- Email: [email protected]
Mathematical conversions of the thermodynamic excess functions represented by the Redlich-Kister expansion, and by the Chebyshev polynomial series to power series representations and vice-versa.
CALPHAD Vol.8, No.4, Printed in the USA.
pp. 283-294,
0364-5916/84 $3.00 + .OO (c) 1984 Pergamon Press Ltd.
1984
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS REPRESENTED BY THE REDLICH-KISTER EXPANSION, AND BY THE CHEBYSHEV POLYNOMIAL SERIES TO POWER SERIES REPRESENTATIONS AND VICE-VERSA.
Josef TOMISKA Institut fuer Physikalische Chemie der Universitaet Wien A-logo Wien, Waehringerstrasse 42 (AUSTRIA) (Partly presented at CALPHAD X, Wien, Austria, July 1981) ABSTRACT. The equivalence of the Redlich-Kister expansion, and the Chebyshev polynomial expansion to power series in regard with the representation of thermodynamic excess properties is proved mathematically. Simple conversion formulas are derived for the computation of the of the coefficients of the power series,sk, from the parameters and from the parameters of the ChebyRedlich-Kister expansion, B shev polynomial expansion, t;, and vice-versa. The essential feature of the proposed conversion procedures is the independence of the conversion coefficients b(n,k) and a(l,k) from the actual thermodynamic excess properties. 1.
INTRODUCTION
Many mathematical expansions in terms of various polynomials have been proposed for the a proximative representation of the thermodynamic excess functions (compare r I] - [?I). Several of these proposals, specially the expansions after Redlich and Kister [4], have been applied successfully for the description of the behavior of many binary systems [3]. Various authors, in particular Bale [z] promote the socalled "orthogonal" series as attractive expansions to represent approximatively the thermodynamic excess quantities. Computer experiments showed the equivalence between various expansions (Redlich-Kister expansion, Legendre polynomial series, a-formalism of Kortuem, Margules equations) to simple power series in regard with the fitting of experimental data of the first derivative of the excess Gibbs energies [3]. Consequently it should be possible to prove this equivalence by mathematical reasoning, and simple conversion formulas should be derivable which are independent from the actual shape of the considered curve. Recently the mathematical proof of this equivalence between the Legendre polynomial series and the ower series as well as the desired conversion formulas have been published P 51. In this paper the corresponding relationship between the Redlich-Kister expansions, and the Chebyshev polynomial series to power series will be presented. 2. EQUIVALENCE BETWEEN THE REDLICH-KISTER EXPANSION AND THE POWER SERIES Redlich and Kister [4] suggested to repres y"t ihe theEfmodynamic integral excess properties of binary systems, Z(x) (Z = G , H , or S ), by
z(x)
= x(1-x)
E
B, (2x-l)"-'
(1 1
n=l
where x = mole fraction of the component 2, N = total number of parameters B,, Bn = adjustable parameters. Received December 27, 1983
283
284
J. TOMISKA
The right hand side of E (1) is a polynomial in (2x-l) of the degree (N-l) multiplied by the term x ? l-x). A polynomial in z is a sum of multiples of powers of z the degree being the highest power. The function Q(z)
= q,
+
q, z + q2 z2 + . .. + qk zk
(2)
for any values of q,, q,, q2, ....qk (except qk= 0) is a polynomial of the degree k [6]. On the other hand any term B,(2x-l)"-' represents a polynomial in x of the degree (n-l). So the sum over n in the right hand side of Eq(1) can be expressed as a polynomial in x of the order (N-l)
! n=l
B, (2x-l)"-' = IL n=l
(3)
where rn denotes the corresponding coefficient of the (n-1)-th power of x. Substituting by Eq(5) in Eq(1) and applying the distributive law of real numbers [6] yields N 1 rn x" - E n=l n=l
Z(x) =
r, xn+' .
(4)
With respect to the associative law of the addition of real numbers [6] the sum of two polynomials in x (of arbitrary degrees j and 1) gives also a polynomial in x (of the degree 1, if jcl). So, Eq(4) can be rewritten into z(x)
N+l 1 k=l
=
Sk xk
(5)
with Sk
:=
rk
-
rk-1
(ro = rN+l
= 0;
k=l,2
,...,
N+l).
Corresponding to the N independent Redlich-Kister arameters B the N coefficients r, are also linear independent. Whereas the PN+l) coefficients sk are linear dependent, as it can be proved easily from Eq(6): Each sk represents the difference of two successive coefficients r, and so the sum over all (N+l) coefficients sk vanishes necessarily. With other words, the (N+l) coefficients sk has been built by linear combinations of only N coefficients sk. The right hand side of Eq(5) represents a simple power series in x of the degree (N+l), corresponding to the highest power of x in Eq( 1). Consequently Eq(5) is the representation of the considered integr,al excess function Z(x) in terms of a power series. Since E (5) has been derived from Eq(1) by applying only equivalent transformations 61 the Eqs (1) and (5) are necessarily equivalent over the real numbers. And the equivalence between the Redlich-Kister expansion and the power series for the representation of the thermodynamic excess properties Z(x) follows finally from the law of transitivity of equations [6].
