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物理化学电子教案 —第四章
Gaseous
solutions
Solid
solutions Liquidsolutions
Solutions of
non-electrolytes
Solutions of
electrolytes
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Chapter 4 Multi-component Systems
4.1 Introduction
4.2 Expressions of concentration
4.3 Partial molar properties
4.4 Two empirical laws
in dilute liquid solutions
4.5 Chemical potential of each component
in gaseous mixtures
4.6 Liquid mixtures
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Contents
4.11 Distribution law
4.10 Non-ideal liquid solutions
4.9 Gibbs-Duhem relations
4.8 Colligative properties in dilute liquid
solutions
4.7 Chemical potential of each component
in dilute liquid solutions,
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4.1 Introduction
Solutions can be gaseous,liquid,or solid,but
we will be concerned in this chapter primarily
with liquid.
It is sometimes convenient to designate as
solvent that component that is present in
highest concentration.
The remaining components are then called
solutes.
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Expressions of concentration
1,mole fraction
Bx B
B
d e f
(
nx
n 总 ）
We have used mole fraction most often as a
composition measure.When it is convenient
to designate one component as a solvent,it
may also be convenient to express the
composition with respect to each solute as a
ratio relative to the amount of solvent.
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4.2 Expressions of concentration
2,molality mB
B
B
A
d e f nm
m
The molality mB of a solute B is defined as
the amount of substance of solute per unit
mass of solvent.
The relationship between the mole fraction and
the molality can be obtained from the definition
of mole fraction.
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4.2 Expressions of concentration
3,molarity cB,or concentration of B
B
d e f Bnc
V
amount - of - substance
where V is the volume of the
solution,usually in cubic decimeters.
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4.2 Expressions of concentration
4,wB mass fraction of B
)(
B
B
总m
m
w ?
The ratio of mass of solute to the
total mass of solute and solvent.
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The important relations
),( WxZZ ? ),( yxWW ?
dw
w
Zdx
x
ZdZ
xw )()( ?
??
?
?? dy
y
wdx
x
wdw
xy )()( ?
??
?
??
])()[()()( dy
y
wdx
x
w
w
Zdx
x
Z
xyww ?
??
?
?
?
??
?
??
dy
y
w
w
Zdx
x
w
w
Zdx
x
Z
xxyxw )()()()()( ?
?
?
??
?
?
?
??
?
??;
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The chain relation
),( yxZZ ?
dy
y
Z
dx
x
Z
dZ xy )()(
?
?
?
?
?
?
By comparing with the coefficients of dx and dy
in two dZ’s eqs gives,
xxx y
w
w
Z
y
Z )()()(
?
?
?
??
?
?
yxwy x
w
w
Z
x
Z
x
Z )()()()(
?
?
?
??
?
??
?
?
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A better method
dw
w
Zdx
x
ZdZ
xw )()( ?
??
?
??
By dividing by ―dy‖ at const,x;
xxx y
w
w
Z
y
Z )()()(
?
?
?
??
?
?
By dividing by ―dx‖ at const,y;
yxwy x
w
w
Z
x
Z
x
Z )()()()(
?
?
?
??
?
??
?
?
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B
*
Bm,n
VV ?
molar volume
B
*
Bm,n
UU ?
molar thermodynamic energy
For a pure substance
For a homogeneous system:
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The thermodynamic functions
B
*
Bm,n
HH ?
molar enthalpy
B
*
Bm,n
S
S ?
molar entropy
B
*
Bm,n
AA ?
molar Helmholz free energy
B
*
Bm,n
G
G ?
molar Gibbs free energy
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Open systems and changes of composition
The formulation presented so far is obviously
incomplete because it has been tacitly
assumed that composition remains constant,
Furthermore,the expression dG = VdP – SdT
suggests that G is a function of only P and T,and
that when these are held constant dG = 0,this
conflicts with the result derived on p,149 that at
constant P and T we have dG = ?Wmax,
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The two points are connected,
for we shall now see that taking composition
variation into account allows us to identify a source
of non-pV work.
Now we define partial molar quantity and
describe how any thermodynamic quantity depends
on the composition of the system.
Almost everything we do will be confined to
systems with only two components,these are
binary mixtures.
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Preparing to study mixtures:
Imagine an indefinitely large volume of water,
When a further 1 mol of H2O is added the volume
increases by 18 cm3, The quantity 18 cm3 ·mol-1
is the molar volume of pure water,
Now suppose that 1 mol of water is added
to a large volume of pure ethanol,It is found that
the volume increases by only 14 cm3,
The quantity 14 cm3 ·mol-1 is
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the partial molar volume of water in
pure ethanol,The definition of partial molar
volume has depended on the constancy of the
original composition of the solution.
The system is taken to be so large that the
addition of A does not change the mole fractions.
The same constancy can be assured if
the sample is finite,but the addition of A
is limited to an infinitesimal amount.
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V is a state function that depends on
the amounts of A and B present.
Therefore dV is an exact differential,and can be
written
B
nPTB
A
nPTA
dn
n
V
dn
n
V
dV
AB,,,,
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
The partial molar volumes may be identified with
the partial derivatives:
VB,m =
AnPTBnV,,)/( ??
