Physical Chemistry
Lecture 30
Gibbs’s Phase Rule, Activity
Components and phases
Component -- a compound in a solution
whose amount can be independently varied

Example: Aqueous NaCl solution
 2 components, NaCl and H2O
Phase – a macroscopic piece of matter that is
homogeneous with respect to physical
attributes

Example: a system of oil and vinegar has two
liquid phases because oil and vinegar (an aqueous
acetic acid solution) are immiscible
How many components?
How many phases?
Equilibrium mixture of N2O4 (g) and NO2 (g)


c = 1 because the equilibrium links the concentrations of the
two species
p = 1 (All gases are miscible.)
COCl2 + CO + Cl2 at equilibrium with no excess CO or
Cl2


c = 1 because the equilibrium is determined by
decomposition of phosgene
p = 1 (All gases are miscible.)
COCl2 + CO + Cl2 at equilibrium with excess CO
added (i.e. the concentrations of CO and Cl2 are
unequal)


c =2 because there are three materials but only one
relationship between their concentrations
p=1
How many components?
How many phases?
NH4Cl solid sits in an evacuated chamber. After a
while, some ammonia and HCl appear in the gas
phase above it.


c = 1 because there is an equilibrium relationship AND the
number of moles of NH3 and HCl must be equal. (Why?)
p = 2, solid and gas
NH4Cl (s) in contact with an arbitrary amounts of NH3
(g) and HCl (g) in a chamber


c = 2 (There are three materials, but there is an equilibrium
expression between them that links the concentrations.)
p = 2, solid and gas
Gibbs’s phase rule
Applies to multicomponent systems
Describes the number of degrees of
freedom, f


f = number of intensive variables required to
specify the state
Homogeneous single-component system has two
degrees of freedom, T and P
Depends on the number of components, c,
and the number of phases present, p
f
= c −
p + 2
How many degrees of
freedom?
Equilibrium gas phase containing N2O4 and NO2

f=1 – 1 + 2 = 2
Phosgene, CO and Cl2 (with CO and Cl2 derived from
decomposition exclusively)

f=1 – 1 + 2 = 2
Phosgene, CO and Cl2, with excess CO added

f=2 – 1 + 2 = 3
NH4Cl, initially in an evacuated chamber, with NH3
and HCl present from decomposition

f=1 – 2 + 2 = 1
Water, water vapor and ice in equilibrium


f=1 - 3 + 2 = 0
Triple point of water
Phase rule applied to H2O
F=2

Points in the regions
F=1

Points that are on a
boundary line
F=0


Point at an
intersection
Triple point
There are several points
where H2O has f = 0
There are many
phases of water
I count six points at
which f = 0 on this
graph for H2O
Other materials, like
CO2 have triple
points
From P. Atkins, Physical Chemistry
Equilibrium activity in solution
θ
µ liq (aliq , T ) = µ liq
(T ) + RT ln aliq
For any liquid system
Gas-phase activity is
easily defined
Solution activity defined
with the concept of
µ liq (aliq , T ) = µ gas (a gas , T )
phase equilibrium
Activity in solution is
determined relative to
the activity of the vapor aliquid = exp(− (µ liqθ − µ θgas )/ RT ) = K (T )
a gas
with which it is in
equilibrium
Phase equilibrium with an
ideal-gas phase
Generally activities in
phases are related by
equilibrium condition
Vapor pressure is a
measure of the gasphase activity in ideal
limit
Ideal-gas-limit activity
depends on the gasphase standard state
aliq
ideal gas limit
liq
a
= K a gas
 P 
= K θ
P 
Summary
Gibbs’s phase rule allows estimation of phase
equilibrium


Number of components and phases gives the number of
degrees of freedom, f
Related to the regions, lines and points on a phase diagram
Determination of number of degrees of freedom
depends on equilibrium requirements

Care in expressing the requirements essential
Equality of chemical potentials in two phases relates
the equilibrium activities of materials in the two phases