2.1) CALCULATION OF THE COEFFICIENTS sk FROM THE REDLICH-KISTER PARAMETERS
B,,
The most general conversion between two different, but equivalent representations of the same thermodynamic excess function Z(x) is a procedure which is independent from the actual shape of the considered function. This means, the parameters of the power series (Eq(5)), sk, should be computable from the
285
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS
parameters of the Redlich-Kister expansion (Eq(1 “conversion coefficients” which are only dependent total number (N) of the expansion parameters
‘k
= b& B1
+
But, the conversion of the B, and the The
explicit
method form of
(u+v)l =
b$
sk’ of the
B2
N X n=l
+
:
B,
n=l
BN =
have
to
be
polynominals
2j
.j+2
)n-l
(n;l)
(-1 y-j
r;l) .
2j
B,
n 1 k=l
xk
2k-1
(-1 )n-k
+
: n=l
B,
n+l I: k=2
xk
2k-2
(_l )n-k-2
the
from
the
actual
values
to determine theorem [6]
and
multiply
out.
Then
Eq(1 )
+
theorem
t:;,
part to
of
Eq(9)
of
(i-l)
index
a = arbitrary j = arbitrary
real number, non-negative
integer,
k =
(10) is
of
associated the k-th
(11 1
.
binomial
by the
+
obvious that the term which contribute to the coefficient
)n-kl [2 (knI])+ (;I;)
addition
first leads
coefficients
[6]
(12)
where
the
j=O
Considering Eq(l0) it becomes the parameter B, in Eq(1) will in x, sk, with:
Applying
.
is a simple way Use the binomial
(-1 )n-1-j
; n=l
(-1
B,
independent
(2x-l
=
Bn[ 2k-2
bt
(8)
Replacing of the summation index j in the and in the second part by k = (j+2) (j+l),
Z(x)
N X n=l
comparison coefficients:
n-l 1 .j+l j=O
1
bi
by means of socalled indices (n,k) and the
J the
n-l
B,
+
coefficient conversion
jio(j) d-j
=
a..
coefficients
in Eq(1 ) to substitute can be written as
Z(x)
+
)), B on ti%
with power
-220 364
72 -98
11
-13
15
1
-1
1
-1
5
6
7
8
-9
7
-1
-840
400
1232
-432
112
-160
120
48
-56
32 -50
-16
-8
20
-18
-4
4
8
-5
1
3
-2
3
-1
2
-1
1
1
-1120
256
-32
576
-64
-128 4
Numerical values of the conversion coefficients b(n,k) for the determination of the coefficients of a series, sky from the parameters of a Redlich-Kister expansion, B,, and vice-versa.
TABLE 1 power
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS
transforms the expression (11) into a form definitions for negative vapes of (k-2). n=(k-I) contribute to the x -term, the 1 ,2,...,N+l) is given by
Sk
n=max(l with
B
=
max(1 ,k-I
)
,k-1 )
[2k-2
(-l)n-k
’ (:I;)
n for for
= 1 I = k-l
k=l k>l
which renders unnecessary Since all B,-terms of corresponding coefficient
+ (,“_I)
I3
287
additional Eq(1) with sk (k =
(13)
’
(14)
.
The Def. (14) has been introduced to make obvious that the first (k-l ) B,-terms do not contribute to the xk -term. But, with respect to the definition of the polynomial coefficients (compare [6]) the summation in Eq(l3) can also run over all n with l&n&N. No factor put in square brackets on the right hand side of Eq(l3) depends on the value of the parameter B,. The values of all these mentioned factors do only depend on the indices n and k. Therefore these factors can be summarized to
b”k :=
with
2k-2
(-1 )n-k
n = 1 ,2,. . . ,N;
Sk
1
t::;)
k = 1,2,.
=
n=max( 1 ,k-1 )
Bn bt
+(k”-,)
’
(15)
’
. . ,N+l ; and Eq(l3)
(k=l ,2 ,...,
N+l )
can
be
simplified
.
to
(16)
The expression (16), a set of (N+l ) simultaneous equations, represents the desired conversion procedure for the computation of the coefficients s of the equivalent power series from the parameters Bn of the corresponding & edlichKister expansion. The numerical values of the conversion coefficients b(n,k) for n = 1 ,2 ,..., 8 and k = 1 ,2 ,..., 9 are given in Table 1.
2.2)
CALCULATION OF THE REDLICH-KISTER
The conversion the determination ding power series,
‘N+l
SN
= BN
PARAMETERS B,
FROM THE COEFFICIENTS ski
coefficients b(n,k), defined by Eq(l5), of the parameters B, from the coefficients as sk. The Eqs (16) can be written
can
be used also for of the correspon-
b!+l
= BN b# + BN_,
bi-’ (17)
52
= BN b!$ + BN_l
b$!-’
+ . . . + B,
b;
s1
= BN by + BN_,
by-’
+ . . . + B 1 bl
-
288
J. TOMISKA
Solving the system of equations (17) with respect to the parameters BN yields
BN = ‘N+l
bt+l
/
(18) B, =
F
bn+, -
Bj
j=n+l
b$+l1 / bk+l
(n=1,2 ,...,N-I).
