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THE CATASTROPHE MODEL FOR THE ACID-BASE PROPERTY OF THE HYDROXIDES OF ELEMENTS IN AQUEOUS SOLUTION
4(1983),1,97-106
It¥~JI,m JfalIw,JfffatIycimliJl
THE CATASTROPHE MODEL FOR THE ACID-BASE P.ROPERTY OF THE HYDROXIDES OF ELEMENTS IN AQUEOUS SOLUTION· J in Guantao
(~~~)
Association for the Journal of Natural Dialectics Academia sinica
Dedicated to professor Lee K wok- ping( Li Guoping) On the Occasion of his 50th Year of Educational and Scientific Work Abstract
In this paper the acid-base property of various hydroxides of elements has been studied by applying the catastrophe theory. It is indica ted tha t the topological structure of the plane diagrams formed by bond-parameters may be deduced by using the catastrophe theory. The author considers that the application of catastrophe theory may be open a"new way of thinking for the theoretical derivation of plane diagrams of bond-parameters. The discussion in this paper is only an illustration
§ 1 Bond---parameter diagrams and catastrophe theory At recent years the diagrams formed by bond-parameters have been e xtensi vely used to in vest i ga te the various property of various substances and their r e g ular i t i s , The most applied are the plane diagrams which is formed by using two suitable bond-parameters as coordinates. The different regions of the plane often represent the sets of substances with different property. Some efforts have been made to interpret and predicate the forms of those empirical diagrams • Received 1 Oct.,1981; in revised form 9 March,1983.
98
ACTA MA THEMATICA SCIENTIA
vei.,
by the investigators from the Quantum Chemistry. However the great difficulty for such complex problems have not been yet overcomed at present. Now some attempts have been made in this paper to study the bond-parameters diagrams for the acid-base property of hydroxides of various elements with application of catastrophe theory which was firstly proposed by Rene Thorn at 1972. Tb e method proposed in this paper may probably possess the general signification for deriving the other bond-parameters diagrams. The catastrophe theory considers that if a system assumes some property Xl which is stable and not changed under small disturbance, there must exist a potential function V(x,s,P) in the space of variables representing the propertis, and the property Xl is corresponding to the minimum of the potential function (ie Xl is situated in the basin of the poten t ia l curve). 5 and P are the parameters influencing the property ie the form of curve V(x,s,P) versus X and the position ofm in irnu m and number of minimum(basin) in space of property. For instance, the system may assume one two or three minima, each of which corresponding to properties Xl, X 2, X3, according to different range of the values of 5 and p. If the system is in the first minimum (Xl state) (basin of curve) initially, which will flatten with the change of Sand P, as soon as the values of Sand Preach to certain value (51, PI), the first minimum (first basin of curve) disappears and the catastrophe of the system happens, the sudden transition from Xl state (property Xl) to X2(2nd minimum) .51 and PI are called conditions for catastrophe. Those point at which occur catastrophe andthe regions where the potential function assume one two or three m in imarn ay be drawn on the space of conditions (or controlling space) 5 xP, They may be called catastrophe set, which represents the topological structure for the distribution for various properties of the system in the space of conditions. In spite of the various different potential function V (X, S, P) for different systems, the catastrophe theory has given a mathematical proof for the following conclusion: there are only seven topological types for the catastrophe set of potential function on the conditional space (fold, cusp, swallowtail, butterfly, hyperbolic urnb i lic , elliptic umbi li c , parabolic u mb i l ic) , if the number of conditional par arne ter s is lese than four and number of state variables "is lese than or equals to two As the properties of the substance are generally stable relatively and may be expressed by the minimum of the potential function in the space of property, the controlling parameters such as some kinds of
The Catastrophe Model For the Acid-Base Property
NO,,1
99
"bond parameters dct er m in e the number and position of the m i n i mu rn in the space of property through the potential function. Thus, by . means of catastrophe mod el , we may not only understand why the diagrams' of bond-parameters assume the special structure, bu t also can investigate the relation between the properties and bond-parameters from the model. In the fol1owig section, a catastropne model is given for the acid-base property of hydroxides M COH)m.
