Phase, Phase Equilibrium & Solution
※ Phase, homogeneous, & heterogeneous:
Phase - a form of matter that is uniform throughout in chemical
composition and physical state; i.e. phase is
homogeneous.
Heterogeneous - a system is composed of two or more phases.
※ Criteria of Phase Equilibrium:
Three requirements must be satisfied when system is at phase
equilibria:
★ at the same temperature (T)
★ at the same pressure (P)
★ at the same chemical potential (μ)
※ Phase number, Component number & Degree
of freedom:
★ phase number, p, (相數): the number of homogeneous phase.
通常系統中的氣相、液相和固相的數目，可以下列的方法
來確定：
(1) 氣相：通常氣體能無限互溶(假定氣相不發生化學反應
1
時)，所以系統無論有多少氣體只可能有一個氣相。
(2) 液相：依液體的互溶程度而定。
(3) 固相：一般是有幾種固體便有幾個相，但不同固體若能形
成固體溶液則為一個相。 若同一種物質以不同晶形存在，
則每一種晶型為一相。
★ component number, c, (成份數): the smallest number of
independent chemical constituents needed to fix the
composition of every phase in the system.
Substance number, s, (物種數): the number of substance
involved in the system.
There are two types of relations between substances:
(1) chemical equilibrium relation (n)
(2) independent concentration (m)
◎ The number of component is determined by: c = s - n - m.
ex: A system contains N2, H2, and NH3 mixture without catalyst.
∴c = 3.
ex: A system contains N2, H2, and NH3 mixture at the presence of
iron catalyst. ∴c = 3 - 1 = 2.
ex: A system contains N2, H2, and NH3 mixture at the presence of
iron catalyst with equal concentration of N2 and H2.
∴c = 3 - 2 = 1.
2
★ Degree of freedom, f, (自由度): the smallest number of
independent variables, such as temperature, pressure, and
concentration, that must be specified to describe completely
the state of the system .
◎ f = c - p + 2  phase rule
◎ phase rule: the rule governs the relationship between the
number of phases in equilibrium, the number of components,
and the number of intensive independent variables that must
be specified in order to describe the state of the system
completely.
◎ Derivations:
Consider a system in equilibrium that consists of p phases. If
a phase contains c components, p(c-1) concentration terms
are required to define the composition completely, since
c
 x  1 . In addition, there are two additional variables
i
i
that have to be considered, T and P.
The total number of
independent variables is p(c-1)+2.
At equilibrium, the chemical potentialμfor each component is
the same in each phase; i.e.
3
There are p phases, but only (p-1) equilibrium relationships
for each component.
Thus, there is a total of c(p-1)
equilibrium relations.
The number of degrees of freedom f is equal to the total
number of variables minus the total number of equilibrium
relations:
∴f = [p(c-1) + 2] - c(p-1) = c - p + 2
ex: Determine the number of degrees of freedom for a system with
a fixed amount of an ideal gas?
Sol: ∵PV=nRT
∴f = 1 - 1 +2 = 2.
Either P/V, P/T or V/T
must be specified.
※ One-component System - Water:
From the phase rule, f = c - p + 2.
∵c = 1, ∴f = 3 - p.
For a one-component system,
specification of at most two
intensive variables describes the
4
intensive state.
We can represent any intensive state of system by a point
on a two-dimensional P versus T diagram, where each point corresponds to
a definite T and P.
Such a diagram is called the phase diagram.
◇ Single phase: f = 2; P & T must be known.
◇ Two phases : f = 1. Either P or T is known only.
◇ Three phases: f = 0. Neither P, or T must be known only.
◇ Normal boiling point: the temperature at which a liquid vaporizes at 1
atm.
◇ Normal melting point: the temperature at which a solid melts at 1 atm.
◇ Triple point (三相點) - the point at which vapor, liquid and solid
coexist. For water, the triple point is at 4.58 torr and 0.01℃.
※ The Clapeyron equation:
由水的相圖得知，單成份兩相平衡系統的溫度與壓力不是相
互獨力的。 二者的關係我們可直接用熱力學導出，此關係式
稱之為 Clapeyron 方程式。
If two phases of a pure substance are in equilibrium, the
chemical potentials, or ,molar Gibbs energies, are equal:
G  G
If the pressure and temperature are changed so that equilibrium
is maintained, it is necessary that dG  dG .
∵ G = f(P, T)
5
 G  
 G  
 G 
 G 
dP

dT

dP








 dT
 P  T
 T  P
 P  T
 T  P
V dP  S dT  V dP  S  dT
dP
dT

S  S S
H


V  V V T V
 Clapeyron equation
※The Clausius—Clapeyron equation:
For vapor-liquid equilibrium: if the vapor obeys the ideal gas
law and the liquid volume is neglected in comparison with the
vapor volume,
dP
dT
n

