chemia - Studia

VALENTINA ROXANA DEJEU, SILVIA TOADER, BARABAS RÉKA, PAUL-ŞERBAN AGACHI
are:
where:
- two neighboring polynomials S i
(x)
and S i+ 1(
x)
have the following
properties:
S
i
( xi− 1)
= f ( xi−
1)
; for any i = 1,...,
n
(1)
S '
i
( xi
) = S'
i+
1
( xi
) , for any i = 1,...,
n −1
(2)
General expresions for two adjacent cubic fuctions S i
(x)
and ( )
S i+ 1
x
Si
( x)
= α ⋅ ai
( x)
+ β ⋅bi
( x)
+ γ ⋅ ci
( x)
+ δ ⋅ d
i
( x)
(3)
S x)
= β ⋅ a ( x)
+ γ ⋅b
( x)
+ δ ⋅ c ( x)
+ τ d ( ) (4)
i + 1( i+
1
i+
1
i+
1
⋅
i+
1
x
2
( xi
− x) ( x − xi−
1)
( x − ) 2
i
xi−
1
2
( x − xi−
1) ( xi
− x)
( x − ) 2
i
xi−
1
( x ) 2 i
− x [ 2( x − xi−
1) + ( xi
− xi−
1)
]
( x − ) 3
i
xi−
1
2
( x − xi−
1) [ 2( xi
− x) + ( xi
− xi−
1)
]
( x − x ) 3
a ( x)
=
(5)
i
b ( x)
= −
(6)
i
c ( x)
=
(7)
i
d ( x)
=
. (8)
i
i
'
'
S
i
( xi
) = Si
+ 1(
xi
) = β
(10)
'
S
i
( x i − 1)
= α
(11)
To solve mathematical problems (scientific calculations), advanced
software systems such as Matlab, Mathematica, or Mathcad are used [2,4,5].
It is generally acknowledged that in the crystallization of calcium
phosphate first occurs the formation of a precursor phase (amorphous calcium
phosphate) which is subsequently dissolved or restructured as the precipitation
reaction occurs and turns into hydroxyapatite [6.7]. Transformation kinetics of
amorphous calcium phosphate into hydroxyapatite, which can be described
by a first order reaction law, is only a function of the solution pH at constant
temperature [8.9]. Solution transformation depends on the conditions that
regulate both amorphous calcium phosphate dissolution and formation of the
first nuclei of hydroxyapatite [10]. In a recent study [11,12], the experimental
results concerning the influence of pH and temperature on the transformation
of amorphous calcium phosphate into hydroxyapatite have been presented.
162
i−1
Properties (1) and (2) become:
S x ) γ = f ( x ); S ( x ) = δ = S ( x ) = f ( x ) (9)
i
(
i−1 =
i−1
i i
i+
1 i
i