§2
The model of butterfly type for acid-base property of M(OH)m
Doubtless, the acid-base property of M(OH)m in aqueous solution depends upon the distance R be t w ee n M+ and OH and the distance
Y between M+m and H+ as shown in Fig~-R-~ M+m
0-2
H+
~-------y~ Fig -1
1
•
Ho w ev er , owing to the polarizing effect, Y is not independent of R and is a function of R . .i\.pparently, when R is very large, M(OH)m dissociate into M+m andOH- ions
1"'1 (Ofl)m-+-.!ll +m + m(OH) - , It presents as a base in solution. When R is sufficient short; Y may become so large that M(C)H).m dissociates into M (OH)m_' 0- and H+ It P1;' esen t s a san a c i din sol uti 0 n. -\iV hen R iss i t u ate din a sui tab 1 value, it presents as neutral molecule species M(OH)m. In addition to variable R, the behavior of McOI-I)m in aqueous solution depends a1sou po nco n t roll in g f act 0 r s , s u chas par a met er s S, Pet c. In ot h. er words, the energy' function of molecule M(OH)m in aqueuos solution may be expressed as E(R,S,P) which determines the behavior of molecules M(OH)m in solution. It is illustrated in Fig 2, where RCa), R(h)R(b) represet respectively the meal1 distance of R for acid neutral and 'basic states in solution. (R(a)
100
ACTA MA TI-IEMA TICA SCIENTIA
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system containg numberous molecules reach the following equilibrium: ~~1 COH)m~M+m+m(OH)-
In such case, the system presents as a weak base. If the two minima of the curve correspond to R(a) and R(n). respecive ly the system presents as a weak acid (Fig2(4», according to similar reason. The curve as shown in Fig2(6) corresponds to the coexistence of M"'", OH- and M(OH)m_l 0-, H+, owing to the combination of H+ and OHinto water molecule, it means that the sy stern MCOH)m is dehydratrd part ially into oxides. If the E(R, S,P)curve assumes three minima corresponding to R(a) R(n) and R(b) respec tivele as shown in Fig2 (3),it shows apparently M(OH)m to be amphoteric hydroxides. What Ior m the curve would assumes is dependent upon the energy function EcR,S,P) and the values of controlling parameters Sand p. If the controll parameters S P represent two bond parameters, and the topological structure of EcR,S,P) on the SXP plane is investigated, a plane diagrams of bond parameters may be easily obtained as follows E (J [. -
o
•• - •• _. • - _ ••.• - - - - - •
I
! j
P( c)
L
....-.. --.. --'" _..- '" .-.--..- .---- R
-----------·-·--···---R
• 1 • strong acid
• 4: • weak acid
E
-_-. -. ---__.--_I~/
E
~1\ i -L
._R(") _. .._ • 2 •
'--"'('" .- ..-..- - - -
J
O~-
I
----------------.
L_R(b)
it
_ R
weak base
• 5 • strong base
E
I
(J~
o
.. - -- - -. - - -. -- -- -- - •• -. _.• - -
I I I
!
L ._. _ !-~~(l)
~-----------
3 • am.photeric
• 6 •
R
dehydrated into oxides
Fig z
Now the form of energy function EcR,S,P) is a n a l y z.ad firstly. It is well known that the energy function E for a molecule A +B- with
101
The Catastrophe Model For the Acid-Base Property
NO.1
pure ionic bond may be expressed as
E=
-~q2 +~-,
(1)
Where q 1 ,Q2 is the electric charge of the ions A + and B-, n is the Boh n index n = 5-12. Because M(OH)m is not a compound with pure ionic bond and in solution it is submitted to the action of water molecules, the equation(l) can not be applied to M(OH)m in aqueous solution before making some suitable corrections. Considering the polarizing effect of ions and the action of water, energy function E(R, S,P,) may be reasonable assumed to be expressed as follows:
-~+ E~i~~+ !(S,P) +-~
E=
R
i =
R
2
R "" -
i
1
(2)
R"" ,
where S,P are positive-valued function of bond parameters, f(S,
P) is the f unction of Sand P, Ai-I are constants. - ~ and
f.,. represent
respectively the attracting energy and repulsion energy between the positive and negative ions in aqueous solution. They correspond to two terms of equation(l)for the limiting case of pure ionic bond in v acc um ,
~ 11_i=_~_~ + R
ft-2
~
I
1- 2
iSP -<--'_) Rft-l
means t he correcting terms for . the polarizing
effect ect , From the discussion in above section, it is known that the curve E(R, S,P) should assume only three minima at the most. According to the catastrophe theory, the catastrophe set should satisfy the following equation: [1]
oE=~_~_~iAi_l·S_(n-1)!(S,P)=O
oR
R2
R':: 1
~
R""
R~
1= '/.
