1

THERMODYNAMIC FORM OF KINETIC EQUATIONS AND AN EXPERIENCE

OF ITS USE FOR ANALYZING COMPLEX REACTION SCHEMES

Valentin N. Parmon

Boreskov Institute of CatalysisNovosibirsk State University

Novosibirsk, Russia, 630090, parmon@catalysis.ru

May 30, 2012Ghent, Belgium

2

Novosibirsk is the 3rd largest city in Russia (behind Moscow and St-Petersburg)

Population >1,500,000Universities and academies 30

Great logistic center (Trans-Siberian railway, International airport)High-tech industriesThe highest density of science in Russia

Novosibirsk

Moscow

St-Petersburg

Russia

Siberian Branchof the Russian Academy of

Sciences

SiberiaSiberia

Novosibisrk Scientific Center – Akademgorodok

•Population 130,000

•35 academic research institutes of SB RAS with ca. 9,000 employees

•7 chemical research institutes

•Novosibirsk State University

3

The Siberian school of mathematicians in chemistry (since the beginning of 60ths)

M.G. Slin’ko (1914–2008) – initiator of the wide application of mathematical methods in catalysis

G.S. Yablonsky – generalization of analysis of complex reaction schemes

A.N. Gorban’ – coupling of kinetic analysis with thermodynamics

V.I. Bykov – analysis of reaction schemes with singularitues

M.Z. Lazman

4

G.S. Yablonsky et al. (1970s-2000s)

• Rigorous results based on assumed detailed reaction mechanisms with the ideal mass-action-law dependences

(1) Linear Theory (1970s-1980s)

(2) Non-Linear Theory (1980s-2000s)

5

Linear theory• ONE-ROUTE CATALYTIC REACTION with the linear mechanism

General expression for the steady state reaction rate (Yablonsky, Bykov, 1976)

where Cy is a “cyclic characteristics”

Cy = K+f+(C) – K–f–(C)

Cy corresponds to the overall reaction

presents a complexity of complex reaction

yR C

• MULTI-ROUTE LINEAR MECHANISMS (Yevstignejev, Yablonsky, Bykov, 1979)

ipj ji

j i

K c

6

Non-linear theory

KINETIC POLYNOMIAL(Lazman, Yablonsky, 1980-2000s)

It is considered as the most generalized form which includes Langmuir-Hinshelwood-Hougen-Watson equations and equations of enzyme kinetics as particular cases

The kinetic polynomial of the one-route non-linear reaction scheme is

BmRm +…+ B1R +B0Cy = 0 where R is the steady-state reaction rate, m are the integer numbers

7

V.N. Parmon is a lecturer of Novosibirsk State University in chemical kinetics and both classical (equilibrium) and, since 1995, non-equilibrium thermodynamics

8

Contents of the presentation:

• Introducing the thermodynamic form of kinetic equations

• Some interesting one consequences

• A problem of the “bottle neck” (limiting step) and “rate controlling step” of a stepwise reaction

• Few practical application

9

A Thermodynamic Form of Kinetic Equations is an Inevitable Step for the Successive Unifying

the Languages of Chemical Kinetics and Chemical Thermodynamics

Chemical Kinetics: the main parameters are concentrations, c, of reactants A and rate constants, ki,

v ff ]αα i α i α

dc= = k,c k,[A

dt

Chemical Thermodynamics: the main parameters are chemical potentials, , of thermalized reactants A

α=fNote, however, that

o RTln c α =

where is an activity coefficient

10

Introducing the Thermodynamic Form of Kinetic Equations

01c [A ] exp

RT

iji jA A

For a substance A

For an elementary reaction “ij”

i

i

0

ij ii ij j

1v k c k exp

RT

i

iij ij

0i i

ij iexp nRT

1k exp exp

RT RT

where is a “chemical potential” of reaction group ii i

i

i i

0 00 0ij

iii B

j

0ij

ii

B

j

G Gk T

Gk Texp

h R

1 1k exp exp exp

RT h RT

T

RT

11

Properties of the Thermodynamic Form of Equations for an Elementary Reversible Reaction

ij

i jjiA A

ij ij ij i ji jij

ij ij i jv v n nd

v n ndt

ST

i i j j

If ij is the chemical variable for reaction “ij”

since ij = ji !