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Definition of partial molar quantities
B,,( c B )
B
de f
()
cT p n
Z
Z
n
?
?
?
The partial molar quantity of Z,
The concept of partial molar quantity can be
extended to any of the extensive thermodynamic
state functions – Z.
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Multi-component systems
The state function Z is a function of
composition as well as pressure and temperature,
and should therefore be written
???
1 2 k(,,,,,)Z Z T p n n n? ? ? ?
At const,T and P,
2 k 1 3 k
1 k - 1
,,,,1,,,,,2
12
,,,,k
k
d ( ) d ( ) d
+ ( ) d
T p n n T p n n n
T p n n
ZZ
nn
nn
Z
n
n
Z ??? ???
???
??
??
??
?
? ? ? ?
?
k
,,( B )
B =1 B
= ( )
cT p n c
Z
n ?
?
??
dnB
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Pay attention to const,ZB
cB,,( B )
B
() T p n cZZ n ??? ?
.
1 2 k
1 1 2 2 k k0 0 0d d d
n n nZ Z n Z n Z n? ? ? ? ? ? ?? ? ?
1 1 2 2 k k
k
BB
B = 1
d d d d
= d
Z Z n Z n Z n
Zn
? ? ? ? ? ? ?
?
then
At const,ZB,the foregoing eq.can be integrated to
= Z1 dn1 + Z2 dn2
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The assemble relation of ZB
1 1 2 2 k kn Z n Z n Z? ? ? ? ? ? ?
k
BB
B = 1
Z= nZ?
= n1 Z1 + n2 Z2
One further property of partial molar quantities has
important implications.
Suppose the concentrations are varied by
small amounts we might expect the Z to
change both because the ni and the Zi.
Z
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Gibbs-Duhem equation
? ?1 1 1 1 k k k kd d d d d 1Z n Z Z n n Z Z n? ? ? ? ? ? ? ?
1 1 2 2 k kZ n Z n Z n Z? ? ? ? ? ? ?
But we have seen already that
? ?1 1 2 2 k kd d d d 2Z Z n Z n Z n? ? ? ? ? ? ?
By differentiating,a general change in Z is given by
(1)-(2):
0?? i
i
i dZn
02211 ?? ?? dndn
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Based on four master equations
definition：
B,,( c B )() cS V n
B
U
n
? ?
?
?
?,,( c B )
()
cS p n
B
H
n
?
?
?
?
,,( c B )() cT V n
B
A
n
?
?
?
?,,( c B )() cT p n
B
G
n
?
?
?
?
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The definition of the chemical potential
The concept of partial molar quantity can be
extended to any of the thermodynamic state
functions.
One already encountered is the partial molar Gibbs
function,or the chemical potential.
We know that the direction of spontaneous
change when T and P are held constant is
towards a minimum of the chemical potential.
for the system.
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Multi-component systems
cBB
k
,,,,( c B ) B
B1 B
d ( ) d ( ) d ( ) dV n S n S V nU U UU S V n
S V n ??
? ? ?? ? ?
? ? ??
BB
B
d d d dH T S V p n?? ? ? ?
BB
B
d d d dA S T p V n?? ? ? ? ?
BB
B
d d d dG S T V p n?? ? ? ? ?
BB
B
d d d dU T S p V n?? ? ? ?
U = U (S,V,n1,n2,…)
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The ? depends on the P.
B c c B c
B
,,,,,,
B
( ) [ ( ) ]T n n T p n T n n
G
p p n
?? ??
?
? ? ?
B c c,,,,
B
[ ( ) ]T n n T p n
G
np
??
?
??
c,,B
B
() T p n
V
V
n
?
??
?
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the ? depends on the T.
B c c B c
B
,,,,,,
B
( ) [ ( ) ]p n n T p n p n n
G
T T n
?? ??
?
? ? ?
B c c,,,,
B
[ ( ) ]p n n T p n
G
nT
??
?
??
,,B
B
()
[ ] =
cT p n
S
S
n
??
??
?
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4.4 Ideal solutions,Raoult’s law
*
A A Ap p x?
)1( B*AA xpp ??
*
A
B*
A
App x
p
? ?
The simplest kind of solution is one in which the two
kinds of molecules in the solution have the same size
and in which the interaction between like and unlike
molecules is the same.
Such a solution is called an ideal solution.
This equation was obtained empirically.
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The implication of Raoult’s law
g
l
T
?
AP
??
A
g
l??A,B
AP
BA xx,
AAA xPP /1/ ?
?
AAA xPP
??
B is a involatile solute.
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4.4 Raoult’s Law and Henry’s Law
Henry’s Law states that
at low concentration
the vapor pressure of the solute obeys
BBxB xKP,?
BBC CK,?
BBm mK,?
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Henry’s Law and Raoult’s Law
For solutions sufficiently dilute
so that the solute follows Henry’s law,
it can be shown that the solvent
follows Raoult’s law.
Important concepts,(1) Ideal mixtures and
ideal dilute solutions; (2) Non-ideal
mixtures and non- ideal solutions.
