Colligative properties | Relative lowering of Vapour Pressure | Elevation in boiling point | Depression in freezing point | Osmotic pressure

Four (4) colligative properties

Colligative property-1

Relative lowering of Vapour Pressure Raoult’s Law

When a non-volatile solute is dissolved in a solvent, the vapour pressure of the solvent is lowered.

According, to Raoult’s law, the partial vapour pressure ‘P’ of a constituent of a liquid solution is equal to the product of its mole fraction ‘X₁’ and its vapour pressure in the pure state.

                                              i.e. P = X₁P^o                                         ….(1)

            Where,           X₁        = Mole fraction of solvent

                                    P^o       = the vapour pressure of pure solvent

                                    P         = the vapour pressure of the solvent above a give solution.

            Since X₁ in any solution is less than unity, ‘P’ must be less than ‘P^o’

                                                            OR

            The dissolution of a solute in a solvent leads to lowering the vapour pressure of the latter below that of the pure solvent.

                                    ΔP  = P^o – P

                                           = P^o – X₁P^o

                                                  = P^o (1 – X₁)                   ….(2) 

            For a solution       X₁ + X₂ = 1

                                      1 – X₁ = X₂

                                      ΔP = P^oX₂

            Where, X₂ is mole fraction of solute.

           Thus, the vapour pressure lowering of the solvent depends both on the vapour pressure of the solvent and the mole fraction of solute in solution.

            The relative lowering of vapour pressure is given as,

            Thus, the relative lowering of vapour pressure is equal to the mole fraction of the solute in the solution.

            The relative lowering of vapour pressure is totally independent of the nature of solvent or solute but depends only on the amount of solute in a solution and hence, a colligative property.

Colligative property-2

Elevation in boiling point

Thermodynamically relation between the elevation in the boiling point of the solution and the molecular weight of non-volatile solute dissolved.

The boiling point of solvent is the temperature at which the vapour pressure of the solvent becomes equal to 1 atmosphere. When a non-volatile solute is added, the vapour pressure is lowered and the boiling point of the solution is increased.  The difference in the boiling point of the solution and pure solvent is known as elevation in boiling point

If external pressure is P^o then the solvent boils at T_o. The solution, however, attains this pressure P^o only at temperature T.

            The boiling elevation of the solution ΔT_b is given as.

                        ΔT_b = T – T_o                                                    (1)

            By applying the Clausius Clapeyron equation, we have,

            If ‘n₁’ is no. of a mole of solvent & ‘n₂’ is the number of moles of solute, then the mole fraction of solute is given as,

        Where W₁ & W₂ are weights & M₁ & M₂ are the molecular weights of solvent & solute respectively.

Molality ‘m’ is the number of moles of solute dissolved in 1000 grams of the solvent.

            Since W₁ g of solvent contains W₂/M₂ moles of solute

            Therefore 1000g will contain(W₂/M₂W₁) x 1000 moles of solute which is molality,

Colligative property-3

Depression in freezing point

The relationship between depression in freezing point of the solution and molecular weight of Non-volatile solute dissolved.

The freezing point of the liquid is defined as the temperature at which vapour pressure of the liquid becomes equal to that of solids under one atmospheric pressure.

            When a non-volatile solute is dissolved in a solvent the freezing point of the solvent is lowered.  The difference between the two-freezing point is known as depression in freezing points.

            If we plot a graph of vapour pressure V_s temperature it shows that ‘T_o’ is the freezing point of pure solvent & ‘T’ is the freezing point of the solution.  Then, depression in the freezing point is given as

                        ΔT_f  = T_o – T                                      …. (1)

            Let ‘P_s’ correspond to vapour pressure of solid and pure liquid solvent at To and let ‘P’ be the vapour pressure of solid solvent and solution at temperature ‘T’.  Let Po be the vapour pressure of the pure, super-cooled liquid at temp ‘T’.

            If ‘n₁’ is the number of moles of solvent ₂ ‘n2’ is the number of moles of solute, then the mole fraction of solute. is given as

Where W₁ & W₂ are weights & M₁ & M₂ are molecular weights of solvent & solute respectively.

            Substituting for X₂ in equation (5) we get.

Osmosis

Osmosis is a phenomenon that refers to the movement or passage of solvent molecules from the pure solvent into a solution or from a dilute solution to a concentrated solution by a semipermeable membrane.