Indeed, for partial equilibrium of reaction ij

when orji

i jexp n exp nRT RT

ijv 0

thus ij = ji

12

New proprietary definitions:

iin exp

RT

0ijB

ij ji

k Texp

h RT

– thermodynamic “rush” of reaction group i

Why “rush” ?

– “truncated” rate constant of reaction ij which depends only on the properties of TS

ij i jv 0 if n n

TSii A

jj A

ij i jv 0 if n n

ij i jv 0 if n n

13

Some Related Properties

Inequality

is equivalent to the positive value of affinity Arij of reaction ij :

at

is equivalent to

Direction of reaction ij coincides with the sign of Arij !

i jn n

i j

jii jn exp n exp

RT RT

i jn n

i jn n

rijA 0

rijA 0

i j i j rijA 0

is equivalent to

14

So, the rate of reverse reaction is negligible in respect to

Thermodynamic Criterium of Kinetic Irreversibility of a Reaction i j

j i ji iij ij ij

rijij i

v exp exp exp 1RT RT RT RT

An 1

RT

ijvT

ijv

In this case double can be substituted by single " " " "

rijI f A RT

j i

€

(far from equilibrium)

rijI f A RT (the vicinity to equilibrium) then

ij irij rij

nv A

RT

15

Consequence 1

rAm !

RT

For a set of stationary consecutive reactions which occur from left to right

•obligatory

•the number of single arrows can not exceed the value of" "

Here Ar R – P is affinity of stoichiometric stepwise reaction R P

1 2 N N 1

1 2 NR Y Y ... Y P:

1 2 NR > Y > Y >...> Y >P!

16

Consequence 2

R P

dv n n

dt

For any stoichiometric stepwise reaction “”

which is linear in respect to its reaction intermediates Yi the rate is expressed in the same way as for elementary reaction:

where

R and P are initial and final reaction groups;

is an algebraic combination of ij and, in some cases, thermodynamic rushes of “external reactants” from either initial of final reactions groups

Note: Stoichiometric stepwise reaction means steady state occurrence of the reaction in respect to its intermediates Yi

RRn exp ,

RT

P

Pn expRT

R P

This relation in catalysis is known as a Horiuti–Boreskov equation!

17

Simple Example 1Stepwise reaction

occurs according to scheme

In the steady state in respect to Y

where

1R Y

R P

2Y P

1 2

d[Y]R Y Y P 0

dt

1 2

1 2

R PY

1 2

1 2 1 21

1 2 1 2

d[R] d[P]R Y Y

dv

dt

R

Pdt dt

R PR P PR

1 2

1 2

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Simple Example 2

1

1R Y

Stepwise reaction

occurs according to scheme

In the steady state in respect to Y

1 2R R P

2

2Y R P

where

1 1 2 2

d[Y]R Y Y R P 0

dt

1 1 2

1 2 2

R PY

R

1 1 22 2 2

1 2 2

1 21 2

1 2 2

1 2

R Pd[P]Y R P P

dt R

R R

dv

dt

PR

R R P

1 2

1 2 2R

19

For an arbitrary set of monomolecular transformations of intermediates there is a total

analogy with an electrotechnical equivalent scheme!

is calculated in the same way as :

is an algebraic combination of ij like R is that of Rij

1R

Stepwise process occurs according to scheme

Electrotechnical analog

ij ij i jv Y Y

i

iY ij i j

j

d[Y]v Y Y 0

dt

v R P

ij i jij

1I U U

R

i i jj ij

1I U U 0

R

R P

1I U U

R

R P

iR {Y} P

20

Main basis for “linear” non-equilibrium

thermodynamicsFlux Ji of a parameter ai

where Xi is thermodynamic driving force for ai

ii i

daJ X ,

dt

j ij ij

J L X ,For a complex system

where Lij are the Onsager’ coefficients of interrelation.