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The chemical potential of a perfect gas
A pg，
pTn
G
,
B
)( ????
T
pTT n
G
pp ??
?
??
?
?
?
?
??
?
?
,
B
)()( ?
pT
T
B p
G
n,
)( ?
?
?
??
?
?
?
?
??
pTn
V
,B
?
?
?
?
?
?
?
??
mV? )ddd( pVTSG ???
(,) (,) l n pT p T p R T p?? ?? $ $
md d d
p p p
p p p
RTV p p
p? ??? ? ?$ $ $
(,) (,) l n pT p T p R T p?? ??$$ $
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(,) (,) l n pT p T p R T
p
?? ??$$ $
The chemical potential of a perfect gas
Chemical potential indicates potential for
chemical change.
We have to determine how it varies with pressure,
temperature,and composition.
Then we shall apply it to the discussion of
equilibria in a variety of chemically important
systems.
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A pg B,B
BB(,) (,) l n
pT p T p R T
p?? ??
$$
$
Each component in mixtures of pg
A real gas:
? ? ? ? ?
?
?
?
?
?
?? ?????
P
f
RTPTPT ln,,
A rg B:
? ? ? ? ?
?
?
?
?
?
?? ?????
p
f
RTPTPT BBB ln,,
f,fB - the fugacity It plays the role of the pressure,
is a kind of equivalent pressure.
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( ) l n fT R T
p
???$ $( ) l n( )pT R T p ?????$ $
fp ??
f fugacity
? fugacity coefficient
0,1,p f p?? ? ?则
Fugacity and fugacity coefficient
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The chemical potential of liquids
g
l
T
A
?
AP
??
Consider a container in which
there is a liquid in equilibrium
with its vapour,Fig..From the
uniformity of the chemical
potential throughout the system it
is known that μ (I) = μ (g).
T
Since the chemical potential of a gas can be expressed
in terms of its pressure,this equality lets us relate the
chemical potential of a liquid to its vapour pressure.
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A pure liquid
If only component is present (the component A),
the chemical potential of the pure liquid is related
to that of its pure vapour (which is assumed to
behave as a perfect gas) by
?
?
?
?
?
?
???
?
??
?
????
P
P
RTgl AAAA ln)()(
where is the vapour pressure of the pure
liquid at that temperature (the star * is used to
indicate a pure component).
?AP
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Mixture of ideal liquid
g
lT ??
BA xx,
BA PP,
In mixture of ideal liquid each
component obeys Raoult’s law.
A
?
AP
B
?
BP
BBB
AAA
xPP
xPP
?
?
?
?
)()( gl AA ?? ?
?
?
?
?
?
?
?? ???
P
P
RT AA ln
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Substitute Raoult’s or Henry’s law
?
?
?
?
?
?
? ?
?
?
? ? ?
P
x P
RT pg lA A Aln ) ( ) (
A
A xRT
P
P
RTpg lnln)( ??
?
?
?
?
?
??
?
?
??
AAA xRTp ln)( ??
???)( lA?
AAA xRTLl ln)()( ??
???
∥
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The solute in ideal dilute solution
Mixture of ideal liquid; Ideal dilute solution
(A1,A2,A3···) (A,B1,B2,B3···)
The chemical potential of the solutes:
???BA xx,
???BA PP,
??
)(.)( gs o l BB ?? ?
?
?
?
?
?
?
?? ???
P
P
RTpg Bln)(
)( s o lB?
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Henry’s Law：
B B B Bx m cp k x k m k c? ? ?
B B B
BB
*
BB
( 1 ) (,) ( ) l n( / )
= ( ) l n( / ) l n
= (,) ln
x
T p T RT p p
T RT k p RT
Tx
x
p RT
??
?
?
??
?
??
$$
$$
The chemical potential of solute B
)( s o lB?
BBB xRTs o ls o l ln)()( ??
???
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The chemical potential of solute B
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BB
**
B
= (
(,) = ( ) l n l n
,) l n
m
km m
T p T R T R T
pm
m
T p R T
m
?
??
?
?
?
?
$
$
$$
$
（ 2）
BBmp k m?
**B (,)Tp? mm? $
-11 m ol kgm ??$
The chemical potential of solute B
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The chemical potential of solute B
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*
B
*
B
B
*
= (,) l n
(,) = ( ) l n l n
c
kc c
T p T R T R T
pc
c
T p R T
c
??
?
?
?
?
?
$
$
$$
$
（ 3）
BBcp k c?
***B (,)Tp? cc? $
-31 m o l d m c ?? 。$
The chemical potential of solute B
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The chemical potential of solute B
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4,8 Colligative properties
colligative properties,A property that
depends on the relative number of solute
particles and that is independent of their
nature is called a colligative property,
In addition to vapor pressure lowering,
boiling point elevation,freezing point
depression,and osmotic pressure
are colligative properties.
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B
*
AA
*
A xpppp ????
The decrease in the vapor pressure
AAA xPP ?? )1( BA xP ??
?