P2
P1
H vap
T Vv


P H vap
RT
2
 H vap  1
R

T

2

1

T1 

dP

P
H vap
RT
2
dT
 Clausius-Clapeyron equation
 Determination of vaporization energy or sublimation energy.
 The normal boiling point.
 A simple way to calculate the vaporization energy by
Trouton’s rule:
H vap
Tb
 88 joule
K  mole
for non-hydrogen
bond compound.
※ Variation of Vapor Pressure with External
6
Pressure:
Consider that a closed system is at constant temperature.
The vapor pressure may be affected by the addition of the inert
gas which is insoluble in the liquid.

Vm (v)dP  Vm (  )dP
t
n
P
Pv

Vm (  )
RT
dP
dPt
P  P  
t
v

Vm (  )
Vm ( v )
Gibbs equation
where Pv = the saturated vapor pressure
Pt = the total external pressure
※ Two Components - Liquid and Vapor:
For a two-component system, f = 4 - p.
Since there is at least
one phase, the maximum number of degrees of freedom is 3,
so that the system may be represented by a three-dimensional
plot.
Since three-dimensional plots are generally difficult to
work with, most variable composition equilibrium are
represented either at constant T, with P and C as the variables,
or at constant P, with T and C as the variables.
※ Solutions:
7
A liquid or solid phase containing two or more components is
often called a solution.
※Ideal Solutions:
★Definition of ideal solution:
An ideal solution is defined as one in which the chemical
potential of each component is given by the formula
 ( T , P )   ( T , P )  RT n x
*
i
i
i
 ideal solution
where  ( T , P ) is the chemical potential of the pure
*
i
substance i when it is at the temperature T and pressure P.
xi is the mole fraction of substance i in the solution.
★ Properties of ideal solution:
Property 1: The equilibrium partial vapor pressure of each
component of an ideal solution is equal to the
product of the equilibrium vapor pressure of the
pure substance time its mole fraction.
PPx
*
i
i
i
 Raoult’s law
where Pi = the partial vapor pressure of substance i.
*
P = the vapor pressure of pure substance i.
i
xi = the mole fraction of substance i in the solution.
Property 2: The entropy change of mixing of an ideal solution
8
is the same as the mixing of ideal gases.
S   ( n  n ) R  x nx  x nx
1
2
1
1
2
2

★ Derivation:
Assume that the vapor phase in equilibrium with an ideal
solution is an ideal gas mixture.
The chemical potential of component i in the vapor phase is:
 
i
0 ( gas )
i
 P
 RT n  
P 
i
0
where P0 = the standard state pressure (1 bar)

0 ( gas )
i
= the chemical potential of the gas in the standard
state.
Pi = the partial pressure of component i in the vapor phase.
At the equilibrium, 
 
i(vapor)
solution 
∴  ( T , P ) RT n x = 
*
i
i
 P

P 
 RT n 
0 ( gas )
i
i
0
For pure liquid, xi = 1, equation becomes
P 
 RT n  
P 
*
 (T , P )= 
*
*
i
0 ( gas )
i
0
i
To eliminate the standard chemical potentials of gas and liquid:
P 
 P
RT n x  RT n