9
3. CONVERSION BETWEEN THE CHEBYSHEV POLYNOMIAL SERIES AND THE POWER SERIES 3.1) CHEBYSHEV POLYNOMIALS
The Cheb shev polynomials in the variable w are defined over the real range [-l&w&l y and satisfy the hypergeometric differential equation [7]. There exist Chebyshev polynomials of first and of second kind, but usuallv the snecification "of first kind" is omitted. Following this notation, the "Chebyshev polynomials" of the order 1 in the varible w can be written as [S] T1 := [(w +&G-)1
)l] / 2 .
+ (W-K
(19)
The explicit expressions of the first six polynomials Tl(w) are summarized in Table 2. The Chebyshev polynomials defined in Eq(19) form a orthonormal set with the weight function [2/(Tc,/T)]
2 Tl
_,'(l-~~)-~/~ s
T,(w)
Tl(w)
dw =
6,,,
(n,m=1,2,...)
(20)
with 6
m,n
1
= 6m =
n
0
form=n for m # n 9
(21)
and the 91(w) fulfill the following recurrence relation:
Tl(d
= 2w Tl_, (w) - Tl_2(~)
.
(22)
By means of the transformation (23)
w= 2x-l
the range of definition will be shifted to the interval [04x41] without changing the norm of the Chebyshev polynomials, since
J
ITl(w)112=
-1
s 1
1
2 Tl(w)2
Tt - ~ -
dw = o
2 Tl(2x-1)2 ; T dx = 1Tl(2x-1 )l12. (24)
It should be mentioned that various normalizations of the Chebyshev polynomials exists which differ in constant factors. By means of comparison with
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESSFUNCTIONS
289
TABLE 2 The Chebyshev polynomials Tl(w) for 1 = 0 to 1 = 5
To(w) = 1 T,(w) = w T2(w) = 2w2
- 1
T+)
- 3w
= 4w3
T4(w) = 8~4 TV
the polynomials determined.
- 8~2
+ 1
= 16~5 - 20~3 + 5~
in Table 2 the actual factor of proportional ity can easi_ly be
The approximation theory shows that the Chebyshev polynomials can be applied for the "uniform approximation" (in contrast to the approximation in the quadratic mean) of a periodic continuous function [g]. But the described character can only be used if the function which has to be approximated is well-known over the whole range of definition of the applied Chebyshev polynomials, that means over the continuous interval of [06xLl] in case of the "shifted" polynomials Tl(2x-1) [g]. If only a finite number of points of the function, which has to be approximated, is known then the "shifted" Chebyshev 01 nomials Tl(2x-1) represent only "normal" polynomials in x of the degree 1 t)IO3 . 3.2) EQUIVALENCE IN REGARD WITH THE REPRESENTATION OF THE THERMODYNAMIC PROPERTIES.
EXCESS
The representation of an arbitrary thermodynamic excess function f(x) (integral or partial properties) by means of Chebyshev polynomials can be written as
f(x) =
: l=O
Al
Tl(2x-1) ,
(25)
where Al = adjustable parameters . In contrast to Eq(l), the Redlich-Kister expansion of the integral excess the Gibbs-Duhem quantities Z(x), the Eq(25) will not fulfill automatically conditions. But, the thermodynamic boundary conditions are conditions imposed on the adjustable parameters A which influence only the problem of the fitting of experimental data. There h ore the consistency of Eq(25) can be supposed without loss of generality. With respect to Eq(22) T (w) is a polynomial in w of the degree 1, if Tl ,(w) and Tl_2(w) represent + he corresponding polynomials. Since T (w) and (see &able 2), T,rw) are defined as polynomials of the order l=O and l=l in w
290
J. TOMISKA
al
0
W
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS
291
all
Tl are necessarily polynomials of the degree 1. Therfore, after executing the transformation (23) the Chebyshev polynomials can be expressed as
Tl(2x-1) =
: j=O
q3 (2x-1)j .
(26)
A recurrence relation for the coefficients qi q; = 2 95-1 - 93-2 ,
(27a
oI:=, Qo - q1
(27b
,
9: = 0, if j>l or j f 2n-1, (n=1,2,...) ,
(27~)
can be obtained by substituting in Eq(22) from Eq(26) and applying the method of coefficient comparison, as easily can be verified by explicit computation. The numerical values of the coefficients for 1,j = 0,1,2,...,9are presented in Table 3. Eq(26) makes evident that the shifted Chebyshev polynomials Tl(2x-1) are polynomials in (2x-l) of the degree 1, and consequently a polynomial in x of the same degree
Tl(2x-1) =
ai xk .
: k=O
(28)
Applying the binomial theorem, Eq(8), in Eq(26) yields
Tl(2x-1) =
: j=O
93
ifo
2i (-I )j-i (J)xi ,
(29)
and coefficient comparison between the Eqs.(28) and (29) leads then to a formula for the explicit determination of the a(l,k),
1 _ 2k ak -
(-I )j-k ($1 q+
: j=k
(30)
as easily can be erified from Eq(29): The sum over the index j can only whereas the sum over the index i reduces to contribute to the xyr-termifkLjLl, the term with i=k. Substitution in Eq(23) from Eq(28) makes evident that the Chebyshev polynomial expansion can be expressed as a power series
f(x) =
! l=O
Al
r" sk xk . k=O
(31)
J. TOMISKA
292
v
.-44-l
a0
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESSFUNCTIONS
Similar reasoning power series, sk:
Sk
!