A sequence: existence of the Raleigh-Onsager dissipation function

ii i ij i j

i i j

dST J X L X X 0

dt P

According to the Prigogine theorem, P is the Lyapunov’ function which reaches a positively defined minimum at the stationary state of the system (when Jj = 0)

21

Consequence 3:

Existence of the Lyapunov’ functions which are positively determined and minimazing at the steady state in respect to intermediates even far from equilibrium for any reaction schemes which are linear in respect to intermediates

Example 1: Stepwise reaction occurs via the scheme

2 2 2

Ri i ij i j jR ji i j j

1R Y Y Y Y P

2

i

iY Ri i ij i j iP i

ji

d[Y] 1v R Y Y Y Y P 0

dt 2 Y

R P

iR {Y} P

where {Yi} means an arbitrary set of monomolecular transformations of Yi

22

Physical meaning of the Lyapunov’ function for an arbitrary set of monomolecular reactions

far from equilibria

Stepwise process occurs according to scheme

Electrotechnical analog

ij ij i jv Y Y

i

iY ij i j

j

d[Y]v Y Y

dt

v R P

ij i jij

1I U U

R

i i jj ij

1I U U

R

R P

1I U U

R

Thus, the Lyapunov’ function

corresponds to the power W of the dissipation of Ohmic heat in the electrical circuit

R P

iR {Y} P

2ij i ji j

Y Y

2ij i j i ji j i j ij

1W I U U U U

R

23

Example 2: Stepwise reaction occurs via the scheme

2 22

1 1 2

2

R Y Y R PR

Y 1 1 2 2

d[Y] 1v R Y Y R P 0

dt 2 Y

Conclusions:

1. The Lyapunov function exists for any stepwise reactions which are linear in respect to intermediates

2. Steady state of above reaction is stable

1 2R R P

1

1R Y

2

2Y R P

24

Consequence 4: According to the Prigogine theorem all systems near thermodynamic equilibrium have the stable steady state. All stepwise reactions linear in respect to intermediates have their Lyapunov’ functions and thus are also stable

A contrary example: Stepwise reaction

occurs via the nonlinear autocatalytic scheme in respect to Y:

The steady state in respect to Y can be nonstable !

Thus, the necessary conditions for loosing the stability of the steady state of a kinetic scheme: (1) As least one elementary reaction has to be kinetically irreversible(2) This elementary reaction has to be non-linear in respect to the intermediates

1R Y 2 Y

R P

2Y P

2 2 22

1 1

0, Rd[P]

v Ydt R , R

any

The Lyapunov function does not exist !There are two steady states

25

Necessary conditions for oscillation of the concentration of reaction intermediates:

• far from equilibrium (at least one single in the reaction scheme)

• at least two reaction intermediates

• at least one step which is nonlinear in respect to intermediates

" "

26

Consequence 5: It is possible to write modified Onsager’ (the Horiuti-Boreskov-Onsager) equations of interrelation of parallel stepwise chemical reactions

A simple example:

Parallel step-wise reactions occur via mechanism

R YP1

P2

1

3

2

At the steady state

Thus,

1 2 1 3 2

d[Y]R Y Y P Y P 0

dt

11 = 12/(1 + 2 + 3) > 0

22 = 13/(1 + 2 + 3) > 0

12 = 21 = -23/(1 + 2 + 3)

1

1R P 2

2R P

1111 1 12 22

d[P ]R PP R P

dtY

2212 1 22 23

d[P ]R PP R P

dtY

where

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In a general case for parallel stepwise reactions

Here j ii R i PR exp RT , P exp RT

ii ij

ji j jj j

j

Rd

Xdt

Pv

Note: ij is not obviously symmetrical in respect to indexes i and j as it is the case for reprocisity coefficients Lij in the classic Onsager equations in the vicinity of equilibrium

i

j iR P

ii > 0

i ij jj

J L X

28

An unambigous interpretation of the notion “rate determining” (rate controlling) step by IUPAC

Rate controlling factor

lmij k ,l i,m j

lnvCF

lnk

0r ij

ij ji

Gk k exp

RT

but

In the thermodynamic representation ij ij i jv n n

ofi jB

ij

Gk Texp

h RT

– contains parameters of only the transient states

iiin exp exp

RT RT

– contains parameters of only the thermalized

reactions groups and reactants

Thus, ij iv f ,n

lmij ,l i,m j

lnvCF

ln

– rate controlling factor of transient states

,

lnvCF

ln

– rate controlling factor of the reactant

The problem of “rate controlling” (“rate determining”) step and “rate limiting” step

(“bottle neck” of the stepwise reaction)

29

How to define correctly the “rate limiting” step (the “bottle neck”)?

