1

1

Lecture 4, 5

Physicochemical Principles

of Absorption and Distribution

of Compounds

2

Reaction kinetics

– rates and time-courses of chemical reactions

- mechanisms of chemical reactions

- processes related to reactions: absorption, dissolution, etc.

Chemical kinetics - system is far from equilibrium - reaction rates, concentrations of components, reaction yields change in a time-dependent manner

Catalyzed reactions

– change of reaction rates caused by alteration in the

reaction mechanism (equilibrium remains unchanged)

Distribution - transport (flow, diffusion, partitioning …) - chemical reactions

2

3

Chemical kinetics in pharmacy:

Living organisms – non-equilibrium thermodynamic systems

with a multitude of biochemical reactions, many of which are

catalyzed by enzymes

- chemical stability of drugs

- pharmacokinetics

4

Example of time-dependent drug concentration in organism

3

5

Chemical kinetics

1. Reaction kinetics – classification of chemical reactions,

kinetic equations

2. Theory of chemical kinetics

– effects of molecular mechanism

– effect of temperature on reaction rate

Basic classification of chemical reactions

- homogeneous reactions – in single phase – liquid, gas, …

- heterogeneous reactions – at phase boundaries, including

dissolution

- micro heterogeneous – in colloid systems

6

Rates of chemical reactions:

DCBA 32Reaction:

Instantaneous rate of substance conversion:

Consumption of reactants: dt

Rd

Formation of products: dt

Pd

),( BAR

),( DCP

dt

Bd

dt

Ad

dt

Cd

dt

Dd.

2

1.

3

1

Rate of conversion of each component canbe different:

Extent of chemical reaction:

i

i

i

i0i

ν

Δn

ν

nnξ

[mol]

4

7

Rate of chemical reaction u:

1 1 1 1i i

i i

dn dcdu

V dt V dt dt

dt

dn

dt

du i

i

1 1mol s

Rate of chemical reaction in solution (volume V) expressed via concentration:

3 1mol dm s 1 i

i

dcu

dt

8

...... PASimple reaction:

dt

dc

dt

dcu AA

A

.1

reactant A: 1A

product P: 1Pdt

dc

dt

dcu PP

P

.1

dt

dc

dt

dc PA

Reaction rate is expressed as increase inconcentrations of reactants or products

5

9

Reaction in gas phase:

dt

dpu A

dt

dpu P

pA, pP – partial pressures

1Pa s

Rate of a heterogeneous chemical reaction on the surface (A) (corresponding to reagents in the volume V):

A

nii Surface density: [mol.m-2]

1 i

i

du

dt

[mol.m-2.s-1]Reaction rate:

10

Kinetics of homogeneous reactions

Law of mass actions (C.M. Guldberg and P. Waage, 1867)

.....ba

BAku

Experimental observation: rate of chemical reaction can be expressed as:

is rate constant

rRpPbBaA

... rate of a reaction is proportional to concentrations of reagents (i.e. starting material) at constant temperature

chemical equilibrium

1 3 3 1( a b ) ( a b )k mol dm s k

6

11

a b

u k A B ...

Rate of the chemical reactions:

rRpPbBaA u

rRpPbBaAu

bidirectional reaction

p r

u k P R ...

At equilibrium: uu

a b p r

A B P Rk c c k c c

p r

P R

a b

A B

c ck

c ck

eq

kK

kEquilibrium constant of reaction:

Rate constant & equilibrium constant:

12

C. M. Guldberg and P. Waage, 1867

At equilibrium, the ratio of concentrations of the reaction products

to the power of their stoichiometric coefficients and concentrations

of the reagents to the power of their stoichiometric coefficients

remains constants, independently on the initial composition of the

reaction mixture.

p r

P R

a beq

A B

c ckK

c ck

7

13

Order and molecularity of reaction:

rRpPbBaA u

General rate equation:A B

u k c c

a+b molecularity of reaction (a = A, b = B)

+ order of reaction (also non-integer order)

uu

kk

uu

Reactions: zero order0

Au k c k

first orderA

u k c

second orderA B

u k c c 2

Au k c

third orderA B D

u k c c c 2

A Bu k c c

(in the simplest cases are the exponents , identical with the stochiometric coefficients of reactants A, B)

14

Reaction kinetics of zero order:

katalys.A ... P ... kdt

dcu A

kdtdcA

tc

c

A dtkdcA

A 00

ktcc AA 0

ktcc AA 0

AAP ccc 0Concentration of produkt:

Concentration of reactant:

ktcP

3 1[ mol dm s ]

Reaction rate is independent of the concentration of A:

8

15

Kinetics of first order reactions:

... PAA

A kcdt

dcu

kdtc

dc

A

A

tc

c A

A dtkc

dcA

A 00

ktc

c

A

A 0

ln

kt

AA ecc .0 ktcc AA )(lnln 0

AAP ccc 0 01 A

kt

P cec Concentration of product:

1

A Au k c k c

= concentration ofreactant

16

Reaction rate:0

kt

A Au k c k c e

Half-life of reaction t1/2 – time required for half of the initial amount of A to be converted to P 2

0AA

cc

0

kt

A Ac c e

1 20

02/

ktA

A

cc e

kt

2ln2/1 Half-life:

9

17

Kinetics of second order reactions:

BA ccku ..

... PBA

dt

dc

dt

dcu BA If cA=cB=c 2kc

dt

dc

tc

c

dtkc

dc

0

2

0

ktcc

0

11

ktcc

0

11

0

0

1 ktc

cc

...2 PA

18

Half-life of second order reaction:

Half-life of reaction t1/2 – time needed for half of the initial amount

of A to be converted to P

2

0cc

ktcc

0

11

2/1

00

12kt

cc

0

2/1

1

kct

10

19

BA ccku ..If cA cB:

cA0, cB0 – initial concentrations, x – concentration of molecules converted to P at time t

xcc AA 0

xcc BB 0

dt

dx

dt

dcA

BAA cck

dt

dcu ..

xcxckdt

dxBA 00 .

ktcc

cc

cc AB

BA

AB

0

0

00

ln1

tcckcc

ccAB

AB

BA00

0

0ln

t1/2 is different for A and B

... PBA

20

0th order:

1st order:

2nd order:

ktcc AA 0

ktcc AA )(lnln 0

ktcc

0

11

kdt

dcA

kcku A 0.

Acku .

AA kc

dt

dc

2. Acku 2kc

dt

dc

13. sdmmol

1s

131 sdmmol

11

21

Summary:

Reactions: Zero order kcku A 0.

First order AA ckcku .. 1

Second order BA ccku .. 2. Acku

Third orderDBA cccku ...

BA ccku .. 2

22

Determination of order of reaction:

nkcdt

dcu

cnku logloglog

General shape of kinetic equation:

log c

log

u

log k

ntg

12

23

24

Effect of temperature on reaction rates – Arrhenius equation

Reaction rates and rate constants of chemical reactions depend on temperature

Generally, reaction rates increase with growing temperature, but not always:

Arrhenius equation:

RT

EAk aexp.

Ea – activation energy of reaction, typically 10–200 kJ.mol-1

A – pre-exponential factor (units –same as k, frequency of molecular collisions)

13

25

RT

EAk a lnln

Ea , A – Arrhenius parameters

26

Activation energy:

Large Ea means, rate constant will depend strongly

on temperature.

Ea = 0 – rate of chem. reaction is independent of

temperature

Ea < 0 – with increasing temperature the rate of

reaction drops (reaction

mechanism is complicated)

Activation energy represents the minimumkinetic energy, which the reactants should have so that products will be created

14

27

Kinetics of complex reactions:

In general, they do not obey a kinetics equation of integer order

Types of complex reactions:

- reversible (bi-directional) reactions

- parallel (side) reactions

- consecutive reactions

- chain (radical) reactions

- autocatalytic reactions

28

.....ba

BAku

Rate of chemical reactions :

rRpPbBaA kk

,

Reversible reactions:

.....rp

RPku

At equilibrium: uu

r

R

p

P

b

B

a

A cckcck ....

b

B

a

A

r

R

p

P

cc

cc

k

k

.

.

k

kKeq

Equilibrium constant of reaction:

15

29

Parallel reaction – share common reagent

OHHC 52

OHHCk

2421

232 HCHOCH

k

APPP uk

,

RRR uk ,

APP cku

ARR cku R

P

R

P

u

u

c

c

R

P

R

P

k

k

c

c

Total rate of depletion of A: RPA uu

dt

dcu

Total increase of products:

RP ccx

ARPARAPA ckkckck

dt

dc

RP kkk AA kc

dt

dc

kt

AA ecc .0 kt

A ecx 10

k

kxc P

P .k

kxc R

R .