BAA xPP
?? ??
of the solvent is proportional to
the mole fraction of solute and independent of
the nature of the solute,
Thus,for an ideal solution,or for
the limiting case of a real solution that is
sufficiently dilute,…
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freezing point lowering coefficients，
fk
1K m ol kg ??? 。
f f BT k m?? f
*
ff TTT ???
Freezing point depression
freezing point depression constant or
cryoscopic constant of solvent A.
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b b BT k m?? *bbb TTT ???
Boiling point elevation
Kb is the boiling point elevation coefficient
Only for solute non-volatile.
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osmotic pressureosmotic pressure
Osmosis is the
spontaneous transfer
of solvent from a
dilute solution to a
more concentrated
solution through a
semipermeable
membrane,which is
permeable to solvent
but not to solute.
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osmotic pressure
AAA xRTPTPT ln),(),( ?? ???
1?Ax
???
AA ?
Solvent transfers from left
to right.
? = CBRT ; CB-mol·m -3
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4.9 Gibbs-Duhem-equation
B B B B
BB
d 0 d 0n z x z???? 或
A A B B
B
AB
A
d d 0
dd
x V x V
xVV
x
??
??
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G,N,Lewis defined the activity ?x,B of
component B by the relation
4.10 Real solutions and activities
For real solutions Raoult’s law is modified as
BB
*
BB xpp ??
,B,B Bxxax??
BB
,B
,B 11
B
l i m ( ) 1l i m xx
xx
a
x? ?? ??
BBP ?
??
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non-ideal solutions
is the activity,Bxa,Bx? is
the activity coefficient of a component in a
real mixture.
How to calculate the activity of a solvent A?
(1) PA is known,assumed the gaseous phase is
an ideal gas mixture;
(2) Assumed the gas obeys Raoult’s law,then
PA / PA *,the result is the activity ?A,
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non-ideal solutions
(1) use
Bx
B B B,BB l n ( / ) ( ) l n ( / ) l n xxR T p p T R T k p R T a?? ?? ? ? ? ?$ $ $ $
The chemical potential of solute B:
*
B,B = (,) l n xT p R T a? ?B?
The key is using Henry’s law!!
BBxB kP ?,?
Substituting
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non-ideal solutions
（ 2） use
Bm
B,BB
**
B,B
( ) l n l n
(,) l n
m
m
mT
km
T R T
pR
Ra
Ta
T
p
?
?
???
??
?
$
$
$
B
m,B m,B m
ma ???
$
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non-ideal solutions
（ 3） use
Bc
B,BB
***
B,B
( ) l n l n
= (,) l n
c
c
c
kc
TR
T p R T
TT
p
a
Ra?
?
?
?
? ? ?
$
$
$
B
,B,Bcc
ca
c??? $
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m i x m i x
m i x m i x
0,0
0,0
VH
GS
? ? ? ?
? ? ? ?
Ideal solutions,Raoult’s law
There is no volume change on mixing,so △ V = 0;
and no enthalpy change on mixing,so △ H = 0.
BBAAm i x xRnxRnS lnln ????
m i xm i x STG ????
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The thermodynamics of mixing process
m ix id
1 1 2 2 1 1 2 2
BB
B
1 1 2 2
( ) (
=( ) ( )
l = n l n nl nR
G G G
n n n n
n RT x n RT x Tx
? ? ? ?
??
? ? ?
??
?
?
混合后 混合前)
$$
?? idm i x S
? ?
P
idm i x
T
G
??
?
??
?
?
??
??
???
B
BB xRn ln
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4.11 Distribution equilibrium
B
B
c
K
c
?
?
?
If solute B dissolves in two solvents,A and C,
which are immiscible,and if B follows Henry’s law
in each solvent,we can derive an expression for the
equilibrium distribution of B between A and C
when the two solutions are in contact,
The condition of equilibrium is
CBAB,,?? ?
,or
CBCBABAB xRTxRT,,,,lnln ???
?? ?? The end
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FRANCOLS-MARIE RAOULT
FRANCOLS-MARIE RAOULT (1830-1901)
French chemist,was a pioneer in solution
chemistry,His work on vapor-pressure lowering and on
freezing-point depressions was fundamentally important
in the chemistry of solutions,and his fealization that both
were a function of the number of moles of dissolved
solute contributed greatly to the determination of molar
masses and the theory of ionic solutions,
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FRANCOLS-MARIE RAOULT
His contributions to cryoscopy,or freezing-point
depressions,were prodigious,and Victor Meyer used his
data as early as 1886 for determining molar masses,
Raoult built for these measurements what he called a
cryoscope of precision,and the accuracy of his work was
unsurpassed,agreeing in many instances with modern
measurements within 0.0001 ℃,
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FRANCOLS-MARIE RAOULT
His thermometer was considered by some to be
antediluvian,but as van’t Hoff expressed it,―With
this antediluvian thermometer the world was
conquered.‖
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WILLIAM HENRY
WILLIAM HENRY(1775-1836)
was an English chemist,textbook writer,and
translator of Lavoisier,While he was engaged in
experiments of the amount of gas absorbed by water,he
discovered what is now known as Henry’s law; the
amount of gas absorbed is directly proportional to the
pressure,Henry committed suicide in a fit of
melancholia.