=
as
yields
ai ,
Al
l=k
above
the
explicit
form
of
the
293
coefficients
(k=0,1,2,...,N).
of
the
(32)
The power series representation of the considered thermodynamic excess function f(x), Eq(31), has been derived from the Chebyshev polynomial expansion of f(x), by applying only equivalent transformations. Consequently the Eqs(25) Eq(25), and (31) are necessarily equivalent in regard with the representation of f(x) (similar reasoning as in section 2). 3.3)
CALCULATION OF THE sL FROM THE CHEBYSHEV PARMAETERS Al2
Eq(30) shows that the a(l,k) are independent from the parameters Al which characterize the shape of the actual thermodynamic excess function f(x) in the representation by means of Chebyshev polynomials. Therefore Eq(32) represents already the desired conversion formula for the detemination of the coefficients of a power series, sk, from the parameters of the Chebyshev pol nomial expansion, A . The computation of the conversion coefficients a(l,k the 9 requires explici 2 values of the q(l,j). Numerical data of a limited number of q(l,j) are listed in Table 3. If a higher number of data is required, the determination of the desired values by applying the recurrence ralation, Eq(27), will not cause any difficulties. But, the conversion coefficients a(l,k) can also be determined without any recurrence relation: Executing the transformation (23) the definition of the Chebyshev polynomials and use of the binomial Eq(8), yields
Tl(2x-1)
Since
[1+(-l
=
: i=O
2i-’
Ii]
= 2 bi,2j
Tl(2x-1
) =
1niki’2)
Int(l/2)
=
(i)
for
) 1-i
(2x-l
j
= 1,2...
t2$ $j
(x2-x)i/2
[1+(-l
, Eq(33)
(2x-1 )1-2j
can
(x2-x)j
)i]
.
be written
,
directly in Eq(lg), theorem,
(33)
as
(34)
with
Applying
the
binomial
Tl(2x-1)
Comparison
l/2
(l-l )/2
=
of
if if
theorem,
i:.
1 = even number, 1 = odd number. Eq(8),
$1’
and
(,;)
(i)
rearrangement
(‘i’j)
21Dn
yields
(-1 )i+n
X1-i-nm
j=O
coefficients
in
Eqs(28)
and
(36)
yields
finally
(36)
the
desired
J. TOMISKA
294
formula for the direct calculation of the conversion coefficients a(l,k) l"k -
1;;;l'2) j0
(ij) (;)($2_;)
2i+k
(-IF
.
(37)
3.4) DETERMINATION OF THE CHEBYSHEV PARAMETERS Al FROM THE skz
The parameters Al can be determined from re resentation (Eq(28)) Eq P32), with respect to
of Eq(25), the Chebyshev polynomial expansion of f(x), the Coefficients sk of the corresponding power series by solving the system of (N+l) simultaneous equations, the parameters Al:
AN = SN / a{ (38)
Al = [sl -
!Z j=l+l
Aj ai] / ai .
4. DISCUSSION
The utility of orthogonal polynomials which are defined over a continuous range of definitions (Chebyshev polynomials, etc) in regard with the fitting of a finite number of experimental points of a thermodynamic excess function will be discussed elsewhere [lo]. This work proved mathematically the equivalence of the Chebyshev polynomial expansion, and the Redlich-Kister expansion to power series in regard with the representation of thermodynamic excess properties. Consequently any deviations among best fit curves of the same experimental data which have been determined by fittings based upon the Redlich-Kister expansion, the Chebyshev polynomial series and the power series can only be caused by insufficient accuracy of computation. By means of the derived conversion formulas, together with the system-independent conversion coefficients presented in Table 1, and Table 4 the values of the thermodynamic excess quantities can easily be converted among the discussed representations. ACKNOWLEDGEMENTS I would like to thank Prof. Dr. A. Neckel very much for his interest and discussion of this work. Grateful acknowledgement is made for the financial support of the "Fends zur Foerderung der wissenschaftlichen Forschung in Oesterreich". REFERENCES 1.
E.HALA, J.PICK, V.FRIED, and O.VILiM, Vapor-Liquid Equilibrium, Pergamon Press, London, 1958. C.W.BALE and A.D.PELTON, Met. Trans. 5, 2323 (1974). ;: J.TOMISKA, CALPHAD 4, No. 2, 63 (19807. O.REDLICH and A.T.KISTER, Ind. Eng. Chem. 40, 345 (1948). 4. J.TOMISKA, CALPHAD 2, No. 2, 93 (1981). 65: I.N.BRONSTEIN and K.A.SEMENDJAJEW. Taschenbuch der Mathemathik. Nauka, Moskau, & BSB B.G.Teubner,'Leipzig, 1979. I.N.SNEDDON, Spezielle Funktionen der Mathematischen Physik, 7. Bibliographisches Institut, Mannheim, 1961. 8. R.COURANT and D.HILBERT, Methoden der Mathematischen Physik I, Springer, Berlin, 1931.. in der Mathematischen Phvsik, 9. H.MESCHKOWSKI. Reihenentwicklunaen _. Bibliographisches Institut, Mannheim, 1963. 10. J.TOMISKA, CALPHAD, to be published.
pp. 283-294,
0364-5916/84 $3.00 + .OO (c) 1984 Pergamon Press Ltd.