(3)
,
(4) 2
Because~(R""+l CJlJ)=Rf&+l. 0 E +oE.(n+1)R ok oR o R 2 iJR ' ft
and
2
aE = 0 0 ~= oR
oR 2
0
R~O,
hence
~(Rf&+l ~E)= oR oR
0
Therefore, the equation (3) and (4) may be transformed into the following equation:
R":' -
EiAi-l ·R"i-2
i -
(n
-1)! (~ ,P).R -~~ = 0 S
S
(5)
102
ACT A MATHEMATICA SCIENTIA
VOl.4
In order to satisfy the condition described above for the curve E(R, S, P) assuming only three minima at most, the mathematical theory indicates that equation (3) should assumws only five positive roots at the most, and equ at ionj a) assumes only four smaller positive roots to satisfy the condition for c at ast roph e set, though equation (4) may contain five positive. Consequently, there are only four positive roots which satisfy the cqu at ion s r g ) and (6). It may be further proved in mathematics that the catastrophe set sat isfy ing equations (5) and (6) with only four positive roots possesses the same topological structure as the catastrophe set of butterfly type which assumes the potential function:
R6+tR4+ uR 3+ vR2+ CiJ R . [2]
s a t i sf y i n g the following equation:
6R6 + 4tR3 + guR2 + 2vR + CiJ = 0,
(7)
30R4 + 12tR2 + 6uR + 2v = o.
(8)
It is apparent that equations (5) and (6) are identical to equations (7) and (8) as n = 6, if let
-6·n·p S
(9)
=(i),
-6-(n-1)!(x,P)
S
= 2v
·
(10)
When n>6, they are not identical each other, and yet possess still same topological structure, if equations (5) and (6) assume only four posi ti v e roots. According to the catastrophe theory, the topological structure of catastrophe set of butterfly (,) type on w X v plane is shown in 1 Fig3, if the parameters t and u are constants [ 3 ] In the district A or B the potential function possesses only one m i n i rnu m w h ic h is situv ated in the region of R (a) or R(b). In the district w(a) or w Cb) the potential function possesses two minima which correspond to R(n) R(a) or R(n) R(b) respecFig 3 tively _ In the district Q there are
The Catastrophe Model For the Acid-Base Property
NO.1
103
also two minima which correspond to .l?(a) and R(b) region respectively. In the district T, there are three minima corresponding to R(n) R(a) and R(b) respectively. Because co and v are function of S,P (function of bond parameters) as shown by equations (9) (10), the mathematical theory shows that the catastrophe set on S x P plane would preserve the same form as shown on Fig 3 if the systems of coordinates w, v is transformed into that of and S,p. From the discussion about F'i g z, we obtain the various representation of different district of Fig3. That is, A and B represent the regions of strong acid and strong base respectively, W(a) and W(b) are the regions of weak acid and w eak base respectively. T is the regions for amphoteric hydroxide, and in the region
z r~
- ',(i
-Nb .;+
+
Ti ~ H r
M
Sc
AI Ii
4+ S 4-+ Os
,~
Z r GaP b
v~+ ;~ Cr~+' Fe 3 +
Cr l
+
Mn 7n
2+ Pb Mn
3 +- 3+
In
Sb
TI
Nl Sn Hg Ag
2
+'
104
ACTA MA THEMATICA SCIENTIA
Vol.4
x ~
tI
C\ J
N
u-
!
I
J
~1;:*1 c.
(,1·
i,":
p~-+----------
~ ~.1 r '>b t ~ (h .
,...../
M.
"u (.~ V ~ ~. ~ •
Sn ~
F... · .
N\~ (.6
iJ
..,~.
Bi f,
rl Ph
...
•• .. Ir~ )
Z.«
..
