General Physical Chemistry I Lecture 14 Aleksey Kocherzhenko April 9, 2015"
Last time "
Chemical potential " Partial molar property the contribution per mole that a substance makes to an overall property of a mixture " Chemical potential = partial molar Gibbs free energy:!µ i = @G @ i T,p, j6=i The total Gibbs free energy:"g = X i i µ i Ø µ i is a measure of the ability of substance i to bring about chemical or physical change in a mixture" Ø The chemical potential depends on the conditions of a mixture (pressure, temperature, composition)" Ø We used chemical potentials to study the mixing of perfect gasses "
Raoult s law " Raoult s law: p J = x J p indicates property of pure substance" J," the partial vapor pressure of a substance in a liquid mixture is proportional to its mole fraction in the mixture and the vapor pressure of the pure substance" Ø Solutions that obey Raoult s law are called ideal solutions! Ø Most reliable for mixtures of substances with molecules of similar shape held together by intermolecular interactions of similar type and strength! Real mixtures show deviations from Raoult s law"
Chemical potentials for mixtures of liquids " Ø A liquid in a closed vessel is in dynamic equilibrium with its vapor" There is as much liquid evaporating per s as there is vapor condensing" Ø Since the system is in equilibrium, the difference in the Gibbs free energy between the two phases is:" G l/g = G l G g =0 ) µ (l) = µ (g) = µ (g) + RT ln p p Ø For a component in a mixture, we can apply Raoult s law:" p J = x J p J ) µ (l) J = µ (g) J + RT ln x Jp J p J = µ (g) J Independent of the composition of the mixture" + RT ln p J p {z J } = µ J +RT ln x J Chemical potential of pure liquid"
Mixtures of liquids (continued)"
Henri s law " Ø Raoult s law works well for the solvent (more abundant component of the solution), but not for the solute" Henri s law:" The vapor pressure of a volatile solute B is proportional to its mole fraction in" a solution: "p B = K H x B Henri s constant (solute property)" Values in the table are for Henri s law written in terms of mole fractions:" p B = K H [B] Henri s law does not work if vapor interacts with solution"
Real solutions: Activities " Ø To preserve the form of the equation for the chemical potential, write: " For ideal solutions: " (obey Raoult s law)! a J = x J µ J = µ J + RT ln a J activity" For ideal-dilute solutions:" a B = [B] c (solute obeys Henri s law)! Standard molar concentration, 1 mol/l" Activity coefficients (describe deviation from ideal solution behavior)" [B] For non-ideal solutions: a A = A x A and" a B = B c (for solvent)" (for solute)" x A! 1 ) A! 1 [B]! 0 ) B! 1 Ø Both activities and activity coefficients are dimensionless"
Colligative properties"
Adding solute changes properties of solution " Ø Adding solute to solvent introduces disorder à increases entropy" Ø Consider solute that is:" o non-volatile (no solute in vapor phase, only solvent);" o not an electrolyte (does not dissociate into ions)." Example: sucrose (C 12 H 22 O 11 ) in water" Ø We will assume that the solution is ideal and dilute" Solvent obeys Raoult s law" Ø Presence of non-volatile solvent decreases vapor pressure:" Total vapor pressure, since solute is not volatile" p solvent = x solvent p solvent < 1 ) p solvent =(1 x solute ) p solvent =1 x solute ) p = p solvent p solvent = x solute p solvent Reduction in vapor pressure proportional to mole fraction of solute"
Modification of phase transition temperatures " In lecture 13 (slides 18 19) we showed that" à in the presence of solute, "µ A <µ A Depression of freezing point! µ A = µ A + RT ln x A solvent" Elevation of boiling point! < 1 < 0 Freezing point reduced by:" T f = K f b B solute" Boiling point increased by:" cryoscopic constant" ebullioscopic constant " molality" T b = K b b B
Modification of phase transition temperatures " At the boiling temperature (for a given pressure and solution composition):" µ (g) A (p, T b )=µ (l) A (p, T b) Equilibrium condition" = µ (l) A (p, T b )+RT b ln x A The molar Gibbs free energy of vaporization" Chemical potential = partial molar Gibbs free energy" (g) µ A (p, T b ) µ (l) A (p, T b ) ) ln x A = = vap G m (p, T b ) RT b RT b For the pure solvent,"x A =1) 0= vap G m (p, Tb ) RTb ) ln x A = vap G m (p, T b ) vapg m (p, Tb ) RT b Take the difference" RTb (Vaporization enthalpy and vaporization entropy approx. independent of temperature)" = vap H m T b vap S m vaph m Tb vaps m RT b RT b = vap H m 1 1 R T b Tb = =
Modification of phase transition temperatures " ln x A = vap H m 1 1 We found:" R T b Tb = vap H m (Tb T b ) RT b Tb By definition: T b = T b Tb and" x A =1 x B For a small amount of solute (dilute solution):" T b T b (T b ) 2 ) ln (1 x B ) vaph m R (T b )2 From the expansion of the natural logarithm into a Maclaurin series it follows that for"x B ln (1 x B ) x B ) x B vap H m T b ) T b R (T b ) 2 R (Tb )2 T b vaph m x B Similarly, the freezing temperature is reduced upon the addition of the solute by " T f R (T f )2 fush m x B Since" vaph m > fus H m ) T f >T b
Osmosis " Osmosis diffusion of solvent molecules through a semipermeable membrane (a membrane that can be permeated by small solvent molecules, but not by larger solute molecules), so as to equilibrate solute concentration of solute on both sides " Osmotic pressure pressure that must be applied to stop the flow of solvent into the volume with larger solute concentration"
Osmosis " The variation of chemical potential with pressure:" ) µ A (p + ) = µ A (p)+v A ) µ A (p) =µ A (p)+v A {z } =µ A (p+ ) +RT ln x A ) 0=V A +RT ln x A Osmotic pressure" Equilibrium condition:" µ A (p) =µ A (x A,p+ )= = µ A (p + ) + RT ln x A We showed in lecture 12 (slide 6) that the variation in the Gibbs free energy can be written as follows:" G = V p S T =0 Osmosis does not involve temperature variation"
Osmosis " We found:" 0=V A +RT ln x A We also know that" ln x A =ln(1 x B ) x B ) V A RT x B Jacobus Henricus van t Hoff, Jr." x B = B B A ) V A RT B A ) V RT B Van t Hoff s equation" For dilute solutions, the total amount of the solution is almost the same as the amount of solvent" ) V A A {z } =V A V RT B Notice the similarity to the perfect gas law!"
Osmosis " V RT B Van t Hoff s equation" ) RT B V = RT [B] Osmometry: measurement of molar masses of large molecules (proteins, polymers) from osmotic pressures" Ø Non-ideal solutions à virial expansion" =RT [B](1+B [B]+...) ) [B] Osmotic virial coefficient" = RT + RT B [B]+... See textbook/homework for calculation of molar masses based on osmotic pressure"