Colligative Properties Slides PDF

BCHEM259: Physical Chemistry I Solutions I & II Dr. Elliot Sarpong Menkah Faculty of Physical and Computational Sciences Chemistry Department...

BCHEM259: Physical Chemistry I Solutions I & II Dr. Elliot Sarpong Menkah Faculty of Physical and Computational Sciences Chemistry Department February 14, 2024 Elliot S. Menkah, Ph.D. (KNUST) Solutions I January, 2024 1 / 13  Colligative Properties Colligative Properties - Freezing Point Depression - Boiling Point Elevation - Osmotic Pressure Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 2 / 13  Colligative Properties - Characterization In an ideal-dilute solution, when an involatile solute, B, is added to a solvent, A, the chemical potential of the liquid solvent, A, is lowered. µi = µΘ i + RT ln ai (7) This effect is independent of the chemical nature of B, but only depends upon the number of moles of B added, determined by its mole fraction. xA = 1 − xB (7b) This brings about a thermodynamic stabilization and low activity of the solvent which eventually results in measurable changes, such as the: depression of the solvent vapor pressure. decrease in the freezing temperature. an increase in the boiling point, which only depends on the number of moles of B added and not its chemical structure. Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 3 / 13  Colligative Properties ⋄ A colligative property of a solution is a property that depends only on the number of solute molecules present. ⋄ Measurement of one of these properties allows the determination of the molality or the concentration of the solute, ⋄ from which the molecular mass of the solute can be calculated if the mass of solute in solution and the volume of the solution is known. ⋄ Example, given that a known amount of substance B is added to A, the molecular mass is unknown, and colligative property and molality can be determined. ⋄ With known molality; nA mA = (8) M ⋄ With known total mass of solution, xA = mA , M, mole fraction is found Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 4 / 13  Colligative Properties - Freezing Point Depression For the equilibrium corresponding to the physical transition of A between solid and liquid A(s) ⇌ A(l) (⋆) When little B is added: At the freezing point, temperature equilibrium exists; Below this temperature, liquid A exists as solid A. Above this point, solid A exists as liquid A (melting) K = xA (9) The equilibrium constant, K, is dominated by A, Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 5 / 13  Colligative Properties - Freezing Point Depression The temperature variation of the equilibrium constant is given by the Van’t Hoff equation; Θ     KT 1 −∆Hfus 1 1 ln KT 1 − ln KT 2 = ln = − (10) KT 2 R T1 T2 Θ −∆Hfus : is the standard enthalpy of fusion: the enthalpy required to melt a mole(1) of solid, A, under standard conditions. Θ     KT 1 −∆Hfus 1 1 ln KT 1 − ln KT 2 = ln = − (11) KT 2 R T1 T2 Given that for pure liquid A, KT 2 = xA = 1 and T2 = Tf◦ But when, B, is added, KT 1 = xA and Tl = Tf Θ   −∆Hfus 1 1 ln(xA ) = − ◦ (12) R Tf Tf Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 6 / 13  Colligative Properties - Freezing Point Depression From the expression, Tf − Tf◦ = −ve Tf for solution is lower than Tf◦ for pure liquid A. Θ   −∆Hfus 1 1 xB ≈ ln(1 − xB ) = − ◦ (13) R Tf Tf when a small amount of B is added and xB is small compared to 1. Under these conditions, the change in the freezing temperature is also small and: 1 1 T ◦ − Tf ∆Tf − ◦ = f ◦ ≈ (14) Tf Tf Tf Tf (Tf◦ )2 where ∆Tf = Tf◦ − Tf (15) Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 7 / 13  Colligative Properties - Freezing Point Depression An expression for mole fraction is given as: Θ ∆Hfus xB = 2 ∆Tf (16) R (Tf◦ ) ∆Tf Freezing point depression(the decrease in the freezing temperature of the solvent A on adding B: a positive (+) value) nA mA = ; (17) M xB = mB M; (18) The equation becomes: R(Tf◦ )2 M ∆Tf = Θ mB = Kf mB · i (19) ∆Hfus R(Tf◦ )2 M where Kf = Θ (20) ∆Hfus Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 8 / 13  Colligative Properties - Freezing Point Depression Kf is the cryoscopic constant or the freezing point depression constant of the solvent A Kf can, in principle, be calculated from the enthalpy of fusion and the freezing point of pure liquid A. Still, in practice, values have been measured experimentally and tabulated for various solvents. This value can be used, along with the measured value of ∆Tf , to determine mB and/or xB for any solute. Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 9 / 13  Colligative Properties - Boiling Point Elevation A(l) ⇌ A(g ) Boiling point equilibrium corresponds to the transition of A from liq to gas for a given ideal-dilute solution. aA = xA (21) ρA K= Θ (22) ρ ρA K= Θ (23) ρ xA Where P θ , the standard pressure, has a value of 1 atmosphere. At the boiling point, the pressure of the solvent, P A , is also 1 atmosphere, and so K = x1B. If T b is for solution and Tbo is pure solvent. Θ     KT 1 −∆Hfus 1 1 ln KT 1 − ln KT 2 = ln = − (24) KT 2 R T1 T2 Θ  −∆Hvap  1 1 − ln xA = − (25) R T1 T2 Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 10 / 13  Colligative Properties - Osmotic Pressure Osmotic Pressure Osmosis is the movement of solvent molecules From a solution of low solute concentration to a high solute concentration (lower solvent concentration). In osmotic pressure measurements, a semi-permeable membrane, which is permeable only to solvent (as it has holes that are small enough to prevent large solute molecules from passing through), separates two liquids. This means that this technique is only used for relatively large solute molecules, often polymers or biological macromolecules. Typically, one liquid is pure water, and the other is the solute solution of interest. This produces a water flow from the solvent to the solute solution. Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 11 / 13  Colligative Properties - Osmotic Pressure The experimental apparatus incorporates two identical vertical columns, one for each liquid, The height of the liquid in the solution column increases relative to that in the pure solvent column due to this net flow. The extra height (and mass of water) in the solute column compared to the solvent column produces an excess gravitational force (and pressure) in the solute compartment. Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 12 / 13  Colligative Properties - Osmotic Pressure Excess pressure is generated in the solute compartment as volume occupied reduces. At equilibrium, the pressure exerted at the foot of a column of solvent is proportional to its height, so a reading of the difference in heights of the two columns then leads directly to the difference in pressure, which is itself equal to π πV = nB RT - Osmotic press. of an ideal solution. where nB is the number of moles of solute B. Thus: π = cB RT - Difference in press. cB , the concentration of solute B in the solution in the right-hand compartment at equilibrium, to be measured at any temperature T. Elliot S. Menkah, Ph.D. (KNUST) Solutions II January, 2024 13 / 13