Lecture 23: Colligative Properties

5.60 Spring 2007 Lecture #23 page 1 Colligative Properties These are properties of solutions in the dilute limit, where there is a solvent “A” and a solute “B” where nA >> nB. pure These properties are a direct result of µ Amix (l, T, p) < µ A (l, T, p) Use two measures of concentration: a. Mole Fraction: xB = nB/(nA+nB) ~ nB/nA b. Molalility: mB = (moles solute)/(kg solvent) = nB/(nAM A) Where MA is the mass in kg of one mole of solvent. There are FOUR Colligative Properties: 1. Vapor pressure lowering: "pA = pA ! pA* = !xB pA* 2. Boiling point elevation: "Tb = Tb ! Tb* = Kb mB Kb = 3. Freezing point depression: Osmotic pressure: !Hvap "Tf = Tf ! Tf* = !Kf mB Kf = 4. 2 ( ) MA R Tb* ! = RTc~ 2 ( ) MA R Tf* !Hf ~ = nB is concentration of solute where c V 5.60 Spring 2007 Lecture #23 page Let’s go through these one at a time: 1. Vapor pressure lowering: This is just Raoult’s Law. pA = xA pA* = (1 ! xB )pA* 2. So "pA = pA ! pA* = !xB pA* ( < 0) Boiling point elevation: µ A (l, T, p) = µ A (g, T, p) Let’s derive this. Start with So, µ A* (l, T, p) + RT ln xA = µ A (g, T, p) And ln xA = !Gvap 1 [µ A (g, T, p) " µ A* (l, T, p)] = RT RT But lnxA = ln(1-xB) ~ -xB = -nB/(nA+nB) ~ -nB/nA = -(MnB)/(MnA) Where M is the total mass of A, So, lnxA ~ m BMA, where MA is the molar mass of A. mB = Putting it all together then, But we need ΔT in there! … ( ) " !Hvap " 1 (& ) !Gvap T %# ( )mB % = & # = 2 # )T M RT ' )T $ p MA R &' A $p which gives us !T = MA RT 2 "Hvap !mB " !Gvap MA RT 2 5.60 Spring 2007 Lecture #23 page If δmB = mB – 0 (mixed – pure) and mB is very small #T = Then 3. Tb " Tb* !Hvap mB = Kb mB Freezing point depression: Same arguments as above, replacing 4. = MA R(Tb* ) 2 ΔGvap with –ΔGf ΔHvap with –ΔHf Tb with Tf Kb with Kf Osmotic Pressure: The pressures at points: pext h α• β• A+B β: p = pext + ρg α: p + π = pext + ρg + h ρg A At equilibrium: µ A (l, p + !, T ) = µ A* (l, p, T ) Using Raoult’s Law: RT ln xA + µ A* (l, p + ", T ) ! µ A* (l, p, T ) = 0 At constant T: dG = Vdp, or dµ A* = VA* dp 3 5.60 Spring 2007 Integrating… Lecture #23 µ A* (l, p + !, T ) " µ A* (l, p, T ) page p+ ! = * 4 * # VA dp = VA ! p (this assumed an incompressible liquid, where volume does not depend on p) So then RT ln xA + VA* ! = 0 Again using lnxA ~ -nB/nA Then RT(-nB/nA) + (VA/nA)π = 0 But VA ~ VA + VB = V So finally, (VB < < VA) πV = RTnB This is the Van’t Hoff Equation. It looks like the ideal gas law! ~ = n /V in the Van’t Hoff Eq., then we get the osmotic If we replace c B pressure relation: ! = RTc~