν − • Assuming dilute solutions where the activities are approximately equal to the concentrations: ν + ν −  c+   c− *  Eqn 33   c+ = *     c−   Lecture 9 – polyelectrolyte hydrogels 14 of 17