物理化学电子教案 —第四章
Gaseous
solutions
Solid
solutions Liquidsolutions
Solutions of
non-electrolytes
Solutions of
electrolytes
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Chapter 4 Multi-component Systems
4.1 Introduction
4.2 Expressions of concentration
4.3 Partial molar properties
4.4 Two empirical laws
in dilute liquid solutions
4.5 Chemical potential of each component
in gaseous mixtures
4.6 Liquid mixtures
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Contents
4.11 Distribution law
4.10 Non-ideal liquid solutions
4.9 Gibbs-Duhem relations
4.8 Colligative properties in dilute liquid
solutions
4.7 Chemical potential of each component
in dilute liquid solutions,
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4.1 Introduction
Solutions can be gaseous,liquid,or solid,but
we will be concerned in this chapter primarily
with liquid.
It is sometimes convenient to designate as
solvent that component that is present in
highest concentration.
The remaining components are then called
solutes.
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Expressions of concentration
1,mole fraction
Bx B
B
d e f
(
nx
n 总 ）
We have used mole fraction most often as a
composition measure.When it is convenient
to designate one component as a solvent,it
may also be convenient to express the
composition with respect to each solute as a
ratio relative to the amount of solvent.
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4.2 Expressions of concentration
2,molality mB
B
B
A
d e f nm
m
The molality mB of a solute B is defined as
the amount of substance of solute per unit
mass of solvent.
The relationship between the mole fraction and
the molality can be obtained from the definition
of mole fraction.
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4.2 Expressions of concentration
3,molarity cB,or concentration of B
B
d e f Bnc
V
amount - of - substance
where V is the volume of the
solution,usually in cubic decimeters.
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4.2 Expressions of concentration
4,wB mass fraction of B
)(
B
B
总m
m
w ?
The ratio of mass of solute to the
total mass of solute and solvent.
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The important relations
),( WxZZ ? ),( yxWW ?
dw
w
Zdx
x
ZdZ
xw )()( ?
??
?
?? dy
y
wdx
x
wdw
xy )()( ?
??
?
??
])()[()()( dy
y
wdx
x
w
w
Zdx
x
Z
xyww ?
??
?
?
?
??
?
??
dy
y
w
w
Zdx
x
w
w
Zdx
x
Z
xxyxw )()()()()( ?
?
?
??
?
?
?
??
?
??;
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The chain relation
),( yxZZ ?
dy
y
Z
dx
x
Z
dZ xy )()(
?
?
?
?
?
?
By comparing with the coefficients of dx and dy
in two dZ’s eqs gives,
xxx y
w
w
Z
y
Z )()()(
?
?
?
??
?
?
yxwy x
w
w
Z
x
Z
x
Z )()()()(
?
?
?
??
?
??
?
?
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A better method
dw
w
Zdx
x
ZdZ
xw )()( ?
??
?
??
By dividing by ―dy‖ at const,x;
xxx y
w
w
Z
y
Z )()()(
?
?
?
??
?
?
By dividing by ―dx‖ at const,y;
yxwy x
w
w
Z
x
Z
x
Z )()()()(
?
?
?
??
?
??
?
?
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B
*
Bm,n
VV ?
molar volume
B
*
Bm,n
UU ?
molar thermodynamic energy
For a pure substance
For a homogeneous system:
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The thermodynamic functions
B
*
Bm,n
HH ?
molar enthalpy
B
*
Bm,n
S
S ?
molar entropy
B
*
Bm,n
AA ?
molar Helmholz free energy
B
*
Bm,n
G
G ?
molar Gibbs free energy
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Open systems and changes of composition
The formulation presented so far is obviously
incomplete because it has been tacitly
assumed that composition remains constant,
Furthermore,the expression dG = VdP – SdT
suggests that G is a function of only P and T,and
that when these are held constant dG = 0,this
conflicts with the result derived on p,149 that at
constant P and T we have dG = ?Wmax,
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The two points are connected,
for we shall now see that taking composition
variation into account allows us to identify a source
of non-pV work.
Now we define partial molar quantity and
describe how any thermodynamic quantity depends
on the composition of the system.
Almost everything we do will be confined to
systems with only two components,these are
binary mixtures.
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Preparing to study mixtures:
Imagine an indefinitely large volume of water,
When a further 1 mol of H2O is added the volume
increases by 18 cm3, The quantity 18 cm3 ·mol-1
is the molar volume of pure water,
Now suppose that 1 mol of water is added
to a large volume of pure ethanol,It is found that
the volume increases by only 14 cm3,
The quantity 14 cm3 ·mol-1 is
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the partial molar volume of water in
pure ethanol,The definition of partial molar
volume has depended on the constancy of the
original composition of the solution.
The system is taken to be so large that the
addition of A does not change the mole fractions.
The same constancy can be assured if
the sample is finite,but the addition of A
is limited to an infinitesimal amount.
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V is a state function that depends on
the amounts of A and B present.
Therefore dV is an exact differential,and can be
written
B
nPTB
A
nPTA
dn
n
V
dn
n
V
dV
AB,,,,
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
The partial molar volumes may be identified with
the partial derivatives:
VB,m =
AnPTBnV,,)/( ??