  RT n
P 
P 
*
i
i
i
0
0
9
∴ x 
i
Pi
Pi
*
※The mixing of an ideal solution:
For a component i in an ideal solution, the chemical potential
is given by:
 ( T , P )   ( T , P )  RT n x
*
i
i
i
For c components in an ideal solution, the Gibbs energy of
the solution is equal to
  n    RT n x 
c
G
( solution )
i 1
*
i
i
i
The Gibbs energy of the unmixed components is a sum of
contributions, one for each component:
 n 
c
G
( u n m )i x e d
i 1
i
*
i
The Gibbs energy change of mixing is defined as the Gibbs
energy change of forming the solution from the unmixed pure
components at the same P and T:
G  G
( solution )
mix
 G 
∵ S   
 T 
S  S
mix
( solution )
c
G
( unmixed )
 RT  n n x
i 1
i
i
∴The entropy of mixing is given by:
P
S
c
( unmixed )
  R n n x
i 1
i
i
This is the same as the ideal gas of mixing.
The enthalpy change of mixing for a solution is given by:
10
H  G  TS
m i x
m i x
 G 
∵ V  
 P 
m i x
∴ H  0
mix
∴ V  0
mix
 H
 V
( solution )
c
H
( solution )
i 1
c
 V
i 1
*
i
*
i
T
Although the entropy change of mixing, the Gibbs energy
change of mixing, the enthalpy change of mixing, and the volume
change of mixing for an ideal solution are given by the same
formulas as the corresponding quantities for a mixture of ideal
gases, an ideal solution does not resemble a mixture of ideal gases
in its molecular structure.
The ideal gas is a model system in
which the molecules do not interact with each other.
In a liquid
or solid solution, a large intermolecular attraction holds the
system together and a large intermolecular repulsion keeps the
system from collapsing to a smaller volume.
The ideal solution could be formed when substances have
similar size, shape, and polarity in the mixture.
The similarity
between the molecules allows them to mix randomly in a solution,
just as non-interacting molecules mix randomly in an ideal gas
mixing, so the formulas for the entropy changes of mixing are
identical.
※ Two-component Phase Diagrams of Ideal
Solutions:
11
There are two principal types of two-dimensional phase
diagrams for a two-component solution:
the
pressure-composition and temperature-composition phase
diagram.
(A) Pressure-Composition Phase Diagram:
In the pressure-composition phase diagram, the temperature
is held fixed.
The mole fraction of one component and the
pressure are the two variables plotted.
Consider a system with two components.
The mole fraction
of components 1 and 2 is y1 and y2 in the vapor phase.
The mole
fraction of components 1 and 2 in the liquid phase is x1 and x2.
By Dalton’s law: y 
P1
1
P2
and y 
2
P
P
By Raoult’s law: P  x P and P  x P
*
1
1
1
2
2
*
2
The total pressure in the vapor phase, P, is equal to:
P = P1 + P2 = x P  x P  P   P  P  x
*
1
∵y 
1
P1
P

*
1
2
*
2
P1
P2   P1  P2  x1
*
*
*
2
*

*
1
2
x 1 P1
1
*
P2   P1  P2  x1
*
*
*
*
yP
x 
P  P  P  y
1
1
*
1
2
*
2
*
1
1
*
yP
 P = P   P  P x  P   P  P 
P  P  P  y
*
2
*
1
*
2
*
1
*
2
1
*
1
2
*
1
12
2
*
2
*
1
1
*
*
PP
P 
P  P  P  y
1
*
1
2
*
2
*
1
1
The area below the curves represents possible equilibrium
intensive states of the system when it is a one-phase vapor, and
the area above the curves represents possible equilibrium states of
the system when it is a one-phase liquid.
Points in the area
between the curves do not represent possible intensive states of a
single phase.
This area represents two phases. Since the total
pressure of the vapor and the pressure of the liquid solution at
equilibrium with each other are the same, a horizontal line
segment, or tie line, between the two curves connects the state
points for the two phases at equilibrium with each other.
13
★The Level Rule :
The relative amounts of the components in the two phases at
equilibrium can be determined by the level rule.
Consider that the amounts of vapor and liquid at point p:
The total amount is: n  n  n
l
For component 1: n  n  n
1
∵ x 
n1
T
n
n1,1
, x 
1
nl
∴ nx  n x  n y
T
l
1
v
1
l,1
v
v,1
, and y 
n v,1
1
 n
l
nl
y  xT pv
 1

nv xT  x1 lp
nv
 nv  xT  nl x1  nv y1
 The Level rule
★Volatile Compound (揮發物質):
For a binary system,
m o l fe r a c t iofo component
n
1in vapor y1
1


