BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 (△4)2 RT V,RT Eqn 13 E-RT>x:= ∑ R o The first approximation holds if Ex is small o Fourth equality holds because we assume in the liquid lattice model that the molar volume of all species is the same, thus Vm n=V, the total volume of the system o Chemical potential change in gel all-ions Eqn 14 (△A)mn=H-A0= RT In a≡-VmRT∑c Eqn 15 (A)m-(A4)m=mRr∑(-c) o The electrolyte dissolved in water provides mobile cations and anions in the solution and in the gel o E.g. NaCl: Na vcr v+()->v Na(ag) vcr(ag) o v=v=1 stoichiometric coefficients CA→vC+vA eg.CaCl2:v+=1,V=2,z=2,z=1 v*+v=v . for a 1: 1 electrolyte for a 1: 1 electrolyte Eqn 19 =(v+v)c total concentration of ions o We will derive equations for an anionic network o Assuming activities- concentrations o Inside gel Egn 20 Ean 21 C.=v. Cs+ ic lz. o C2 is the moles of ionizable repeat groups on gel chains per volume o First term comes from electrolyte anions in gel, second term from ionized groups on the polymer chains o The degree of ionization i can be related to the pH of the environment and the pKa of the network groups K= IRCOOH'] [RCOOHT rcoo RCOO RCOOH K K 10 [ RCOOH+[FoO],风R o0 1+K ph 10PH+10-pa Lecture 9-polyelectrolyte hydrogels 3 of 6BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 all _ ions all _ ions all _ ions all _ ions Eqn 13 ( ) ∆µ1 * ion ≅ −RT ∑x * j = − RT ∑n * j = − vm,1RT ∑n * j ≅ −vm,1RT ∑c * j j n j vm,1n j j o The first approximation holds if Σxj * is small o Fourth equality holds because we assume in the liquid lattice model that the molar volume of all species is the same, thus vm,1n = V, the total volume of the system o Chemical potential change in gel: all−ions Eqn 14 (∆µ1 )ion = µ1 − µ1 0 = RT ln a1 ≅ −vm,1RT ∑c j j all−ions ( )* ( ) * Eqn 15 ∆µ 1 ion = vm,1RT ∑(c j − c 1 ion − ∆µ j) j o The electrolyte dissolved in water provides mobile cations and anions in the solution and in the gel: o E.g. NaCl: Na+ ν+Cl- ν+ (s) → ν+ Na+ (aq) + ν- Cl- (aq) o ν+ = ν- = 1 stoichiometric coefficients Eqn 16 Cν + z+ z− Aν − →ν +Cz+ + ν −Az− • e.g. CaCl2: ν+ = 1, ν- = 2, z+ = 2, z- = 1 Eqn 17 ν+ + ν− = νˆ …for a 1:1 electrolyte ˆ Eqn 18 ν+ = ν− = ν …for a 1:1 electrolyte 2 * * * ˆ * Eqn 19 c+ + c− = (ν+ + ν− )cs = νcs …total concentration of ions o We will derive equations for an anionic network o Assuming activities ~ concentrations o Inside gel: Eqn 20 c+ = ν+cs Eqn 21 c- = ν-cs + ic2/z￾o c2 is the moles of ionizable repeat groups on gel chains per volume o First term comes from electrolyte anions in gel, second term from ionized groups on the polymer chains o The degree of ionization i can be related to the pH of the environment and the pKa of the network groups: [ ] Eqn 22 Ka = [RCOO− ] H+ [RCOOH] 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] [ ] Lecture 9 – polyelectrolyte hydrogels 3 of 6