(4
r,
\
Fig 5
Q, the hydroxi ds are dehydrated into oxides and water. If S, P are two bond parameters such as the elec tr ongat iv ity x (Bauanoe) and ZITk(Z = ionic valence of cation, Tic = ionic radiu of cation (Pauling», or two functions of x and z/r», then the plane x x Lor the plane formed by the functions of x and ZITk' the same T k
district distribution of bu tterfly type as f i gu r-e-S may be observed. It has been proved mathematically that taking the different functions of bond parameters doesn't change its topological structure, but change only the geometric form and orientation of the catastrophe set. The above deduction of plane diagram of bond parameters for the acid-base property of hydroxides have been verified with the known experimental facts as shown in Fig4, FigS and Fig6.If the similar kinds of bond parameters x and rk' or x and x , rk are used as coordinates, the similar Butterfly types of diagrams with differing 'only the geometric form and orien tation of catastrophe set of Butterfly type are observed as shown in Fig5 and :Fig6. It gives an effective support to the u nc h an g e.ab i li ty of their topological structure. Besides,it may be deduced from the catastrophe theory that the basin of potential function for neutral M(OH)m would become shallower
105
The Catastrophe Model For the Acid-Base Property
NO.1
All
./
(:11
ri
IT"
<.-::'j PL
In
Rh
K
-I/,
BR
).. .+ Hi Sn ''J-f.
ZrZ.)""b·~ .. ~-+_ Sf
Ca
c. .~
M>+ ~; n
Ti
F"e \/
Br
C\ l.i
.. .
\'
Fig 6
and sh al lower with the decrease of v on the plane v X ill (Fig3), ie M(OI-I)m situated in the top or the brim area (w it h small value of v, of pocket district T) is unstable and tends to decompose This deducatlon may be observed in the Fig 4-6. It is well known that the hydroxides of Au Ag Hg Cu ect are unstable in aqueous solution and easily dehydrated to oxides.
References ( 1)
(2) ( 3)
Woodcock A, E, R; Bulletin of Mathematical Biology Volume 40, (1978), 1-25. Th, Brocker; Differentiable Germs And Catastrophes (1975) 128. Th , Brocker; Differentiable Germs And Catastrophes (1975) 163-164.
It¥~JI,m JfalIw,JfffatIycimliJl
THE CATASTROPHE MODEL FOR THE ACID-BASE P.ROPERTY OF THE HYDROXIDES OF ELEMENTS IN AQUEOUS SOLUTION· J in Guantao
(~~~)
Association for the Journal of Natural Dialectics Academia sinica
Dedicated to professor Lee K wok- ping( Li Guoping) On the Occasion of his 50th Year of Educational and Scientific Work Abstract
In this paper the acid-base property of various hydroxides of elements has been studied by applying the catastrophe theory. It is indica ted tha t the topological structure of the plane diagrams formed by bond-parameters may be deduced by using the catastrophe theory. The author considers that the application of catastrophe theory may be open a"new way of thinking for the theoretical derivation of plane diagrams of bond-parameters. The discussion in this paper is only an illustration
§ 1 Bond---parameter diagrams and catastrophe theory At recent years the diagrams formed by bond-parameters have been e xtensi vely used to in vest i ga te the various property of various substances and their r e g ular i t i s , The most applied are the plane diagrams which is formed by using two suitable bond-parameters as coordinates. The different regions of the plane often represent the sets of substances with different property. Some efforts have been made to interpret and predicate the forms of those empirical diagrams • Received 1 Oct.,1981; in revised form 9 March,1983.