*
mole fraction of component 1in liquid x1 x  x  P2 
1
2
 P1* 
If P  P then y  x , the vapor contains more volatile
*
1
*
2
component.
1
1
Application of this principle is called isothermal
distillation.
(B) Temperature-Composition Phase Diagram:
■
Mole fraction of one component versus temperature with
fixed pressure.
14
 The upper line called dew point line; the lower line called
bubble point line.
Benzene - Toluene System
 The phase above the dew point line is vapor and the liquid
phase below the bubble point line.
 Used for distillation.
Each step is corresponding to one
theoretical plate.
※ Henry’s Law:
Since there is an interaction between liquid molecules, most liquid
and solid solutions are not well described by Raoult’s law.
15
★ Types of real solution:
(A) Positive deviation:
If the interaction between like molecules is greater than
unlike molecules, the tendency will be to force individual
components into the vapor phase.
This increases the pressure
above what is predicted by Raoult’s law and is known as a
positive deviation.
Ideal Solution
Positive Deviation
(B) Negative deviation:
If the interaction between
unlike molecules is greater
than like molecules, the
interaction will be to force
individual components into
the liquid phase.
16
This
decreases the pressure above what is predicted by Raoult’s law
and is known as a negative deviation.
★Two features of real solutions:
(1) For small xi, the curve representing pressure is nearly
called Henry’s law (ki is Henry
straight, i.e. xi→0, Pi = kixi
constant, which depends on temperature and the identity of
the other substances present)
(2) For xi near unity, the pressure nearly coincides with the
Raoult’s law, i.e. xi→1, P  x P called Raoult’s law.
*
i
i
i
★ Properties of real solutions:
(1) A solution contains a solvent and solutes. The large
amount of portion is called the solvent, and the other
substances present are called the solutes.
(2) A nearly pure component approximately obeys Raoult’s
law.
(3) A dilute component approximately obeys Henry’s law.
(4) A solution in which the solvent obeys Raoult’s law and the
solutes obey Henry’s law is called an ideally dilute solution.
17
※ Real Solutions:
★Activity and activity coefficient:
Most solutions are neither ideal or ideally dilute.
For
non-ideal solution, the chemical potential of substance i is defined
by:
 ( T , P )   ( T , P ) RT n a
*
i
i
(definition of ai)
i
where  is the chemical potential of the pure substance i.
*
i
ai is the activity.
im a  x  1
xi 1
i
i
at constant T and P.
The extent to which the activity of a non-ideal solution
deviates from that of an ideal solution is specified by the activity
coefficient,  :
i
 
i
ai
(definition of  )
i
xi
If the solution is ideal, then a  x and   1 .
i
∴ a x
i
i
i
i
(definition of ideal solution)
★ Property of activity coefficient:
The conditions under which  becomes equal to 1:
i
Convention I:
If the components of the solution are liquids,
the activity coefficient of each component may be taken to
approach unity as its mole fraction approaches unity:
18
 
i
ai
xi

Pi
*
Pi x i
Convention II:
It is convenient to use this convention if it is
not possible to vary the mole fractions of both components up to
unity, e.g. gas-liquid solution or liquid-solid solution, etc.
For
such solutions a different convention is applied solvent and solute.
The activity coefficient of the solvent is given by:

solvent

a solvent
x solvent

Psolvent
*
Psolvent x solvent
The activity coefficient of the solute is taken to approach
unity as its mole fraction approaches zero based on Henry’s law:

solute
Example

a solute
x solute

Psolute
k solute x solute
Given that P*(H2O) = 0.02308 atm and P(H2O) =
0.02239 atm in a solution in which 0.122 kg of a nonvolatile
solute
(M = 241 g/mole) is dissolved in 0.92kg of water at 293K,
calculate the activity and activity coefficient of water in the
solution.
Solution:
The activity is determined by:
a 
i
Pi
Pi
*