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Definition of partial molar quantities
B,,( c B )
B
de f
()
cT p n
Z
Z
n
?
?
?
The partial molar quantity of Z,
The concept of partial molar quantity can be
extended to any of the extensive thermodynamic
state functions – Z.
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Multi-component systems
The state function Z is a function of
composition as well as pressure and temperature,
and should therefore be written
???
1 2 k(,,,,,)Z Z T p n n n? ? ? ?
At const,T and P,
2 k 1 3 k
1 k - 1
,,,,1,,,,,2
12
,,,,k
k
d ( ) d ( ) d
+ ( ) d
T p n n T p n n n
T p n n
ZZ
nn
nn
Z
n
n
Z ??? ???
???
??
??
??
?
? ? ? ?
?
k
,,( B )
B =1 B
= ( )
cT p n c
Z
n ?
?
??
dnB
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Pay attention to const,ZB
cB,,( B )
B
() T p n cZZ n ??? ?
.
1 2 k
1 1 2 2 k k0 0 0d d d
n n nZ Z n Z n Z n? ? ? ? ? ? ?? ? ?
1 1 2 2 k k
k
BB
B = 1
d d d d
= d
Z Z n Z n Z n
Zn
? ? ? ? ? ? ?
?
then
At const,ZB,the foregoing eq.can be integrated to
= Z1 dn1 + Z2 dn2
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The assemble relation of ZB
1 1 2 2 k kn Z n Z n Z? ? ? ? ? ? ?
k
BB
B = 1
Z= nZ?
= n1 Z1 + n2 Z2
One further property of partial molar quantities has
important implications.
Suppose the concentrations are varied by
small amounts we might expect the Z to
change both because the ni and the Zi.
Z
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Gibbs-Duhem equation
? ?1 1 1 1 k k k kd d d d d 1Z n Z Z n n Z Z n? ? ? ? ? ? ? ?
1 1 2 2 k kZ n Z n Z n Z? ? ? ? ? ? ?
But we have seen already that
? ?1 1 2 2 k kd d d d 2Z Z n Z n Z n? ? ? ? ? ? ?
By differentiating,a general change in Z is given by
(1)-(2):
0?? i
i
i dZn
02211 ?? ?? dndn
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Based on four master equations
definition：
B,,( c B )() cS V n
B
U
n
? ?
?
?
?,,( c B )
()
cS p n
B
H
n
?
?
?
?
,,( c B )() cT V n
B
A
n
?
?
?
?,,( c B )() cT p n
B
G
n
?
?
?
?
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The definition of the chemical potential
The concept of partial molar quantity can be
extended to any of the thermodynamic state
functions.
One already encountered is the partial molar Gibbs
function,or the chemical potential.
We know that the direction of spontaneous
change when T and P are held constant is
towards a minimum of the chemical potential.
for the system.
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Multi-component systems
cBB
k
,,,,( c B ) B
B1 B
d ( ) d ( ) d ( ) dV n S n S V nU U UU S V n
S V n ??
? ? ?? ? ?
? ? ??
BB
B
d d d dH T S V p n?? ? ? ?
BB
B
d d d dA S T p V n?? ? ? ? ?
BB
B
d d d dG S T V p n?? ? ? ? ?
BB
B
d d d dU T S p V n?? ? ? ?
U = U (S,V,n1,n2,…)
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The ? depends on the P.
B c c B c
B
,,,,,,
B
( ) [ ( ) ]T n n T p n T n n
G
p p n
?? ??
?
? ? ?
B c c,,,,
B
[ ( ) ]T n n T p n
G
np
??
?
??
c,,B
B
() T p n
V
V
n
?
??
?
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the ? depends on the T.
B c c B c
B
,,,,,,
B
( ) [ ( ) ]p n n T p n p n n
G
T T n
?? ??
?
? ? ?
B c c,,,,
B
[ ( ) ]p n n T p n
G
nT
??
?
??
,,B
B
()
[ ] =
cT p n
S
S
n
??
??
?
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4.4 Ideal solutions,Raoult’s law
*
A A Ap p x?
)1( B*AA xpp ??
*
A
B*
A
App x
p
? ?
The simplest kind of solution is one in which the two
kinds of molecules in the solution have the same size
and in which the interaction between like and unlike
molecules is the same.
Such a solution is called an ideal solution.
This equation was obtained empirically.
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The implication of Raoult’s law
g
l
T
?
AP
??
A
g
l??A,B
AP
BA xx,
AAA xPP /1/ ?
?
AAA xPP
??
B is a involatile solute.
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4.4 Raoult’s Law and Henry’s Law
Henry’s Law states that
at low concentration
the vapor pressure of the solute obeys
BBxB xKP,?
BBC CK,?
BBm mK,?
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Henry’s Law and Raoult’s Law
For solutions sufficiently dilute
so that the solute follows Henry’s law,
it can be shown that the solvent
follows Raoult’s law.
Important concepts,(1) Ideal mixtures and
ideal dilute solutions; (2) Non-ideal
mixtures and non- ideal solutions.