30

The “rate determining step” and “bottle neck” in a consequtive monomolecular reaction

1 2 3 n 1

1 2 nR Y Y ... Y P

d d[R] d[P]v R P

dt dt dt

wheren 1

i 1 i lim

1 1 1

lim imin

So rate-determining step is the step with minimal i

Note:

in the steady state for i = 0,…, n+1 i i i 1v Y Y

Thus i i 1i

vY Y

For i = lim the value of is maximal ! i i 1Y Y

It means that the “bottle neck” (limiting step) is the step with the

maximum drop of !i i 1Y Y

31

Application to catalytic reactionsA simple example:

R P

occurs via catalytic Michaelis–Menten scheme1

1R K K 2

1K K P

where K and K1 are free catalytically active site and the catalytic intermediate

The balance equation

can be rewritten:

1 0[K] [K ] [K]

1

00 0KK K

1 0Kexp K exp K expRT RT RT

Here 1

0 0K K K K KRTln 1 RTln

1 1 1

0K K KRTln

Thus

1 0K K K

1

1

0 0K K

K expRT

where and corresponds to at 0K K K 1

32

Finally, at the steady state in respect to K1

1

0 01 2

K1 2 1 2

K1 2

K R P K R Pdv

dt R P11

• At small extent of occupation of the active site with catalytic intermediates

1K K 1

1 20

1 2

v K R P

and does not depend on standard thermodynamic parameters of K1 1

0K

• At large extent of occupation of the active site with catalytic

intermediates 1K K 1

1

0 0K K1 2

0

1 2

v K R P expRTR P

depends on1

0K

33

Note that for one can have the situation when the rate determining step does not coincide with the bottle neck !

1K K 1

1

0 0K K1 2

0

1 2

v K R P expRTR P

Let: 1 2R P R P and

In this situation

1

0 0K K

2 0v K R P expRT

independently on whether 1 2 1 2 ! or

But obligatory the “bottle neck” is the step with minimum i !

So, the rate determining step is always step 2

34

An unexpected conclusion:there are situations when the rate-limiting step can not be the rate-controlling step!

35

Conclusions

• The thermodynamic form of kinetic equations allows a dramatic simplification of analysis of complex reaction schemes

• Indeed, the main application of this approach is possibility to extend fruitful and systematic analysis of chemical reaction schemes for the area “far from equilibrium”

• Among few principal problems which are resolving via this approach this is a mathematically correct definition of the “bottle neck” (the limiting step) of a stepwise reaction and “rate determing step”

Unexpectedly, for some particular cases (e.g. for catalytic reactions) these steps can not coinside

36

Few examples of practical interest

37

An example of a practical application:Super low temperature of melting of active

component of operating metal catalysts due to their oversaturation with carbon

O.P.Krivoruchko, V.I.Zaikovskij, K.I.Zamaraev, Dokl.Akad.Nauk, v.329, 744 (1992) (in Russian)

300 A

The melting temperature is 500 °C (!) lower than that of the Fe–C eutectics

time (sec):

time (sec):

Electron microscopy “in situ” videotape of Fe–C fluidized particles migration over amorphous carbon support at 650 °C

38

Formation of Metastable Oversaturated Solutions of Carbon in Metals at Catalytic

Graphitization of Amorphous Carbon

V.N.Parmon. Catalysis Letters, 42, 195 (1996), O.P.Krivoruchko, V.I.Zaikovskij (1995)

Camorph Cgraphite G –12 kJ/mol ( >RT)

c(amorf) > c(in metal) > c(graphite)

Melting Temperatures, oC

Camorphsolution of C

in metal Cgraphite

met

1539 1145 640

1493 1320 600

1453 1318 670

metalpuremetal

equilibriumeutectics

with graphite

steadystate

Fe

Co

NiResult: steady-state concentration of xC in metal >> concentration of C in stable eutectics

If the rate determining step is formation of graphite from the melt:

xC (eq. with amorph. C) = xC (eq. with graphite)exp(–GR/RT) 4xC(eq. with graphite) 4xC (eq. eutectics)

Hm and To are the melting heat and melting temperature for pure metal

meut sC t s

0t

Hln 1 T T T 500 – 900

1 1C

RX

T!