E.g. 1st order:

30

Consecutive reactions: ZXAkk 21

AA

A ckdt

dcu 1 tk

AA ecc 1.0

Depletion/decay of A:

Production of X:

XAX

X ckckdt

dcu 21 tktk

AX eekk

kcc 21

12

10

intermediate X is produced as fast as A decays (= k1cA) and converts to Z with the rate = k2cX

Production of Z:

0AZXA cccc XAAZ cccc 0

tktkAZ ekek

kk

cc 21 11 12

12

0

16

31

Time-dependent concentrations of A, X a Z:

Fig. 10.1, p. 409

32

Limiting cases of k1 and k2:

:21 kk

A is rapidly converted to X, which slowly yields Z

tktkAZ ekek

kk

cc 21 11 12

12

0

tk

AZ ecc 210

Kinetics of the whole process is determined by the rate constant of the slow reaction (X Z)

ZXAkk 21

17

33

:12 kk First reaction is slow: concentration of X is low and it instantaneously decomposes to Z – steady state

ZXAkk 21

tktk

AX eek

kcc 21

2

10

tktk

AX eekk

kcc 21

12

10

2

10

k

kcc AX

tk

Az ecc 110

Z Z

X

Limiting cases of k1 and k2:

34

Catalysis

Berzelius (1835): Catalysts are compounds which cause by its presence chemical reactions, which otherwise would not proceed

W. Ostwald: Catalyst is a compound which affects the rate of chemical reaction but does not appear among the final products

18

35

Catalysis

Catalyst is a compound which affects the mechanism of the reaction but is not consumed during the reaction

Positive catalysis – reaction rate increasesNegative catalysis – inhibition – reaction rate decreases

Selective catalysis – only one out of several simultaneous reactionsis catalyzed

Autocatalysis – product of the reaction acts as an catalyst

36

RT

Ea

Aek

ART

Ek a lnln

Catalyzed reaction:

noncatcat EE

RT

E

RT

EE

RT

E

RT

E

cat ee

eA

eA

k

k cat

cat

*

.

.

noncatcat kk

19

37

Enzyme catalyzed reactions

saccharose glucose + fructose-fructofuranosidase

H2O

If the initial concentration of reagent (substrate) is held constant

and the concentration of enzyme [E] varies, then the dependence of initial rate of the reaction upon the enzyme concentration is linear

If the concentration of enzyme is held constant and the substrate concentration varies, then the dependence of initial rate of reaction on the substrate concentration [S] is hyperbolic

38

Michaelis – Menten kinetics:

PEESSEkkk

211 ,

0211 ESkESkSEk

dt

ESd

ESEE 0

Total concentration of enzyme [E]0

Free enzyme [E]

02101 ESkkSESEk

Skkk

SEkES

121

01

Complex [ES]

20

39

Rate of product formation:

ESkdt

Pdu 2

Skkk

SEk

Skkk

SEkkESku

121

02

121

0122

/.

SK

SVu

m maxCommon form of the rate eq.:

where 02max EkV 1

21

k

kkKm

Michaelis

constant

40

Limiting cases:

If: mKS

SK

SVu

m max

then

02max EkVu

If: mKS

0th order

then

mm K

SEk

K

SVu 02max

SK

SVu

m max

SK

S

V

u

m

max

1st order

21

41

Rate of enzyme catalyzed reaction depends:

-concentration of enzyme

-concentration of substrate

-conditions (temperature, pH, ionic strength, etc..)

-presence of compounds that affect the rate

Effect of temperature:

42

Transport of matter

Transport properties of compounds – ability of molecules to transport mass, energy, (other property) from one spot to another

Flow of a property

- Rate of transport of property across an unit surface

mass flow: mass of substance, which passes through a plane of unit area in a unit of time

- Observation: flow is proportional to gradient of property along the direction of movement , mass flow (diffusion): thermodynamic driving force of diffusion:

N = No /V number of molecules in

unit volume (number density) [m-3]

z coordinate of movement of molecules

D = kT/f diffusion coefficient

f friction coefficient

Stokes–Einstein Fick 1st law of diffusion

][ 12 smkgJ m

dz

dNDJ m

z

),,( m

z

m

y

m

x

m JJJJ

0m

zJ 0dz

dN

TpTpTp x

c

c

RT

x

cRT

xF

,,,

ln

22

43

Diffusion

Fick first law of diffusion in the units of amount

of substance is proportional

to concentration gradient:

Diffusion is time-dependent process, which equalizes

inhomogeneous distribution of mass in given volume

Rate of change of concentration by diffusion in thin layer

Input per one s: output:

Concentration change in the layer volume

due to flow of substance:

dx

dcD

dx

dN

VNDJ

A

n

x

1][ 12 smmolJ n

AxJ n AlxJ n

AlV

dx

lxdc

dx

xdc

l

D

l

lxJxJ

Al

AlxJ

Al

AxJ

dt

xdc )()()()()()()(

2

2

2

2 )()()()(

)()(

dx

xcdD

dx

xcdl

l

Dl

dx

xdcxc

dx

d

dx

xdc

l

D

dt

xdc

differential: f(xo+Δx)=f(xo)+f’(xo).Δx

44

Diffusion

Fick second law of diffusion:

Including flow of substance, flow rate:

Time-dependent change of concentration in a layer of small thickness is

proportional to curvature (second derivative) of concentration in the

direction of the substance movement

Solution of diffusion equation:

- First order - time 1 initial condition:

- Second order - coordinate 2 boundary cond.:

- Separation of variables:

2

2

x

cD

t

c

l

x

x

cv

x

cD

t

c

2

2

v

xt 01 )0,( cxc

11 ),( ctxc 22 ),( ctxc

txtxc ),(

2

2 ),(),(

x

txcD

t

txc

)()()()(2

2

txx

Dtxt

2

2 )(

)(

1)(

)(

1

x

x

xt

t

tDconstant

23

45

Solution of general diffusion equation – two ordinary linear diff. eq.

initial condition

boundary cond.

Solution for time-dependent eq.:

Solution for one-dimensional case of volume dependent eq., special case Dirichlet bound. cond. and eigenvalue of const.

characteristic equation

a) Solution of ch. eq.

general solution:

0)()(

tDdt

td

0)0( c

0)()(

2

2

xdx

xd

11)( cx 22 )( cx

Ddtt

td

)(

)(CDtt )(ln Dtect 0)(

0)0( 0)2(

)sin()cos()( 21 xkxkx

0'' 02 r

0 ir 2,1

46

For D. boundary cond. we obtain:

then or and const.

Solution for assumes the shape:

b) Solution for the boundary problem: with bound. cond.

has the shape:

For bound. conditions:

and

c) Solution of characteristic eq.:

general solution:

0)0( 0)2(

1)0(0 k

)2sin()2(0 2 k

02 k n2 ,...3,2,1n

2

2

n

0

x

nkx

2sin)( 2 ,...3,2,1n

0 0''

0)0( 0)2( xkkx 21)(

1)0(0 k

22)2(0 k 02 k

0 2,1r

)sinh()cosh()( 21 xkxkx

24

47

For boundary cond.: we obtain:

and

since then

Final solution for will be:

Complete solution of diffusion eq.:

where is a joint constant dependent on and initial and boundary conditions, ….

0)0( 0)2(

121 )0sinh()0cosh()0(0 kkk 01 k

)2sinh()2(0 2 k 0)2sinh( 02 k

)(x

2

2

n ,...3,2,1n

x

nkx

2sin)(

t

nD

n exn

Btxtxc

2

2

2sin),(

,...3,2,1n

nB n

48

Example of solution of diffusion eq. – beaker with dissolved substance, at the time the substance forms a film on the bottom of the beaker, surface area of the bottom , amount of dissolved substance

Solution:

Dependence of concentration

on distance from the

bottom for various times

Unit less parameter:

Concentration at the bottom decreases with increasing time and increases for distances

A

A

x

0t

0t

Dt

x

/e

DtA

n)t,x(c 4

21

0

2

0n

x)t,x(c

t

2x

Dt

2x

Dt)t,(c 0

0x

25

4949

Transport of non-electrolytes across biological membrane:

Passive transport of matter across a membrane bilayer of thickness concentration on outside and inside

steady state

Boundary condit.:

Solution of diffusion equation:

- concentration in membrane decreases linearly

- Flow of matter across membrane is constant:

biologicalmembrane

lipidbilayer

passivetransp.

diffusion

activetransp.