0 . 02239
0 . 02308
 0 . 9701
19
x
H2O

920
920
18

H2 O

a
xH O
2
18
 122

 0 . 9902
241
0 . 9701
0 . 9902
 0 . 98
※Azeotrope: (共沸混合物)
事實上，絕大多數的混合溶液，並不是理想溶液，因此
不會遵守勞特定律。 由於受到不同液體分子之間的作用力影
響，因此溶液在相圖的表現常會有極大點(maximum)或極小
點(minimum)的現象產生，此時溶液的沸點曲線(bubble point
line)會與露點曲線(dew point line)相交在一起，因此混合溶液
在這溫度下蒸發，此時蒸發的氣體會與溶液具有相同的組
成，就如
同純物質般，我們就將這溶液稱之為共沸混合物
(Azeotrope)。儘管共沸混合物其在某一溫度下具有純物質的
性質，然而受到系統的壓力改變，其組成也會受到影響，此
時溶液的蒸發其蒸汽的組成與溶液的組成就無法相同，因此
共沸混合物不能稱之為純物質。
當共沸混合物在共同的沸點蒸發時，此時蒸汽的組成或
溶液的組成只和物質成份的純蒸汽壓有關，其關係式可表示
為：
20
y1 x1 P1*
 
y2 x2 P2*
此種方法常被應用來分離具有揮發性性質且易在其沸點下裂
解的物質。
Azeotrope with minimum
Azeotrope with maximum
Table: Azeotropes with minimum b.p. at 1atm
Table: Azeotropes with maximum b.p. at 1 atm
21
因此，我們將利用此共沸特性來分離或純化物質的方法稱之為
蒸汽蒸餾(steam distillation)。
※Colligative Properties (依數性):
  ( T , P )   ( T , P )  RT n x
*
i
i
i
 Chemical potential of the solution will be lowered by the
addition of the solute
 the lowering of vapor pressure.
 The lowering of chemical potential is related to the number
of solute molecules, but not dependent on the identity of
solute
 the colligative property.
◆Vapor pressure lowering
◆Freezing point depression
◆Boiling point elevation
◆Osmotic pressure
22
★Vapor pressure lowering:
For a nonvolatile solute and a volatile solvent that obeys
Raoult’s law, the total vapor pressure of solution is equal to the
partial vapor pressure of the solvent:
P
vap ( total )
xP
1
*
1
*
where P = the vapor pressure of the pure solvent and x =
1
1
the mole fraction of the solvent.
The lowering of the vapor pressure is given by:
P = P - x P  1 - x  P  x P
*
v a p
1
*
1
*
1
1
1
2
*
1
★Freezing point depression:
Consider a system with two phases at equilibrium with each
other: a liquid solution with a single solute and a pure solid
solvent.
The solute is completely insoluble in the frozen solvent.
For solvent (subscript 1):
 ( T , P, x )   ( T , P )

s
1
1
1
Assumed solution is ideal,  ( T , P ) RT n x   ( T , P )
*, 
s
1
 n x 
1

   ( T , P )   ( T , P )
*, 
s
1
1
RT
1

 G
   fus Gm  


T 
 H
d n x1
1 
 
  fus 2 m
dT
R
T
RT


 P
    G  
 H
       2 
T 
  T  T   P
23
fus
RT
m
1
 n x
1
 fus Hm  1 1 
  
R  Tf* T 


 fus H m  T - Tf* 
n 1- x2  


R  TTf* 

 H  T
x 


R  T 
fus
m
fus
*2
2
f
1
1


n
1

x


x

x

x

.....


x



 and
2
3
2

2
2
3
2
2
2
T  T  T
*
fus
f
  T
fus
*2
M 1 RTf
 H
fus
m  k m
2
f
2
m




n2
n2 m2W1
 
 m2 M1 
 x2 
W
n

n
n
1
1
2
1


M1


M1 RTf*
where k f 
 fus H m
2
called freezing point depression constant
★Boiling point elevation:
 x 
2
 vap H m   vapT 
 2 
R  Tb* 
2
M1 RTb*
 m2  kb  m2
  vapT 
 vap Hm
M1 RTb*
 kb 
 vap H m
2
called boiling point elevation constant
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  T  T T
vap
*
b
★Ideal solubility equation:
 pure solid 1  ideal solution of 1 and 2
 a solution of solid 1 in solvent 2 with solid 1 crystallized.
 n x 
1
 fus Hm  1 1 
*
 *   where Tf is the temperature of the
R  Tf T 
saturated solubility (i.e. x1=1)
 Allowed to calculate the solubility at different temperature.
 This equation is called the ideal solubility equation because
of independent of solute identity.
★Osmotic pressure (滲透壓):
 A system is involved in a semipermeable membrane which
permeable to the solvent but impermeable to the solute.
 A tendency for the solvent to flow from the pure solvent
through the membrane into the solution because of low
chemical potential of the solution.
 The pressure that must be applied to the solution to produce
equilibrium is called the osmotic pressure,π.
 