A semipermeable membrane is a membrane that is permeable to solvent molecules, but not to the solute particles

Colligative property-4

Osmotic pressure (π)

Osmotic pressure is the pressure that must be applied to the solution side to stop the inward flow of solvent.

Van’t Hoff equation

Consider the pure solvent in compartment I is at constant T and not subjected to any pressure change of solute addition.

                        Therefore d (𝓊_A) Pure solvent = O                         (I)

                      i.e. d (G_A) pure solvent = O                           (II)

                        where,

𝓊_A = chemical potential

                        G  = free energy/mol          

                        The solvent in the solution of compartment (II) is at constant ‘T’ but subject to the addition of a solute and a ‘P’ change.  The free energy changes due to of mole fraction ‘X_A’ and pressure change ‘dp’ is given as.

                        dG_A = V_Adp + RTdℓnX_A                              (III)

                        where,

                        X_A = mole fraction of solvent

            To maintain equilibrium,

   dG_A = O                            (IV)

            Therefore V_Adp = -RTdℓnX_A                              (V)

            Integrating eq. (V) between limit X_A = 1 to X_A pressure P_initial and P_final respectively we get.

            V_A = (P_final – P_initial) = – RTℓnX_A                         (VI)

            The excess of ‘P’ is osmotic pressure ‘π‘ and XA = 1 – XB for dilute solution.

            X_B << 1

            Therefore ℓnX_A @ – X_B

            Substituting in equation (VI)

π = CRT

where,

            C = concentration in terms of moles per liter of solution (molarity of solution)

            Equation (XI) is valid for dilute solution only.

Click the diagram below and read about “ROTATIONAL SPECTROSCOPY“

Molar mass of solute from osmosis

π = CRT

            π = n₂ /v . RT

            n₂ = no. of moles of solute

            v = volume

            n₂ = w₂/ m₂

            where,

w₂ = weight of solute

            m₂ = molar mass of solute

π = .w₂ /m₂v . RT 

            m₂ = w₂ RT/ π v            

By measuring ‘π’, the molar mass can be calculated. 

Isotonic Pressure

 When two solutions have the same osmotic pressure at a given temperature, they are said to be Isotonic pressure.  They are of equimolar concentration.

Vant’s Hoff factor (i)  

It is defined as the ratio of the experimental value of the colligative property to the calculated value of the property assuming ideal behaviors.

Ideal behavior = no association and dissociation.     

Physical Significance

1.  i = 1

            When solute particles, neither dissociates and associates. E.g. Urea in water.

2. i < 1

            When solute particles associate in solution. E.g. Acetic acid in benzene.

3. i > 1             When solute particles dissociate. Eg. NaCl in water

Abnormal Molar masses of solute

The expression of different collective properties is valid for a dilute solution.  Hence it is possible to calculate the molecular weight of different solutes by experimentally determining the magnitude of the colligative property for a solution containing a known amount of solute.  However, the molecular weight of the solute determined is found to be the same as the expected value for it is an abnormal value.  The reason for such abnormality is as follows,

Association in Solution

The same solute like acetic acid, and benzoic acid when dissolving in a solvent like benzene, carbon tetrachloride exists as a dimer in solution. Such association reduces the number of particles in the solution.  Thus the value of the colligative property which depends upon the number of particles is found to be less than the theoretical value.  As a result molar mass calculated is much higher than expected.

    Where,

                   n = no. Of particles produced during associating.

II. Vant’s Hoff factor (i) and degree of dissociation of solute in solution.

Let an electrolyte _Ax_y undergoes dissociation which is represented as-

Where,

 Z+ and Z- are charged on cation and anion respectively.

Therefore x^z+ = y^z- for electrical neutrality.

If ‘C’ molar concentration of electrolyte and ‘∝’ degree of dissociation, it follows that at equilibrium.

Concentration of undissociate  A_xb_y = (1 – ∝) c.

Concentration of dissociated ions = xc_∝ + yc_∝

Therefore Total concentration of solute particles = C [(1 – ∝) + x∝ + y∝]

Relation Between Vant’s Hoff factor (i) and degree of association of solute in solution

Let the association reaction of solute ‘A’ can be represented as

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