T

39

Metastable Phase Equilibria for Fe–C Systems during Occurrence

of the Catalytic Reaction

1. Oversaturation: the Schröder equation

Tx = TomH/{mH – RToln(1 – x)}

2. Metal particle size r:m

rm

2 VT T exp

r H

3. Size r’ of a crystallization center (= size of the catalyst particle)

mr

2 Vx x exp

r RT

Parameters, influencing the melting temperature:

1600

800

1200

Solution of Fe in C

eutectics (T = 1420K, x = 0.173)

Content of C (mol. %)

T, K T

steady state (920K)Solution of C in Fe

2000 Graphite liquids

Schröder

Fe3C Fe2C

0 50 100

40

Formation of Filamentous Carbon together with Hydrogen at the Moderate-Temperature Catalytic

Pyrolysis of Methane and Low Hydrocarbons

The weight of the catalyst can be increased by a factor of 400 due to formation of carbonaceous filamentous material

The growth of the filament corresponds to diffusion of carbon through the active component with D>10–10 cm2/s

L.B.Avdeeva, V.A.Likholobov, G.G.Kuvshinov, at al. (1994)

500Ao

Ni-catalyst Ni/Cu-catalyst 1000 A

o

catalysisn m 2450 650 CC H C 2H

41

Size effects in catalysis over metal nanoparticles

0 2 4 6 8 10

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

TO

F (

s–1)

<d>, nm

TO

F (

s–1)

10–4

<d>, nm

Conclusion: There may occur size effects in catalytic reactions, which are many time increase in the activity of metal catalysts upon a decrease of the active component particles in size to several nanometers

Pt/Al2O3CН4 + 2 O2 CO2 + 2 Н2О430 °C

Au/Al2O3CО + O2 CO2400 °C

I. Beck, V.I. Bukhtiyarov, I.Yu. Pakharukov, V.I. Zaikovsky, V.V. Kriventson, V.N. Parmon, Journal of Catalysis 268 (2009) 60-67

42

Influence of the active component particle size on the catalytic activity

(an energy correlation approach)

V.N. Parmon, Doklady Physical Chemistry, vol. 413 (2007) 42-48

æ < 1 is the Brønsted-Polyany correlation coefficient

A B

Mechanism:

A + K K1 (1)

K1 B + K (2)

r

Σ

rΣ

(1 )Δv exp

RTΣr Δ

v expRT

d[B]TOF vdt æ

æat low coverage with K1

at large coverage with K1

Result:Result:

The increment of chemical potential of a nanoparticle of radius rThe increment of chemical potential of a nanoparticle of radius r

= æ r

r =r =2 V

r2 V

rHere – surface excess energy, V – molar volume of the catalyst active phase,

Here – surface excess energy, V – molar volume of the catalyst active phase,

TS1 TS2

43

Correlation of the measured TOF values for the complete CH4 oxidation over Pt/Al2O3

with the Pt size

at temperature 700 K and of the apparent activation energies Ea with the reciprocal to the active component size (diameter) d

at temperature 700 K and of the apparent activation energies Ea with the reciprocal to the active component size (diameter) d

I. Beck, V.I. Bukhtiyarov, I.Yu. Pakharukov, V.I. Zaikovsky, V.V. Kriventson, V.N. Parmon, Journal of Catalysis 268 (2009) 60-67

For both lines the correlation coefficient is the same: æ 0.75

0,0 0,4 0,5 0,6 0,70,30,20,1 0,0

0,0 0,4 0,5 0,6 0,70,30,20,1 0,0

–1,0

0,0

–2,6

–2,2

–1,8

–1,4

100

0

20

40

60

80

140

120

1/d, nm–1

log

(TO

F)

(TO

F in

s–1)

Ea,

kJ/

mol

lg (TOF) = 3,304 (1/d) – 2,981

44

Thermodynamic conjugation of parallel chemical transformations via a common catalytic

intermediate

CA

BA

C

B

K

K

{X} – catalytic intermediates

Conclusion: To change selectivity of a catalytic process one has to generate some thermodynamic driving forces

The Horiuti-Boreskov-Onsager coupling equations:

V.N. Parmon, Thermodynamics of Non-Equilibrium Processes, Elsevier, 2010

Reaction coordinate

C*

B

C

{X}

A

C

B

X

A

kij – formal rate constantskB and KC – equilibrium constants

kij – formal rate constantskB and KC – equilibrium constants

BB BCB C

d[B] [B] [C]k [A] 1 k [A] 1

dt K [A] K [A]