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The chemical potential of a perfect gas
A pg，
pTn
G
,
B
)( ????
T
pTT n
G
pp ??
?
??
?
?
?
?
??
?
?
,
B
)()( ?
pT
T
B p
G
n,
)( ?
?
?
??
?
?
?
?
??
pTn
V
,B
?
?
?
?
?
?
?
??
mV? )ddd( pVTSG ???
(,) (,) l n pT p T p R T p?? ?? $ $
md d d
p p p
p p p
RTV p p
p? ??? ? ?$ $ $
(,) (,) l n pT p T p R T p?? ??$$ $
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(,) (,) l n pT p T p R T
p
?? ??$$ $
The chemical potential of a perfect gas
Chemical potential indicates potential for
chemical change.
We have to determine how it varies with pressure,
temperature,and composition.
Then we shall apply it to the discussion of
equilibria in a variety of chemically important
systems.
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A pg B,B
BB(,) (,) l n
pT p T p R T
p?? ??
$$
$
Each component in mixtures of pg
A real gas:
? ? ? ? ?
?
?
?
?
?
?? ?????
P
f
RTPTPT ln,,
A rg B:
? ? ? ? ?
?
?
?
?
?
?? ?????
p
f
RTPTPT BBB ln,,
f,fB - the fugacity It plays the role of the pressure,
is a kind of equivalent pressure.
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( ) l n fT R T
p
???$ $( ) l n( )pT R T p ?????$ $
fp ??
f fugacity
? fugacity coefficient
0,1,p f p?? ? ?则
Fugacity and fugacity coefficient
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The chemical potential of liquids
g
l
T
A
?
AP
??
Consider a container in which
there is a liquid in equilibrium
with its vapour,Fig..From the
uniformity of the chemical
potential throughout the system it
is known that μ (I) = μ (g).
T
Since the chemical potential of a gas can be expressed
in terms of its pressure,this equality lets us relate the
chemical potential of a liquid to its vapour pressure.
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A pure liquid
If only component is present (the component A),
the chemical potential of the pure liquid is related
to that of its pure vapour (which is assumed to
behave as a perfect gas) by
?
?
?
?
?
?
???
?
??
?
????
P
P
RTgl AAAA ln)()(
where is the vapour pressure of the pure
liquid at that temperature (the star * is used to
indicate a pure component).
?AP
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Mixture of ideal liquid
g
lT ??
BA xx,
BA PP,
In mixture of ideal liquid each
component obeys Raoult’s law.
A
?
AP
B
?
BP
BBB
AAA
xPP
xPP
?
?
?
?
)()( gl AA ?? ?
?
?
?
?
?
?
?? ???
P
P
RT AA ln
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Substitute Raoult’s or Henry’s law
?
?
?
?
?
?
? ?
?
?
? ? ?
P
x P
RT pg lA A Aln ) ( ) (
A
A xRT
P
P
RTpg lnln)( ??
?
?
?
?
?
??
?
?
??
AAA xRTp ln)( ??
???)( lA?
AAA xRTLl ln)()( ??
???
∥
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The solute in ideal dilute solution
Mixture of ideal liquid; Ideal dilute solution
(A1,A2,A3···) (A,B1,B2,B3···)
The chemical potential of the solutes:
???BA xx,
???BA PP,
??
)(.)( gs o l BB ?? ?
?
?
?
?
?
?
?? ???
P
P
RTpg Bln)(
)( s o lB?
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Henry’s Law：
B B B Bx m cp k x k m k c? ? ?
B B B
BB
*
BB
( 1 ) (,) ( ) l n( / )
= ( ) l n( / ) l n
= (,) ln
x
T p T RT p p
T RT k p RT
Tx
x
p RT
??
?
?
??
?
??
$$
$$
The chemical potential of solute B
)( s o lB?
BBB xRTs o ls o l ln)()( ??
???
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The chemical potential of solute B
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BB
**
B
= (
(,) = ( ) l n l n
,) l n
m
km m
T p T R T R T
pm
m
T p R T
m
?
??
?
?
?
?
$
$
$$
$
（ 2）
BBmp k m?
**B (,)Tp? mm? $
-11 m ol kgm ??$
The chemical potential of solute B
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The chemical potential of solute B
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*
B
*
B
B
*
= (,) l n
(,) = ( ) l n l n
c
kc c
T p T R T R T
pc
c
T p R T
c
??
?
?
?
?
?
$
$
$$
$
（ 3）
BBcp k c?
***B (,)Tp? cc? $
-31 m o l d m c ?? 。$
The chemical potential of solute B
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The chemical potential of solute B
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4,8 Colligative properties
colligative properties,A property that
depends on the relative number of solute
particles and that is independent of their
nature is called a colligative property,
In addition to vapor pressure lowering,
boiling point elevation,freezing point
depression,and osmotic pressure
are colligative properties.
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B
*
AA
*
A xpppp ????
The decrease in the vapor pressure
AAA xPP ?? )1( BA xP ??
?