98
ACTA MA THEMATICA SCIENTIA
vei.,
by the investigators from the Quantum Chemistry. However the great difficulty for such complex problems have not been yet overcomed at present. Now some attempts have been made in this paper to study the bond-parameters diagrams for the acid-base property of hydroxides of various elements with application of catastrophe theory which was firstly proposed by Rene Thorn at 1972. Tb e method proposed in this paper may probably possess the general signification for deriving the other bond-parameters diagrams. The catastrophe theory considers that if a system assumes some property Xl which is stable and not changed under small disturbance, there must exist a potential function V(x,s,P) in the space of variables representing the propertis, and the property Xl is corresponding to the minimum of the potential function (ie Xl is situated in the basin of the poten t ia l curve). 5 and P are the parameters influencing the property ie the form of curve V(x,s,P) versus X and the position ofm in irnu m and number of minimum(basin) in space of property. For instance, the system may assume one two or three minima, each of which corresponding to properties Xl, X 2, X3, according to different range of the values of 5 and p. If the system is in the first minimum (Xl state) (basin of curve) initially, which will flatten with the change of Sand P, as soon as the values of Sand Preach to certain value (51, PI), the first minimum (first basin of curve) disappears and the catastrophe of the system happens, the sudden transition from Xl state (property Xl) to X2(2nd minimum) .51 and PI are called conditions for catastrophe. Those point at which occur catastrophe andthe regions where the potential function assume one two or three m in imarn ay be drawn on the space of conditions (or controlling space) 5 xP, They may be called catastrophe set, which represents the topological structure for the distribution for various properties of the system in the space of conditions. In spite of the various different potential function V (X, S, P) for different systems, the catastrophe theory has given a mathematical proof for the following conclusion: there are only seven topological types for the catastrophe set of potential function on the conditional space (fold, cusp, swallowtail, butterfly, hyperbolic urnb i lic , elliptic umbi li c , parabolic u mb i l ic) , if the number of conditional par arne ter s is lese than four and number of state variables "is lese than or equals to two As the properties of the substance are generally stable relatively and may be expressed by the minimum of the potential function in the space of property, the controlling parameters such as some kinds of
The Catastrophe Model For the Acid-Base Property
NO,,1
99
"bond parameters dct er m in e the number and position of the m i n i mu rn in the space of property through the potential function. Thus, by . means of catastrophe mod el , we may not only understand why the diagrams' of bond-parameters assume the special structure, bu t also can investigate the relation between the properties and bond-parameters from the model. In the fol1owig section, a catastropne model is given for the acid-base property of hydroxides M COH)m.
§2
The model of butterfly type for acid-base property of M(OH)m
Doubtless, the acid-base property of M(OH)m in aqueous solution depends upon the distance R be t w ee n M+ and OH and the distance
Y between M+m and H+ as shown in Fig~-R-~ M+m
0-2
H+
~-------y~ Fig -1
1
•
Ho w ev er , owing to the polarizing effect, Y is not independent of R and is a function of R . .i\.pparently, when R is very large, M(OH)m dissociate into M+m andOH- ions
1"'1 (Ofl)m-+-.!ll +m + m(OH) - , It presents as a base in solution. When R is sufficient short; Y may become so large that M(C)H).m dissociates into M (OH)m_' 0- and H+ It P1;' esen t s a san a c i din sol uti 0 n. -\iV hen R iss i t u ate din a sui tab 1 value, it presents as neutral molecule species M(OH)m. In addition to variable R, the behavior of McOI-I)m in aqueous solution depends a1sou po nco n t roll in g f act 0 r s , s u chas par a met er s S, Pet c. In ot h. er words, the energy' function of molecule M(OH)m in aqueuos solution may be expressed as E(R,S,P) which determines the behavior of molecules M(OH)m in solution. It is illustrated in Fig 2, where RCa), R(h)R(b) represet respectively the meal1 distance of R for acid neutral and 'basic states in solution. (R(a)
100
ACTA MA TI-IEMA TICA SCIENTIA
Vol."
system containg numberous molecules reach the following equilibrium: ~~1 COH)m~M+m+m(OH)-
In such case, the system presents as a weak base. If the two minima of the curve correspond to R(a) and R(n). respecive ly the system presents as a weak acid (Fig2(4», according to similar reason. The curve as shown in Fig2(6) corresponds to the coexistence of M"'", OH- and M(OH)m_l 0-, H+, owing to the combination of H+ and OHinto water molecule, it means that the sy stern MCOH)m is dehydratrd part ially into oxides. If the E(R, S,P)curve assumes three minima corresponding to R(a) R(n) and R(b) respec tivele as shown in Fig2 (3),it shows apparently M(OH)m to be amphoteric hydroxides. What Ior m the curve would assumes is dependent upon the energy function EcR,S,P) and the values of controlling parameters Sand p. If the controll parameters S P represent two bond parameters, and the topological structure of EcR,S,P) on the SXP plane is investigated, a plane diagrams of bond parameters may be easily obtained as follows E (J [. -
o
•• - •• _. • - _ ••.• - - - - - •
I
! j
P( c)
L
....-.. --.. --'" _..- '" .-.--..- .---- R
-----------·-·--···---R
• 1 • strong acid
• 4: • weak acid
E
-_-. -. ---__.--_I~/
E
~1\ i -L
._R(") _. .._ • 2 •
'--"'('" .- ..-..- - - -
J
O~-
I
----------------.
