NUMERICAL MODELING IN CHEMISTRY

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NUMERICAL MODELING IN CHEMISTRY

• NUMERICAL KINETIC CHEMISTRY
• (Lorentz JÄNTSCHI, Elena Maria PICA)
• NUMERICAL DESCRIPTION OF TITRATION
• (Lorentz JÄNTSCHI, Horea Iustin NASCU)
• CORRELATIONS AND REGRESSIONS WITH MATHLAB
• (Lorentz JÄNTSCHI, Mihaela UNGURESAN)

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NUMERICAL KINETIC CHEMISTRY(Lorentz JÄNTSCHI,
Elena Maria PICA)
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Concentration gradient in an oscillating reaction

• The oscillating reactions are more than a
  laboratory curiosity.
• If in the industrial processes they appear in few
  cases, in biochemical systems there are numerous
  examples of oscillating reactions.
• For instance, the oscillating reactions maintain
  the rhythm.
• All live processes are based on one or more
  oscillating reactions.

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The MathCad parameters for the animation
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Lotka Volterra autocatalytic oscillator model

• Phenomena
• complex reaction, in homogeneous phase (stage),
  which shows damped or undamped oscillations
• Kinetic model
• R X ? 2X, ? ?1RX (a)
• X Y ? 2Y, ? ?2XY (b)
• Y ? P, ? ?3Y (c)
• P ? , ? ?4P (d)

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Mathematical model

• ?(a) - ?(b) ?1RX - ?2XY (1)
• ?(b) - ?(c) ?2XY - ?3Y (2)

Numerical model
xn1 xn (tn1-tn)xn(?1R-?2yn) (3)yn1
yn(tn1-tn)yn(?2xn-?3) (4)x0 X0 1,
y0 Y0 1, ?1 3, ?2 4, ?3 5, R 2
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Graphs produced/generated the numerical
series/systems (xn)n0 and (yn)n0 corresponding
to the temporal series (tn)n0
The oscillation of the intermediaries in L-V
mechanism
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• The variation path (X,Y) in the L-V mechanism

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Product evolution
The variation of the product concentration in
L-V mechanism
Product storage in L-V mechanism
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A model of damped oscillations

• Phenomena
• let it be a chemical process that takes place
  according to the following model of a reaction
  mechanism
• Kinetic model
• R1 ? X, ? ?1R1 (a)
• 2X Y ? 3Y, ? ?2X2Y (b)
• R2 X ? Y P1, ? ?3R2X (c)
• Y ? P2, ? ?4Y (d)

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• Observation
• As in Lotka Volterra, model, the concentrations
  of the R1 and R2 reacting substances remain
  constant during the process.
• Mathematical model

?(a) - 2?(b) - ?(c) (1) ?1R1 -
2?2X2Y - ?3R2X 2?(b) ?(c) -
?(d) (2) 2?2X2Y ?3R2X -
?4Y
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• Numerical model
• xn1 xn(tn1-tn)(?1R1-xn(2?2xnyn?3R
  2))
• (3)
• yn1 yn(tn1-tn)(xn(2?2xnyn?3R2)-?4y
  n)
• (4)
• Data output
• x0 0, y0 1, ?1 3, ?2 4, ?3 5, ?4 7,
• R1 2, R2 2,
• tn n/100000 with n 0,1..300000

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The damped oscillations in chemical reactions
(a) the conc. of the intermediary X
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The damped oscillations in chemical reactions
(b) the concentration of the intermediary Y
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The damped oscillations in chemical reactions
(c) the damped oscillation path (X,Y)
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• Data results
• equilibrium concentration are
• X 2.315 and Y 0.176
• and the equilibrium ratio are
• X/Y 13.53
• the dependence on time (tn)n0 of the
  accumulation of the
• reaction products P1 (p1n)n0 and
  P2 (p2n)n0
• is linear (see next graphs)

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The linear variation of the amount of products in
damped oscillating reactions (products P1 and P2)
Conclusion the concentration of the reaction
products changes linearity even if the
concentrations of the agents X and Y
oscillate towards the equilibrium value
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The brussel model of autocatalytic oscillation