BAA xPP
?? ??
of the solvent is proportional to
the mole fraction of solute and independent of
the nature of the solute,
Thus,for an ideal solution,or for
the limiting case of a real solution that is
sufficiently dilute,…
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freezing point lowering coefficients，
fk
1K m ol kg ??? 。
f f BT k m?? f
*
ff TTT ???
Freezing point depression
freezing point depression constant or
cryoscopic constant of solvent A.
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b b BT k m?? *bbb TTT ???
Boiling point elevation
Kb is the boiling point elevation coefficient
Only for solute non-volatile.
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osmotic pressureosmotic pressure
Osmosis is the
spontaneous transfer
of solvent from a
dilute solution to a
more concentrated
solution through a
semipermeable
membrane,which is
permeable to solvent
but not to solute.
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osmotic pressure
AAA xRTPTPT ln),(),( ?? ???
1?Ax
???
AA ?
Solvent transfers from left
to right.
? = CBRT ; CB-mol·m -3
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4.9 Gibbs-Duhem-equation
B B B B
BB
d 0 d 0n z x z???? 或
A A B B
B
AB
A
d d 0
dd
x V x V
xVV
x
??
??
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G,N,Lewis defined the activity ?x,B of
component B by the relation
4.10 Real solutions and activities
For real solutions Raoult’s law is modified as
BB
*
BB xpp ??
,B,B Bxxax??
BB
,B
,B 11
B
l i m ( ) 1l i m xx
xx
a
x? ?? ??
BBP ?
??
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non-ideal solutions
is the activity,Bxa,Bx? is
the activity coefficient of a component in a
real mixture.
How to calculate the activity of a solvent A?
(1) PA is known,assumed the gaseous phase is
an ideal gas mixture;
(2) Assumed the gas obeys Raoult’s law,then
PA / PA *,the result is the activity ?A,
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non-ideal solutions
(1) use
Bx
B B B,BB l n ( / ) ( ) l n ( / ) l n xxR T p p T R T k p R T a?? ?? ? ? ? ?$ $ $ $
The chemical potential of solute B:
*
B,B = (,) l n xT p R T a? ?B?
The key is using Henry’s law!!
BBxB kP ?,?
Substituting
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non-ideal solutions
（ 2） use
Bm
B,BB
**
B,B
( ) l n l n
(,) l n
m
m
mT
km
T R T
pR
Ra
Ta
T
p
?
?
???
??
?
$
$
$
B
m,B m,B m
ma ???
$
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non-ideal solutions
（ 3） use
Bc
B,BB
***
B,B
( ) l n l n
= (,) l n
c
c
c
kc
TR
T p R T
TT
p
a
Ra?
?
?
?
? ? ?
$
$
$
B
,B,Bcc
ca
c??? $
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m i x m i x
m i x m i x
0,0
0,0
VH
GS
? ? ? ?
? ? ? ?
Ideal solutions,Raoult’s law
There is no volume change on mixing,so △ V = 0;
and no enthalpy change on mixing,so △ H = 0.
BBAAm i x xRnxRnS lnln ????
m i xm i x STG ????
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The thermodynamics of mixing process
m ix id
1 1 2 2 1 1 2 2
BB
B
1 1 2 2
( ) (
=( ) ( )
l = n l n nl nR
G G G
n n n n
n RT x n RT x Tx
? ? ? ?
??
? ? ?
??
?
?
混合后 混合前)
$$
?? idm i x S
? ?
P
idm i x
T
G
??
?
??
?
?
??
??
???
B
BB xRn ln
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4.11 Distribution equilibrium
B
B
c
K
c
?
?
?
If solute B dissolves in two solvents,A and C,
which are immiscible,and if B follows Henry’s law
in each solvent,we can derive an expression for the
equilibrium distribution of B between A and C
when the two solutions are in contact,
The condition of equilibrium is
CBAB,,?? ?
,or
CBCBABAB xRTxRT,,,,lnln ???
?? ?? The end
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FRANCOLS-MARIE RAOULT
FRANCOLS-MARIE RAOULT (1830-1901)
French chemist,was a pioneer in solution
chemistry,His work on vapor-pressure lowering and on
freezing-point depressions was fundamentally important
in the chemistry of solutions,and his fealization that both
were a function of the number of moles of dissolved
solute contributed greatly to the determination of molar
masses and the theory of ionic solutions,
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FRANCOLS-MARIE RAOULT
His contributions to cryoscopy,or freezing-point
depressions,were prodigious,and Victor Meyer used his
data as early as 1886 for determining molar masses,
Raoult built for these measurements what he called a
cryoscope of precision,and the accuracy of his work was
unsurpassed,agreeing in many instances with modern
measurements within 0.0001 ℃,
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FRANCOLS-MARIE RAOULT
His thermometer was considered by some to be
antediluvian,but as van’t Hoff expressed it,―With
this antediluvian thermometer the world was
conquered.‖
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WILLIAM HENRY
WILLIAM HENRY(1775-1836)
was an English chemist,textbook writer,and
translator of Lavoisier,While he was engaged in
experiments of the amount of gas absorbed by water,he
discovered what is now known as Henry’s law; the
amount of gas absorbed is directly proportional to the
pressure,Henry committed suicide in a fit of
melancholia.