L_R(b)
it
_ R
weak base
• 5 • strong base
E
I
(J~
o
.. - -- - -. - - -. -- -- -- - •• -. _.• - -
I I I
!
L ._. _ !-~~(l)
~-----------
3 • am.photeric
• 6 •
R
dehydrated into oxides
Fig z
Now the form of energy function EcR,S,P) is a n a l y z.ad firstly. It is well known that the energy function E for a molecule A +B- with
101
The Catastrophe Model For the Acid-Base Property
NO.1
pure ionic bond may be expressed as
E=
-~q2 +~-,
(1)
Where q 1 ,Q2 is the electric charge of the ions A + and B-, n is the Boh n index n = 5-12. Because M(OH)m is not a compound with pure ionic bond and in solution it is submitted to the action of water molecules, the equation(l) can not be applied to M(OH)m in aqueous solution before making some suitable corrections. Considering the polarizing effect of ions and the action of water, energy function E(R, S,P,) may be reasonable assumed to be expressed as follows:
-~+ E~i~~+ !(S,P) +-~
E=
R
i =
R
2
R "" -
i
1
(2)
R"" ,
where S,P are positive-valued function of bond parameters, f(S,
P) is the f unction of Sand P, Ai-I are constants. - ~ and
f.,. represent
respectively the attracting energy and repulsion energy between the positive and negative ions in aqueous solution. They correspond to two terms of equation(l)for the limiting case of pure ionic bond in v acc um ,
~ 11_i=_~_~ + R
ft-2
~
I
1- 2
iSP -<--'_) Rft-l
means t he correcting terms for . the polarizing
effect ect , From the discussion in above section, it is known that the curve E(R, S,P) should assume only three minima at the most. According to the catastrophe theory, the catastrophe set should satisfy the following equation: [1]
oE=~_~_~iAi_l·S_(n-1)!(S,P)=O
oR
R2
R':: 1
~
R""
R~
1= '/.
(3)
,
(4) 2
Because~(R""+l CJlJ)=Rf&+l. 0 E +oE.(n+1)R ok oR o R 2 iJR ' ft
and
2
aE = 0 0 ~= oR
oR 2
0
R~O,
hence
~(Rf&+l ~E)= oR oR
0
Therefore, the equation (3) and (4) may be transformed into the following equation:
R":' -
EiAi-l ·R"i-2
i -
(n
-1)! (~ ,P).R -~~ = 0 S
S
(5)
102
ACT A MATHEMATICA SCIENTIA
VOl.4
In order to satisfy the condition described above for the curve E(R, S, P) assuming only three minima at most, the mathematical theory indicates that equation (3) should assumws only five positive roots at the most, and equ at ionj a) assumes only four smaller positive roots to satisfy the condition for c at ast roph e set, though equation (4) may contain five positive. Consequently, there are only four positive roots which satisfy the cqu at ion s r g ) and (6). It may be further proved in mathematics that the catastrophe set sat isfy ing equations (5) and (6) with only four positive roots possesses the same topological structure as the catastrophe set of butterfly type which assumes the potential function:
R6+tR4+ uR 3+ vR2+ CiJ R . [2]
s a t i sf y i n g the following equation:
6R6 + 4tR3 + guR2 + 2vR + CiJ = 0,
(7)
30R4 + 12tR2 + 6uR + 2v = o.