CB CCB C

d[C] [B] [C]k [A] 1 k [A] 1

dt K [A] K [A]

45

Thermodynamic control of selectivity at the decomposition of methanol

2 CH3OH CH3OOCH + 2 H2 (I)

CH3OH CO + 2 H2 (II)

Two independent stepwise reactions – two channels of decomposition:

Conclusion: an increase in the partial pressure of CO has to result in improving the selectivity in respect to methylformate

2 Methanol

{X}

2 CO + 4 H2 (II)

Methylformate + 2 H2 (I)

Reaction coordinate

Gib

bs

ener

gy

2 Methanol

{X}2 CO + 4 H2 (II)

Methylformate + 2 H2 (I)

Reaction coordinate

Gib

bs

ener

gy

{X} – catalytic intermediate

46

An example of an important practical application:

Development of principally new one-step catalytic processes of direct insertion

of methane higher hydrocarbons

Usually: hydrocarbons main products

CH4 as a byproduct

Due to existence of Onsager’s interrelation, one can reverse the direction of the

process of CH4 formation

Now: hydrocarbons + CH4 heavier hydrocarbons

Examples:

Process “Bicyclar” CH4 + C3,C4 alkanes aromatics + 5 H

Process “Biforming” CH4 + linear C5+ aromatics + 5 H2

47

Putative one-stage processes for conversion of light paraffins CH4 and C3–C4 (methane and propane-butanes) to aromatic compounds

Observation: Aromatization of C2–C4 paraffins is accompanied by the methane co-production.

Reactions of light hydrocarbons T*, K

1. 6 CH4 C6H6 + 9 H2 1630

2. 2 C3H8 C6H6 + 5 C2H6 760

3. 2 n-C4H10 p-C6H4(CH3)2 + 5 H2 800

4. C3H8 + n-C4H10 C6H5CH3 + 5 H2 710

5. 3 C2H6 C6H6 + 5 H2 930

6. CH4 + 2 C3H8 C6H5CH3 + 6 H2 880

7. CH4 + C3H8 + n-C4H10 p-C6H4(CH3) + 6 H2 1060

8. CH4 + C2H6 + C3H8 C6H6 + 6 H2 940

9. CH4 + 3 C3H8 C10H8 + 10 H2 830

48

Performance of the BICYCLAR process depending on the C1/C4 ratio

The coupled conversion of butane and methane allows the yield of aromatic hydrocarbons to be 2.5 times increased – up to 1.7 tonn per 1 tonn of involved C4

0

0.5

1.0

1.5

0 3 6 9 12 15 18

Molar ratio C1/C4

Yie

ld o

f ar

om

atic

hyd

roca

rbo

ns,

t/t

C4

CH4 + 2 C3H8 C6H5CH3 + 6 H2

CH4 + 3 C3H8 C10H8 + 10 H2

CH4 + 2 C3H8 C6H5CH3 + 6 H2

CH4 + 3 C3H8 C10H8 + 10 H2

Catalyst Zn-ZSM-5, temperature 550 °CCatalyst Zn-ZSM-5, temperature 550 °C

G.V. Echevsky, E.G. Kodenev, O.V. Kikhtyanin, V.N. Parmon, Appl. Catal. A: General 258 (2004) 159-171

49

An example of a practical application:

V. Parmon, Doklady Phys. Chem., 377, 4 (2001) 510-515

Natural selection in simple autocatalytic systems at diminishing the concentration of food R follows in one-directional progressive evolution of the system

An extremely important conclusion: existence of a prototype of biological memory in the absence of RNA or DNA !

At diminishing the concentration of food R, one proceeds a consecutive and irreversible (due to disappearance of seeds) “death” of all autocatalysts with the larger values of

Thus, a one-directional and progressive (toward diminishing the parameter Rcri

) natural selection

takes place in the system. This is analogous to appearance of a prototype of biological memory

There are two steady states:i i

iR Y 2 Y

i

tii cr

i

(1)Y R R R

)i(2Y 0

icr ti iR

itiY P

iY

2)i(Y

1)1(Y1)

2(Y1)

3(Y

50

Thank you for your attention !