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Colligative Properties
These are properties of solutions in the dilute limit, where there is a
solvent “A” and a solute “B” where n A
n
B
These properties are a direct result of ( ,T,p) ( ,T,p)
pure
A
mix
A
μ l <μ l
Use two measures of concentration:
a. Mole Fraction: x B
= n
B
/(n
A
+n
B
) ~ n
B
/n
A
b. Molalility: m B
= (moles solute)/(kg solvent) = n
B
/(n
A
M
A
Where M A
is the mass in kg of one mole of solvent.
There are FOUR Colligative Properties :
- Vapor pressure lowering:
B A
A A A
"p =p !p =!x p
- Boiling point elevation:
b
b b b
"T =T !T =K m
B
vap
2
A b
b
H
M R T
K
- Freezing point depression:
f
f f f
"T =T !T =!K m
B
f
2
A f
f
H
M R T
K
c
!=RT
V
n
c
~ B
= is concentration of solute
- Osmotic pressure:
where
Let’s go through these one at a time:
- Vapor pressure lowering: This is just Raoult’s Law.
B A
A A A
p =x p =( 1 !x )p So
B A
A A A
"p =p !p =!x p (< 0 )
- Boiling point elevation:
Let’s derive this. Start with ( ,T,p) (g,T,p)
A A
μ l =μ
So,
And
( ,T,p) RTlnx (g,T,p)
A A
A
μ l + =μ
RT
G
[ (g,T,p) ( ,T,p )]
RT
ln x
vap
A A A
= μ "μ l =
But lnx A
= ln(1-x
B
) ~ - x
B
= - n
B
/(n
A
+n
B
) ~ - n
B
/n
A
= - (Mn
B
)/(Mn
A
Where M is the total mass of A,
So, lnx A
~ m
B
M
A,
where M
A
is the molar mass of A.
Putting it all together then,
M RT
G
m
A
vap
B
But we need ΔT in there!
2
A
vap
p
vap
p A
B
M RT
H
T
G T
M R
T
m
which gives us
B
vap
2
A
m
H
M RT
T!
Integrating… μ +! "μ = =!
+!
A
p
p
A
A
A
( l,p ,T) (l,p,T) V dp V
(this assumed an incompressible liquid, where volume does not depend
on p)
So then RT lnx V 0
A A
Again using lnx A
~ - n
B
/n
A
Then RT(-n B
/n
A
) + (V
A
/n
A
)π = 0
But V A
~ V
A
+ V
B
= V (V
B
<<V
A
So finally, πV = RTn B
This is the Van’t Hoff Equation. It looks like the ideal gas law!
If we replace c= n
B
/V in the Van’t Hoff Eq., then we get the osmotic
pressure relation:
c
!=RT