n2 RT
V
called the van’t Hoff’s equation.
(where V = the volume of the solution and n2 = the mole
number of solute)
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※Condensed Phase:
★Two-liquid components:
Three different types of partial miscible liquid are observed.
The
weight of
individual phase in a two-phase region is
determined by the Level Rule:
w1 y2  y

w2 y  y1
The upper consolute temperature (Tuc): the highest
temperature at which two layers can co-exist.
The lower consolute temperature (Tuc): the lowest
temperature at which two layers can co-exist.
★Solid Solution:
◆Eutectic (共晶) diagram:
 liquid  solid    solid  called eutectic reaction.
cooling
Eutectic temperature: the temperature at which both
26
solutions are co-crystallized.
 w/o miscibility:
 w/ limited miscibility:
27
★Eutectoid (共析) diagram:
  solid   solid    solid  called eutectoid reaction.
cooling
※Compound Formation:
Sometimes there are chemical reactions between components
that an actual compound is formed.
Two types of behavior can
then be found: congruent melting and incongruent melting.
★Congruent melting (熔點一致):
Compound formation due to chemical reactions.
 AB  L called congruent melting.
heating
Compound formed melts into liquid having the same
composition as the compound.
28
◇Tl3BO3 is formed.
◇Congruently melts at
725K.
◇Below 627 K,βphase is
formed.
◇αphase is stable above
627K.
◇A eutectic occurs at
682K.
◇The point at ~710K is
called monotectic point.
congruent melting
★Incongruent melting (非熔點一致): also called peritectic
(包晶點) diagram
Compound formation due to chemical reactions.
 AB  L  B called peritectic reaction.
heating
Compound formed melts into liquid which does not
contain the same composition as the compound.
29
incongruent melting
※Ternary system:
To depict the phase behavior of the three-component systems
on a two-dimensional diagram, it is necessary to consider both the
pressure and the temperature as fixed.
The phases of the system
as a function of the composition can then be shown.
The relative
amounts of the three components, usually presented as weight
percentages, can be shown on a triangular plot.
30
Since the mole fraction
of the three components
satisfies xA+xB+xC=1, a
phase diagram is drawn as
an equilateral triangle
because the sum of the
distances to a point inside
triangle measured parallel
to the edges is equal to the length of the side of the triangle, and
each side is taken as a unit length.
The corners of the triangle
correspond to the pure components, A, B, and C, respectively.
The side of the triangle opposite the corner labeled A implies the
absence of component A.
Thus the lines across the triangle show
increasing percentages of A from zero at the base to 100% at the
apex.
In a similar way the percentages of B and C are given by
the distances from the other two sides to the remaining two apices.
The composition corresponding to any point inside triangle can be
read off by drawing the parallel line from this particular point to
the each triangular side and measuring the distance from this
particular point to each corresponding base.
Each distance
measured is the composition of each component opposite the
corresponding base.
31
★Liquid-liquid ternary system:
◇ Toluene-water-acetic
acid system.
◇ Toluene and acetic
acid completely
miscible.
◇ Water and acetic acid
also completely
miscible.
◇ Toluene and water
partially miscible.
◇ Plait point: only one
phase exists.
★Solid-liquid equilibrium: common-ion effect
◇ Water and two salts
with an ion in
common.
◇ NaCl-KCl-H2O system.
◇ Point a is the
maximum solubility of
A in C w/o B.
◇ Point c is the maximum
solubility of A in C w/o
B.
◇ The solubility of A
decreases when B is added to a mixture of A and C, which is shown as
by the line, ab .
◇ Similarly, addition of A to a solution B in C decreases the solubility
32
along the line, cb .
◇ The effect of decreasing solubility by the addition of the common ion is
called the common ion effect (共同離子效應).
◇ Point b represents a solution that is saturated with respected to both
salts.
◇ The region of AbB contains two pure solids A and B and a saturated
solution of composition b.
◇ In the tie line regions, the pure solid and saturated solution are in
equilibrium
◇ Point d gives the composition of the mixture, the amount of solid phase
present is given by the length de and the amount of saturated solution
is given by the length dA .
33