• Remarks
• The brussel model was initiated by a group from
  Bruxelles directed by Ilya Prigogine (Nobel
  Prize) it introduce for the first time, mechanism
  of a reaction whose scheme of evolution converged
  on an attractor.
• More authors have changed this variant and have
  studied the systems running according to this
  mechanism.
• Simplified variant of kinetic model
• R ? X, ? ?1R (a)
• X 2Y ? 3Y, ? ?2XY2 (b)
• Y ? P, ? ?3Y (c)

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Mathematical model

• ?(a) - ?(b) ?1R1 - ?2XY2 (1)
• ?(b) - ?(c) ?2XY2 - ?3Y (2)

Observations Though the equations (1) and (2)
seem simpler, at first sight, they are even more
difficult to be solved by integration than
previous cases. Moreover, the literature has not
recorded their integration into the general case
described by (1-2). Besides, the equations do not
lead to an attractor model not matter by values
of the constants of speed and of the
concentrations R, X0 and Y0.
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• The attempt of solving (1-2) is full of
  surprises. For most of the values a system which
  develops towards a position of equilibrium is
  obtained there are values for which damped
  oscillations to equilibrium are found again the
  undamped periodical oscillations have also an
  important role, which is confirmed by the
  majority of the organisms in which the cellular
  biochemical processes are based on such
  oscillations.
• The processes taking place within the heart are a
  conclusive example the periodical heart beats
  are due to processes of this type. The importance
  of these processes is great.
• This was the reason for which Ilya Prigogine was
  awarded the Nobel Prize for chemistry in 1977,
  namely for his theories on the dissipative
  systems.

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• The equations (1-2) are simplified if
• R 1, ?1 1 and ?3 1,
• are chosen and when the differential system of
  equations becomes
• x 1 ?2xy2 y ?2xy2 y (3)
• However, the numerical simulation is made in the
  same way. Thus the iteration equation of
  variation for (3) is written
• xn1 xn(tn1-tn)(1-?2xnyn2),
• yn1 yn(tn1-tn)(?2xnyn2-yn) (4)
• Now choosing ?2 0.88 and taking into
  consideration two cases, the first one in which
  the initial concentrations of the agents are x10
  X1,0 1.5 and y10 Y1,0 2 and second
  case in which x20 X2,0 2 and y20 Y2,0
  2.5 and the series tn n/100 with n 0,1..150,

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• following representations for the concentrations
  of the
• agents X (xn)n0 and Y (yn)n0 are
  obtained

The concentrations of the intermediaries up to
the attractor for two cases with different I.C.
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• And the variation diagram of Y depending on X
  and the variation in time of the storage of
  reaction product is

(a) The entrance of Y related to X on the
same gravitational orbit for (b) different
product quantities obtained in two cases having
different initial conditions
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• If the previous figures are not very conclusive
  and fig. 10b seems to confirm this, the figures
  shows that, though the two systems start from
  different values of the concentrations of the
  agents, in both cases the system comes to evolve
  rather early on the same trajectory.
• Now, increasing the time interval by choosing
  another n 0,1..3000 the following
  concentrations of the agents are obtained X1
  (x1n)n0, X2 (x2n)n0, Y2 (y2n)n0 and
  Y2 (y2n)n0 for the two cases 1 and 2 of the
  chosen system (see next figures).
• It is noticed that, even if they do not evolve
  according the same values, same period and
  amplitude of the oscillations are recorded.

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The periodical evolution having the same
oscillation periodT 0.226 of (a) X and (b)
Y, for two cases having different initial
conditions
The dependence of Y under X for the cases as
well as the accumulation of the product is
printed in the next figures
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Convergence at atractor of brusselator system
independent from initial conditions and (b)
different quantities of resulted product
start point
atractor
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Conclusion

• The difference between the Lotka-Voltera model
  and Bruxelles model one is the following
• The Lotka-Voltera model oscillates around the
  initial values of the concentrations of the
  agents, whereas the Bruxelles one converges, in
  time on the same variation equation irrespective
  of the initial values of the concentrations of
  the agents.
• In fact the attractor does not appear for any of
  their values for a given k2 there are minimum
  y0,min and x0,min values from which the
  periodical oscillations arise and the system
  tends towards the curve given in previous figures.

