BEH. 462/3.962J Molecular Principles of Biomaterials Spring 2003 o Chemical potential change in solution Eqn 12 (A)m=-A=RTna≡ RTInx=RTlm(-∑x o approximation in third equality is used for dilute solutions RT ∑n V,RT Eqn 13 (4)n三-RT n≡-mRT n 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 Ean 14 (AA)m=-= RTIn a1≡-mRT∑c Egn 15 41 41 RT Cj o The electrolyte dissolved in water provides mobile cations and anions in the solution and in the gel E.g. Nacl: Na vCI a(ag)+ vcr( o v=v=1 stoichiometric coefficients Eqn16C:45→vC++vA- e.g Ean 17 V十V=V for a 1: 1 electrolyte Egn 18 …ora1:1 electrolyte Egn 19 Ci+C=(v+v)c =i total concentration of ions o We will derive equations for an anionic network o Assuming activities- concentrations de Ean 20 C=vCs +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 counter-ions associated The degree of ionization i can be related to the ph of the environment and the pKa of the Ean 22 「 RCOOH Lecture 9-polyelectrolyte hydrogels 130f17BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 o Chemical potential change in solution: all _ solutes ( )* Eqn 12 ∆µ1 ion = µ1 1 * − µ0 = RT ln a1 * ≅ RT ln x1 * = RT ln(1− ∑x * j) j o approximation in third equality is used for dilute solutions 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 (∆µ1 )ion − (∆µ1 )ion = −vm,1RT ∑(c j − c * 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 counter-ions associated with 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] Lecture 9 – polyelectrolyte hydrogels 13 of 17