1984
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS REPRESENTED BY THE REDLICH-KISTER EXPANSION, AND BY THE CHEBYSHEV POLYNOMIAL SERIES TO POWER SERIES REPRESENTATIONS AND VICE-VERSA.
Josef TOMISKA Institut fuer Physikalische Chemie der Universitaet Wien A-logo Wien, Waehringerstrasse 42 (AUSTRIA) (Partly presented at CALPHAD X, Wien, Austria, July 1981) ABSTRACT. The equivalence of the Redlich-Kister expansion, and the Chebyshev polynomial expansion to power series in regard with the representation of thermodynamic excess properties is proved mathematically. Simple conversion formulas are derived for the computation of the of the coefficients of the power series,sk, from the parameters and from the parameters of the ChebyRedlich-Kister expansion, B shev polynomial expansion, t;, and vice-versa. The essential feature of the proposed conversion procedures is the independence of the conversion coefficients b(n,k) and a(l,k) from the actual thermodynamic excess properties. 1.
INTRODUCTION
Many mathematical expansions in terms of various polynomials have been proposed for the a proximative representation of the thermodynamic excess functions (compare r I] - [?I). Several of these proposals, specially the expansions after Redlich and Kister [4], have been applied successfully for the description of the behavior of many binary systems [3]. Various authors, in particular Bale [z] promote the socalled "orthogonal" series as attractive expansions to represent approximatively the thermodynamic excess quantities. Computer experiments showed the equivalence between various expansions (Redlich-Kister expansion, Legendre polynomial series, a-formalism of Kortuem, Margules equations) to simple power series in regard with the fitting of experimental data of the first derivative of the excess Gibbs energies [3]. Consequently it should be possible to prove this equivalence by mathematical reasoning, and simple conversion formulas should be derivable which are independent from the actual shape of the considered curve. Recently the mathematical proof of this equivalence between the Legendre polynomial series and the ower series as well as the desired conversion formulas have been published P 51. In this paper the corresponding relationship between the Redlich-Kister expansions, and the Chebyshev polynomial series to power series will be presented. 2. EQUIVALENCE BETWEEN THE REDLICH-KISTER EXPANSION AND THE POWER SERIES Redlich and Kister [4] suggested to repres y"t ihe theEfmodynamic integral excess properties of binary systems, Z(x) (Z = G , H , or S ), by
z(x)
= x(1-x)
E
B, (2x-l)"-'
(1 1
n=l
where x = mole fraction of the component 2, N = total number of parameters B,, Bn = adjustable parameters. Received December 27, 1983
283
284
J. TOMISKA
The right hand side of E (1) is a polynomial in (2x-l) of the degree (N-l) multiplied by the term x ? l-x). A polynomial in z is a sum of multiples of powers of z the degree being the highest power. The function Q(z)
= q,
+
q, z + q2 z2 + . .. + qk zk
(2)
for any values of q,, q,, q2, ....qk (except qk= 0) is a polynomial of the degree k [6]. On the other hand any term B,(2x-l)"-' represents a polynomial in x of the degree (n-l). So the sum over n in the right hand side of Eq(1) can be expressed as a polynomial in x of the order (N-l)
! n=l
B, (2x-l)"-' = IL n=l
(3)
where rn denotes the corresponding coefficient of the (n-1)-th power of x. Substituting by Eq(5) in Eq(1) and applying the distributive law of real numbers [6] yields N 1 rn x" - E n=l n=l
Z(x) =
r, xn+' .
(4)
With respect to the associative law of the addition of real numbers [6] the sum of two polynomials in x (of arbitrary degrees j and 1) gives also a polynomial in x (of the degree 1, if jcl). So, Eq(4) can be rewritten into z(x)
N+l 1 k=l
=
Sk xk
(5)
with Sk
:=
rk
-
rk-1
(ro = rN+l
= 0;
k=l,2
,...,
N+l).
Corresponding to the N independent Redlich-Kister arameters B the N coefficients r, are also linear independent. Whereas the PN+l) coefficients sk are linear dependent, as it can be proved easily from Eq(6): Each sk represents the difference of two successive coefficients r, and so the sum over all (N+l) coefficients sk vanishes necessarily. With other words, the (N+l) coefficients sk has been built by linear combinations of only N coefficients sk. The right hand side of Eq(5) represents a simple power series in x of the degree (N+l), corresponding to the highest power of x in Eq( 1). Consequently Eq(5) is the representation of the considered integr,al excess function Z(x) in terms of a power series. Since E (5) has been derived from Eq(1) by applying only equivalent transformations 61 the Eqs (1) and (5) are necessarily equivalent over the real numbers. And the equivalence between the Redlich-Kister expansion and the power series for the representation of the thermodynamic excess properties Z(x) follows finally from the law of transitivity of equations [6].