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References

• M. Diudea, I. Gutman, L. Jäntschi (2001),
  Molecular Topology, Nova Science, Huntington, New
  York, 332 p., ISBN 1-56072-957-0.
• M. V. Diudea, Ed. (2001), QSPR / QSAR Studies by
  Molecular Descriptors, Nova Science, Huntington,
  New York, 438 p., ISBN 1-56072-859-0.
• L. Jäntschi, M. Unguresan (2001), Chimie Fizica.
  Cinetica si Dinamica Moleculara, Mediamira,
  Cluj-Napoca, 159 p., ISBN 973-9358-71-3.
• L. Jäntschi (2002), Studii Fitosanitare, Amici,
  Cluj-Napoca, 140 p., in press.

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NUMERICAL DESCRIPTION OF TITRATION(Lorentz
JÄNTSCHI, Horea Iustin NASCU)

• Abstract
• The analytical methods of qualitative and
  quantitative determination of ions in solutions
  are very flexible to automation.
• The process of titration is a recurrent process
  that can be watched by permanent measurement of a
  simple property such as mass, current intensity,
  tension, volume or a complex property such as
  adsorption, heat of reaction, which need a
  complex evaluation.
• The present work is focused on modeling the
  process of titration and presents a numerical
  simulation of acid-base titration. Titration
  parameter selected is the pH.
• The method permits to observe the titration
  process and identify the equivalence point of
  titration.

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Modeled reaction
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• If we consider that initially there exist 100
  mmols of the first solution (NH3) and no reaction
  product (CH3COONH4) and we note with a letter
  quantities from first solution and with b, the
  quantities from second solution, initial
  conditions (IC) can be written as

Integer values of substances from solution let be
evolve by steps of 1 mmol added solution. The
corresponding equation is
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• A correction is now necessary. The concentrations
  are of natural values (not negative numbers) and
  supplementary adding of one reactant in solution
  lead only to deplete the other one reactant. In
  conclusion, must reconsider previous equations
  with following corrections for iteration n

In every moment of titration, the pH is given by
(where another environment condition was
considered, the normal temperature, that make
that pH pOH 14)
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• The following step is now simple. Fitting the
  dependencies pHn for n 0, 1, , 200 with
  MathCad (as example) will be obtain a graphic in
  form

Numerically fitted data in simulated titration
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Conclusions and remarks

• The titration process is simple in appearance but
  shows to be complex in details. Even the simple
  case of titration of monobasic base ammonia with
  a monoprotic acid, such as acetic acid, hangs up
  unexpected difficulties in simulation process.
  The difficulties exists especially because is no
  approximations in model. Only one (so called)
  approximation is sequent adding of titrate, that
  are also perfectly possible in practice.
• The method permits to investigate more complex
  processes such as titration of polyprotic acids
  and polybasic bases. If no approximation will be
  made, more complex equations will be necessary
  and superior rank equations will appear to be
  solved.

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CORRELATIONS AND REGRESSIONS WITH
MATHLAB(Lorentz JÄNTSCHI, Mihaela UNGURESAN)

• Abstract
• A MATHLab computer program is implemented. The
  program presented demonstrates the power of
  MATHLab in working with statistical processing,
  and can be considered as an example for future
  applications.
• The program draws an exportable figure of fitted
  data and regression curve and calculates the
  correlation coefficient r and sum of residues.

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Specifications

• The execution of the program makes the posting
  of a window like in the figures attached, in
  which we can modify the expression of the
  function (y P(x)) or the form of posting (box
  or grid).
• For exemplifying lets consider a correlation
  study of efficiency time executing of numerical
  methods that are using a Runge-Kuta-Niström
  predictor corrector

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There is an automatic calculus for every defined
function the values of the correlation between Y
P(X) and X and the value of residues sum given
by the formula
In first figure are presented fitted dates from
table with a parabolic regression and program
makes the corresponding correlations and sums of
residues. In second figure, in same execution,
with same dates from table are fitted with a
linear regression model and program makes rebuild
corresponding correlations and sums of residues.
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Linear regression demo of the program
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Parabolic regression demo of the program