(8)
It is apparent that equations (5) and (6) are identical to equations (7) and (8) as n = 6, if let
-6·n·p S
(9)
=(i),
-6-(n-1)!(x,P)
S
= 2v
·
(10)
When n>6, they are not identical each other, and yet possess still same topological structure, if equations (5) and (6) assume only four posi ti v e roots. According to the catastrophe theory, the topological structure of catastrophe set of butterfly (,) type on w X v plane is shown in 1 Fig3, if the parameters t and u are constants [ 3 ] In the district A or B the potential function possesses only one m i n i rnu m w h ic h is situv ated in the region of R (a) or R(b). In the district w(a) or w Cb) the potential function possesses two minima which correspond to R(n) R(a) or R(n) R(b) respecFig 3 tively _ In the district Q there are
The Catastrophe Model For the Acid-Base Property
NO.1
103
also two minima which correspond to .l?(a) and R(b) region respectively. In the district T, there are three minima corresponding to R(n) R(a) and R(b) respectively. Because co and v are function of S,P (function of bond parameters) as shown by equations (9) (10), the mathematical theory shows that the catastrophe set on S x P plane would preserve the same form as shown on Fig 3 if the systems of coordinates w, v is transformed into that of and S,p. From the discussion about F'i g z, we obtain the various representation of different district of Fig3. That is, A and B represent the regions of strong acid and strong base respectively, W(a) and W(b) are the regions of weak acid and w eak base respectively. T is the regions for amphoteric hydroxide, and in the region
z r~
- ',(i
-Nb .;+
+
Ti ~ H r
M
Sc
AI Ii
4+ S 4-+ Os
,~
Z r GaP b
v~+ ;~ Cr~+' Fe 3 +
Cr l
+
Mn 7n
2+ Pb Mn
3 +- 3+
In
Sb
TI
Nl Sn Hg Ag
2
+'
104
ACTA MA THEMATICA SCIENTIA
Vol.4
x ~
tI
C\ J
N
u-
!
I
J
~1;:*1 c.
(,1·
i,":
p~-+----------
~ ~.1 r '>b t ~ (h .
,...../
M.
"u (.~ V ~ ~. ~ •
Sn ~
F... · .
N\~ (.6
iJ
..,~.
Bi f,
rl Ph
...
•• .. Ir~ )
Z.«
..
(4
r,
\
Fig 5
Q, the hydroxi ds are dehydrated into oxides and water. If S, P are two bond parameters such as the elec tr ongat iv ity x (Bauanoe) and ZITk(Z = ionic valence of cation, Tic = ionic radiu of cation (Pauling», or two functions of x and z/r», then the plane x x Lor the plane formed by the functions of x and ZITk' the same T k
district distribution of bu tterfly type as f i gu r-e-S may be observed. It has been proved mathematically that taking the different functions of bond parameters doesn't change its topological structure, but change only the geometric form and orientation of the catastrophe set. The above deduction of plane diagram of bond parameters for the acid-base property of hydroxides have been verified with the known experimental facts as shown in Fig4, FigS and Fig6.If the similar kinds of bond parameters x and rk' or x and x , rk are used as coordinates, the similar Butterfly types of diagrams with differing 'only the geometric form and orien tation of catastrophe set of Butterfly type are observed as shown in Fig5 and :Fig6. It gives an effective support to the u nc h an g e.ab i li ty of their topological structure. Besides,it may be deduced from the catastrophe theory that the basin of potential function for neutral M(OH)m would become shallower
105
The Catastrophe Model For the Acid-Base Property
NO.1
All
./
(:11
ri
IT"
<.-::'j PL
In
Rh
K
-I/,
BR
).. .+ Hi Sn ''J-f.
ZrZ.)""b·~ .. ~-+_ Sf
Ca
c. .~
M>+ ~; n
Ti
F"e \/
Br
C\ l.i
.. .
\'
Fig 6
and sh al lower with the decrease of v on the plane v X ill (Fig3), ie M(OI-I)m situated in the top or the brim area (w it h small value of v, of pocket district T) is unstable and tends to decompose This deducatlon may be observed in the Fig 4-6. It is well known that the hydroxides of Au Ag Hg Cu ect are unstable in aqueous solution and easily dehydrated to oxides.
References ( 1)
(2) ( 3)
Woodcock A, E, R; Bulletin of Mathematical Biology Volume 40, (1978), 1-25. Th, Brocker; Differentiable Germs And Catastrophes (1975) 128. Th , Brocker; Differentiable Germs And Catastrophes (1975) 163-164.