2.1) CALCULATION OF THE COEFFICIENTS sk FROM THE REDLICH-KISTER PARAMETERS
B,,
The most general conversion between two different, but equivalent representations of the same thermodynamic excess function Z(x) is a procedure which is independent from the actual shape of the considered function. This means, the parameters of the power series (Eq(5)), sk, should be computable from the
285
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS
parameters of the Redlich-Kister expansion (Eq(1 “conversion coefficients” which are only dependent total number (N) of the expansion parameters
‘k
= b& B1
+
But, the conversion of the B, and the The
explicit
method form of
(u+v)l =
b$
sk’ of the
B2
N X n=l
+
:
B,
n=l
BN =
have
to
be
polynominals
2j
.j+2
)n-l
(n;l)
(-1 y-j
r;l) .
2j
B,
n 1 k=l
xk
2k-1
(-1 )n-k
+
: n=l
B,
n+l I: k=2
xk
2k-2
(_l )n-k-2
the
from
the
actual
values
to determine theorem [6]
and
multiply
out.
Then
Eq(1 )
+
theorem
t:;,
part to
of
Eq(9)
of
(i-l)
index
a = arbitrary j = arbitrary
real number, non-negative
integer,
k =
(10) is
of
associated the k-th
(11 1
.
binomial
by the
+
obvious that the term which contribute to the coefficient
)n-kl [2 (knI])+ (;I;)
addition
first leads
coefficients
[6]
(12)
where
the
j=O
Considering Eq(l0) it becomes the parameter B, in Eq(1) will in x, sk, with:
Applying
.
is a simple way Use the binomial
(-1 )n-1-j
; n=l
(-1
B,
independent
(2x-l
=
Bn[ 2k-2
bt
(8)
Replacing of the summation index j in the and in the second part by k = (j+2) (j+l),
Z(x)
N X n=l
comparison coefficients:
n-l 1 .j+l j=O
1
bi
by means of socalled indices (n,k) and the
J the
n-l
B,
+
coefficient conversion
jio(j) d-j
=
a..
coefficients
in Eq(1 ) to substitute can be written as
Z(x)
+
)), B on ti%
with power
-220 364
72 -98
11
-13
15
1
-1
1
-1
5
6
7
8
-9
7
-1
-840
400
1232
-432
112
-160
120
48
-56
32 -50
-16
-8
20
-18
-4
4
8
-5
1
3
-2
3
-1
2
-1
1
1
-1120
256
-32
576
-64
-128 4
Numerical values of the conversion coefficients b(n,k) for the determination of the coefficients of a series, sky from the parameters of a Redlich-Kister expansion, B,, and vice-versa.
TABLE 1 power
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS
transforms the expression (11) into a form definitions for negative vapes of (k-2). n=(k-I) contribute to the x -term, the 1 ,2,...,N+l) is given by
Sk
n=max(l with
B
=
max(1 ,k-I
)
,k-1 )
[2k-2
(-l)n-k
’ (:I;)
n for for
= 1 I = k-l
k=l k>l
which renders unnecessary Since all B,-terms of corresponding coefficient
+ (,“_I)
I3
287
additional Eq(1) with sk (k =
(13)
’
(14)
.
The Def. (14) has been introduced to make obvious that the first (k-l ) B,-terms do not contribute to the xk -term. But, with respect to the definition of the polynomial coefficients (compare [6]) the summation in Eq(l3) can also run over all n with l&n&N. No factor put in square brackets on the right hand side of Eq(l3) depends on the value of the parameter B,. The values of all these mentioned factors do only depend on the indices n and k. Therefore these factors can be summarized to
b”k :=
with
2k-2
(-1 )n-k
n = 1 ,2,. . . ,N;
Sk
1
t::;)
k = 1,2,.
=
n=max( 1 ,k-1 )
Bn bt
+(k”-,)
’
(15)
’
. . ,N+l ; and Eq(l3)
(k=l ,2 ,...,
N+l )
can
be
simplified
.
to
(16)
The expression (16), a set of (N+l ) simultaneous equations, represents the desired conversion procedure for the computation of the coefficients s of the equivalent power series from the parameters Bn of the corresponding & edlichKister expansion. The numerical values of the conversion coefficients b(n,k) for n = 1 ,2 ,..., 8 and k = 1 ,2 ,..., 9 are given in Table 1.
2.2)
CALCULATION OF THE REDLICH-KISTER
The conversion the determination ding power series,
‘N+l
SN
= BN
PARAMETERS B,
FROM THE COEFFICIENTS ski
coefficients b(n,k), defined by Eq(l5), of the parameters B, from the coefficients as sk. The Eqs (16) can be written
can
be used also for of the correspon-
b!+l
= BN b# + BN_,
bi-’ (17)
52
= BN b!$ + BN_l
b$!-’
+ . . . + B,
b;
s1
= BN by + BN_,
by-’
+ . . . + B 1 bl
-
288
J. TOMISKA
Solving the system of equations (17) with respect to the parameters BN yields
BN = ‘N+l
bt+l
/
(18) B, =
F
bn+, -
Bj
j=n+l
b$+l1 / bk+l
(n=1,2 ,...,N-I).
9
3. CONVERSION BETWEEN THE CHEBYSHEV POLYNOMIAL SERIES AND THE POWER SERIES 3.1) CHEBYSHEV POLYNOMIALS
The Cheb shev polynomials in the variable w are defined over the real range [-l&w&l y and satisfy the hypergeometric differential equation [7]. There exist Chebyshev polynomials of first and of second kind, but usuallv the snecification "of first kind" is omitted. Following this notation, the "Chebyshev polynomials" of the order 1 in the varible w can be written as [S] T1 := [(w +&G-)1
)l] / 2 .
