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5.60 Spring 2008 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 n A >> nB.

These properties are a direct result of Amix( ,T,p) A

pure ( ,T,p)

Use two measures of concentration:

a. Mole Fraction: x B = nB/(n A+nB) ~ nB/n A

b. Molalility: mB = (moles solute)/(kg solvent) = n B/(n AMA)Where M A is the mass in kg of one mole of solvent.

There are FOUR Colligative Properties :

1. Vapor pressure lowering: pA pA pA* xBpA*

2. Boiling point elevation: T b T b T b* Kb mB

*M RT 2

Kb A b

Hvap

3. Freezing point depression: T f T f T f* Kf mB

*

MART f2

Kf Hf

4. Osmotic pressure: RTc~

where c~ nB is concentration of soluteV

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5.60 Spring 2008 Lecture #23 page 2

Lets go through these one at a time:

1. Vapor pressure lowering: This is just Raoults Law.

pA xApA* (1 xB)pA

* So pA pA pA* xBpA* (

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5.60 Spring 2008 Lecture #23 page 3

If mB = mB 0 (mixed pure) and m B is very small

T T b T b* MAR(T b

* )2mB KbmB

HvapThen

3. Freezing point depression:

Same arguments as above, replacing Gvap with GfHvap with Hf

T b with T fKb with Kf

4. Osmotic Pressure:

A

A+B

pext

The pressures at points:

h : p = pext + g

: p + = pext + g + h g

At equilibrium: A( ,p

,T)

A

*

( ,p,T)

Using Raoults Law: RTln xA A* ( ,p ,T) A* ( ,p,T) 0

At constant T: dG = Vdp, or d A* VA

*dp

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5.60 Spring 2008 Lecture #23 page 4

p

Integrating A* ( ,p ,T) A* ( ,p,T) VA*dp VA*

p

(this assumed an incompressible liquid, where volume does not depend

on p)

So then RTlnx A VA* 0

Again using lnxA ~ -nB/n A

Then RT(-n B/n A) + (VA/n A) = 0

But VA ~ VA + VB = V (VB