School of Chemical & iological Engineering, Konkuk University
Lecture 7 Ch. 5 Simple Mixtures Colligative properties Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-2
he presence of a solute in a dilute solution results in the lowering of vapor pressure, the elevation of boiling point, the depression of freezing point, and the osmotic pressure. Such properties depend only on the number of solute particles present, not their identity. For this reason, these properties are called colligative properties (denoting depending on the collection) ssumptions for the colligative properties - he solute is not volatile, so it does not contribute to the vapor. - he solute does not dissolve in the solid solvent. When the solution is frozen, the pure solid solvent separated. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-3
For an ideal-dilute solution (Solvent, Raoult s law; solute, enry s law), the chemical potential of the solvent () is expressed by the Raoult s law: l l R ln x Note that the chemical potential of the pure solvent ( l reduced by R ln x as a result of the presence of solute. l l ) is owever, there is no direct influence of the solute on s and g because the solute appears in neither the vapor nor the solid by the preceding assumptions. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-4
s a consequence, the liquid-vapor equilibrium occurs at a higher and the solid-liquid equilibrium occurs at a lower. b elevated and f depressed. Note that the lowering of the liquid s chemical potential has a greater effect on f than on b. Remember that G. p S Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-5
he molecular origin of the lowering of the chemical potential is not enthalpy effect (interactions between the solute and solvent particles). he lowering occurs even in an ideal solution ( mix =0). he origin must be an entropy effect, if not enthalpy effect. G S When solute is present, the disorder of the liquid phase is higher than that of the pure liquid. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-6
Vapor pressure reflects the tendency of the solution towards greater entropy, which can be achieved by vaporization. When a solute is present, the entropy of the liquid is increased even in an ideal solution. weaker tendency to gas phase lower vapor pressure boiling point elevated. Freezing causes a decrease in entropy of the system: ecause the presence of solute, the freezing requires more decrease in entropy of the system. he entropy of its surroundings can be more increased at lower. dq ds sur freezing point depressed. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-7
Consider a heterogeneous equilibrium between the solvent vapor (g) and the solvent (l) in solution at atm. ecause l l R ln x g l, (Definition of Raoult s law) ln g l R ln x x g R l vap R G Differentiating both sides with respect to and using the Gibbs-elmholtz equation, d ln x d R vapg d d vap 2 R G p 2 Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-8
d ln x d vap 2 vap y multiplying both sides by d, dln x d ln x R y integration, dln x 0 R vap 2 R 2 where is the boiling temperature of pure (x =). ssuming that vap is a constant over the small range of, ln x R d vap x ln For a dilute solution, x and ln x x x vap R Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-9
x vap R x vap and 2 R 2 R Kx where K vap he presence of a solute (x ) causes an increase in normal boiling point from to + (note that K > 0). Note that the above equation makes no reference to the identity of the solute, only to its mole fraction. colligative depends on the properties of the solvent, and the biggest changes occur for solvent with high boiling points (). y routon s rule, herefore vap constant (85 J/K mol) but independent of vap itself. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-0
Kx where K 2 R vap For practical applications of the above equation, x is proportional to its molality (b) in solution. x b herefore K b b where K b is the empirical boiling point constant of the solvent. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-
Consider a heterogeneous equilibrium between pure solid solvent (s) and the solvent (l) in solution at atm. ecause s l l R ln x l, (Definition of Raoult s law) s l R ln x Following the similar procedures to the boiling case, Kx where K 2 R fus Larger depressions are observed in solvents with low fus and high. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-2
Kx where K 2 R When the solution is dilute, x is proportional to its molality (b) in solution. herefore x b K f b fus where K f is the empirical freezing point constant of the solvent. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-3
he solubility is not strictly a colligative property, because it varies with the identity of the solute. owever, the solubility can be thermodynamically estimated by the same techniques. When a solid solute is left in contact with a solvent, it dissolves until the solution is saturated. a state of equilibrium. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-4 s l ecause l l R ln x, s l R ln x s l G fus ln x R R Note that in this case, x means the solubility.
fusg ln x R Differentiating both sides with respect to and using the Gibbs-elmholtz equation, d ln x d fusg d R d Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-5 fus 2 R G y integration from the melting temperature ( f ) of (when x =) to the lower of interest (0 < x < ), ln x 0 R ln x fus d 2 ssuming that fus is a constant over the small range of, ln x fus R f f d p 2
ln x fus R f he solubility (x ) decreases exponentially as is lowered from f of the solute. fus R f t 25 o C, solutes with high f and high fus have low solubility. In the above equation. there is no terms for the identity of solvent. ut solutes generally have different solubilities in different solvents. due to the highly questionable approximation such as the ideality of the solution. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-6
he phenomenon of osmosis (from the Greek word for push ) is spontaneous passage of a pure solvent into a solution separated from it by a semipermeable membrane (permeable to the solvent but not to the solute). he osmotic pressure () is the pressure that must be applied to the solution to stop the influx of the solvent. In this figure, when reaching equilibrium, the hydrostatic pressure of the column of solution matches. he entry of solvent in the solution results in its dilution. Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-7
Consider a simpler model in which there is no flow and the concentrations remain unchanged. he chemical potential of the solvent is lowered by the solute, but is restored to its pure value by the application of pressure, then they are in equilibrium, p x, p x, p p p p p Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-8 R V m ln x dp R ln x G p V where V m is the molar volume of the pure. p p p Vmdp R ln x p
R ln x For dilute solutions ( ), p p ssuming that V m is a constant in the small range of p ~ p+, Rx Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-9 V m dp x ln x ln x x p Vmdp Vm p When the solution is diluted, x, Rx V m van t off equation: where n V R n V n n R n n n R or n n V Rx V m [ ] R is the molar concentration of n n V
R he application example of osmosis: Dialysis: transport of fluids through cell membranes Osmometry: the determination of molar mass by measuring. Widely used for macromolecules such as proteins and synthetic polymers. When a huge molecule is dissolve in a solvent, the macromolecule solution is far from ideal. It is assumed that the van t off equation is only the st term of a virial-like expansion: J R J CJ 2 Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-20
Reading: page 58 ~ 66 Problem set (Ch. 3): Discussion 3.4 3.6a, 3.8a(a), 3.9a(a), 3.20(a) Solutions of the st Exam Due dates: 325 (May ) 3996 (May 2) Problem set (Ch. 4): Discussion 4.4 4.2a, 4.8a, 4.0a Due dates: 325 (May 6) 3996 (May 7) Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 5-2