+ (W-K
(19)
The explicit expressions of the first six polynomials Tl(w) are summarized in Table 2. The Chebyshev polynomials defined in Eq(19) form a orthonormal set with the weight function [2/(Tc,/T)]
2 Tl
_,'(l-~~)-~/~ s
T,(w)
Tl(w)
dw =
6,,,
(n,m=1,2,...)
(20)
with 6
m,n
1
= 6m =
n
0
form=n for m # n 9
(21)
and the 91(w) fulfill the following recurrence relation:
Tl(d
= 2w Tl_, (w) - Tl_2(~)
.
(22)
By means of the transformation (23)
w= 2x-l
the range of definition will be shifted to the interval [04x41] without changing the norm of the Chebyshev polynomials, since
J
ITl(w)112=
-1
s 1
1
2 Tl(w)2
Tt - ~ -
dw = o
2 Tl(2x-1)2 ; T dx = 1Tl(2x-1 )l12. (24)
It should be mentioned that various normalizations of the Chebyshev polynomials exists which differ in constant factors. By means of comparison with
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESSFUNCTIONS
289
TABLE 2 The Chebyshev polynomials Tl(w) for 1 = 0 to 1 = 5
To(w) = 1 T,(w) = w T2(w) = 2w2
- 1
T+)
- 3w
= 4w3
T4(w) = 8~4 TV
the polynomials determined.
- 8~2
+ 1
= 16~5 - 20~3 + 5~
in Table 2 the actual factor of proportional ity can easi_ly be
The approximation theory shows that the Chebyshev polynomials can be applied for the "uniform approximation" (in contrast to the approximation in the quadratic mean) of a periodic continuous function [g]. But the described character can only be used if the function which has to be approximated is well-known over the whole range of definition of the applied Chebyshev polynomials, that means over the continuous interval of [06xLl] in case of the "shifted" polynomials Tl(2x-1) [g]. If only a finite number of points of the function, which has to be approximated, is known then the "shifted" Chebyshev 01 nomials Tl(2x-1) represent only "normal" polynomials in x of the degree 1 t)IO3 . 3.2) EQUIVALENCE IN REGARD WITH THE REPRESENTATION OF THE THERMODYNAMIC PROPERTIES.
EXCESS
The representation of an arbitrary thermodynamic excess function f(x) (integral or partial properties) by means of Chebyshev polynomials can be written as
f(x) =
: l=O
Al
Tl(2x-1) ,
(25)
where Al = adjustable parameters . In contrast to Eq(l), the Redlich-Kister expansion of the integral excess the Gibbs-Duhem quantities Z(x), the Eq(25) will not fulfill automatically conditions. But, the thermodynamic boundary conditions are conditions imposed on the adjustable parameters A which influence only the problem of the fitting of experimental data. There h ore the consistency of Eq(25) can be supposed without loss of generality. With respect to Eq(22) T (w) is a polynomial in w of the degree 1, if Tl ,(w) and Tl_2(w) represent + he corresponding polynomials. Since T (w) and (see &able 2), T,rw) are defined as polynomials of the order l=O and l=l in w
290
J. TOMISKA
al
0
W
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESS FUNCTIONS
291
all
Tl are necessarily polynomials of the degree 1. Therfore, after executing the transformation (23) the Chebyshev polynomials can be expressed as
Tl(2x-1) =
: j=O
q3 (2x-1)j .
(26)
A recurrence relation for the coefficients qi q; = 2 95-1 - 93-2 ,
(27a
oI:=, Qo - q1
(27b
,
9: = 0, if j>l or j f 2n-1, (n=1,2,...) ,
(27~)
can be obtained by substituting in Eq(22) from Eq(26) and applying the method of coefficient comparison, as easily can be verified by explicit computation. The numerical values of the coefficients for 1,j = 0,1,2,...,9are presented in Table 3. Eq(26) makes evident that the shifted Chebyshev polynomials Tl(2x-1) are polynomials in (2x-l) of the degree 1, and consequently a polynomial in x of the same degree
Tl(2x-1) =
ai xk .
: k=O
(28)
Applying the binomial theorem, Eq(8), in Eq(26) yields
Tl(2x-1) =
: j=O
93
ifo
2i (-I )j-i (J)xi ,
(29)
and coefficient comparison between the Eqs.(28) and (29) leads then to a formula for the explicit determination of the a(l,k),
1 _ 2k ak -
(-I )j-k ($1 q+
: j=k
(30)
as easily can be erified from Eq(29): The sum over the index j can only whereas the sum over the index i reduces to contribute to the xyr-termifkLjLl, the term with i=k. Substitution in Eq(23) from Eq(28) makes evident that the Chebyshev polynomial expansion can be expressed as a power series
f(x) =
! l=O
Al
r" sk xk . k=O
(31)
J. TOMISKA
292
v
.-44-l
a0
MATHEMATICAL CONVERSIONS OF THE THERMODYNAMIC EXCESSFUNCTIONS
Similar reasoning power series, sk:
Sk
!
=
as
yields
ai ,
Al
l=k
above
the
explicit
form
of
the
293
coefficients
(k=0,1,2,...,N).
of
the
(32)
The power series representation of the considered thermodynamic excess function f(x), Eq(31), has been derived from the Chebyshev polynomial expansion of f(x), by applying only equivalent transformations. Consequently the Eqs(25) Eq(25), and (31) are necessarily equivalent in regard with the representation of f(x) (similar reasoning as in section 2). 3.3)
CALCULATION OF THE sL FROM THE CHEBYSHEV PARMAETERS Al2
Eq(30) shows that the a(l,k) are independent from the parameters Al which characterize the shape of the actual thermodynamic excess function f(x) in the representation by means of Chebyshev polynomials. Therefore Eq(32) represents already the desired conversion formula for the detemination of the coefficients of a power series, sk, from the parameters of the Chebyshev pol nomial expansion, A . The computation of the conversion coefficients a(l,k the 9 requires explici 2 values of the q(l,j). Numerical data of a limited number of q(l,j) are listed in Table 3. If a higher number of data is required, the determination of the desired values by applying the recurrence ralation, Eq(27), will not cause any difficulties. But, the conversion coefficients a(l,k) can also be determined without any recurrence relation: Executing the transformation (23) the definition of the Chebyshev polynomials and use of the binomial Eq(8), yields
Tl(2x-1)
Since
[1+(-l
=
: i=O
2i-’
Ii]
= 2 bi,2j
Tl(2x-1
) =
1niki’2)
Int(l/2)
=
(i)
for
) 1-i
(2x-l
j
= 1,2...
t2$ $j
(x2-x)i/2
[1+(-l
, Eq(33)
(2x-1 )1-2j
can
(x2-x)j
)i]
.
be written
,
directly in Eq(lg), theorem,
(33)
as
(34)
with
Applying
the
binomial
Tl(2x-1)
Comparison
l/2
(l-l )/2
=
of
if if
theorem,
i:.
1 = even number, 1 = odd number. Eq(8),
$1’
and
(,;)
(i)
rearrangement
(‘i’j)
21Dn
yields
(-1 )i+n
X1-i-nm
j=O
coefficients
in
Eqs(28)
and
(36)
yields
finally
(36)
the
desired
J. TOMISKA
294
formula for the direct calculation of the conversion coefficients a(l,k) l"k -
1;;;l'2) j0
(ij) (;)($2_;)
2i+k
(-IF
.
(37)
3.4) DETERMINATION OF THE CHEBYSHEV PARAMETERS Al FROM THE skz
The parameters Al can be determined from re resentation (Eq(28)) Eq P32), with respect to
of Eq(25), the Chebyshev polynomial expansion of f(x), the Coefficients sk of the corresponding power series by solving the system of (N+l) simultaneous equations, the parameters Al:
AN = SN / a{ (38)
Al = [sl -
!Z j=l+l
Aj ai] / ai .
4. DISCUSSION
The utility of orthogonal polynomials which are defined over a continuous range of definitions (Chebyshev polynomials, etc) in regard with the fitting of a finite number of experimental points of a thermodynamic excess function will be discussed elsewhere [lo]. This work proved mathematically the equivalence of the Chebyshev polynomial expansion, and the Redlich-Kister expansion to power series in regard with the representation of thermodynamic excess properties. Consequently any deviations among best fit curves of the same experimental data which have been determined by fittings based upon the Redlich-Kister expansion, the Chebyshev polynomial series and the power series can only be caused by insufficient accuracy of computation. By means of the derived conversion formulas, together with the system-independent conversion coefficients presented in Table 1, and Table 4 the values of the thermodynamic excess quantities can easily be converted among the discussed representations. ACKNOWLEDGEMENTS I would like to thank Prof. Dr. A. Neckel very much for his interest and discussion of this work. Grateful acknowledgement is made for the financial support of the "Fends zur Foerderung der wissenschaftlichen Forschung in Oesterreich". REFERENCES 1.
E.HALA, J.PICK, V.FRIED, and O.VILiM, Vapor-Liquid Equilibrium, Pergamon Press, London, 1958. C.W.BALE and A.D.PELTON, Met. Trans. 5, 2323 (1974). ;: J.TOMISKA, CALPHAD 4, No. 2, 63 (19807. O.REDLICH and A.T.KISTER, Ind. Eng. Chem. 40, 345 (1948). 4. J.TOMISKA, CALPHAD 2, No. 2, 93 (1981). 65: I.N.BRONSTEIN and K.A.SEMENDJAJEW. Taschenbuch der Mathemathik. Nauka, Moskau, & BSB B.G.Teubner,'Leipzig, 1979. I.N.SNEDDON, Spezielle Funktionen der Mathematischen Physik, 7. Bibliographisches Institut, Mannheim, 1961. 8. R.COURANT and D.HILBERT, Methoden der Mathematischen Physik I, Springer, Berlin, 1931.. in der Mathematischen Phvsik, 9. H.MESCHKOWSKI. Reihenentwicklunaen _. Bibliographisches Institut, Mannheim, 1963. 10. J.TOMISKA, CALPHAD, to be published.