carrier

lA

outside inside

.0 constcc AoAo .0 constcAi

0

t

cAv 0

2

2

2

2

dx

cd

x

cD AvAv

00

AAc)x(c

0 )lx(cA

l

xc)x(c

AA1

0

Aic

l

cD

dx

dcDJ A0

5050

Correction for passive transport of matter

Concentration on membrane surface is different than conc. in solution

Partitioning coefficient:

Rate of passive transport of matter across membrane by diffusion:

Active transport of matter mediated by carriers , equilibrium:

Dissociation constant

Total concentration of

carrier

Rate of active transport:

Rate of transport of matter is given by:

0Aoc Asc

As

Aor

c

cK 0

l

cDK

dx

dcDJ As

r

P

APPA

AP

PA

Pd c

ccK

APPPccc

0

PdA

PA

AP Kc

ccc

0

PdA

A

max

PdA

AP

r Kc

cJ

Kc

c

l

cDKJ

0

A

D rK PdK

26

5151

Material balance:

Consider cell to which active substance enters via diffusion and convection and is eliminated via chemical conversion (reaction of first order). Rate of change of concentration at the site is given by:

diffusion flow (convection) reaction

If rate constant is large, drops fast, if diffusion coef. is large is replenished by diffusion, if convection (flow rate ) is large (mixing), flow can have various effects, depending on the sign of and gradient

If convection is neglected and also reaction , then solution for assumes the shape:

Solution for with an elimination reaction:

B

Bc x

B

BBB kcx

cv

x

cD

t

c

2

2

kB

c D Bc

vv

Dt

x

/Be

DtA

nc 4

21

0

2

kt

B

kt

B

t*

Becdteckc 0

0k0v

Bc

*

Bc

numerical integration

Bc / x

52

Dissolution of a solid:

Heterogeneous process – rate of dissolving is given by the slowest step of the process (solvation of the surface, desorption, diffusion, reaction in the volume phase, ...)

Surface of a solid phase is covered by diffuse layer of solvent molecules

Rate of diffusion in the layer – 1st Fick’s law

Concentration of substance at the phase boundary(saturated solution), thickness of the diffuse layer Concentration of substance in bulk solution

Gradient:

Noyes-Whitney equation

solid phase

Surface layer

diffusion layer

volume phase

concentrationCto(x)

mixing of solvent

dx

dcDJ

dt

dn

S

n

x 1

sc

c

cc

dx

dc s

ccDS

dx

dcDS

dt

dn s

x

t=to

δ

c

cs

27

53

Amount of substance

rate constant of diffusion

diffusion coefficient, surface area of the solid, volume of the solid,thickness of diffusion layer

Solution of diffusion equation:

- dissolution at const. temperature

- constant surface area of solid

- perfect mixing of solution

- constant volume of solution

)cc(k)cc(V

DS

dt

dcsds

Vdcdn

dk

tkcc

cln

d

s

s

tk

s

decc

1

time-dependence of concentration in solution for various values of kd

D S V

.constD

.constS

.const

.constV

54

Dissolution of a substance with reaction:

- Rate constant of diffusion

- Rate constant of reaction

Rate of product P generation:

Solution

CBs = 2, kd = 2 no reaction

CBs = 2, kd = 2, kr = 1

CBs = 2, kd = 2, kr = 0,5

CBs = 2, kd = 2, kr = 0,2

)()()( lPlBsB rd kk

dissolutiondiffusion

reaction

V

DSk

d

rk

tk

BsrBrP deckck

dt

dc 1

Bs

d

rtk

Bs

d

rBsrP c

k

kec

k

ktckc d

Time-dependence of concentration of product P in solution for various kr

28

55

Thank you

5656

Literature

Literature:

• ATKINS, Peter W. – DE PAULA, Julio: Physical Chemistry: Thermodynamics, Structure, and Change, 10th Ed., Oxford University Press, Oxford, UK, 2014.

• BOROUJERDI, Mehdi: Pharmacokinetics and Toxicokinetics, CRC Press, Boca Raton, FL, U.S.A. 2015.

• DOSTÁLEK, Miroslav a kol.: Farmakokinetika, Grada, Praha, ČR, 2006.

• JAMBHEKAR, Sunil S. - BREEN, Philip J.: Basic Pharmacokinetics, 2nd Ed., Pharmaceutical Press, London, UK, 2012.

• KERNS, Edward H. - DI, Li: Drug-like Properties: Concepts, Structure Design and Methods, Elsevier, Burlington, MA, U.S.A., 2008.

• PATRICK, Graham L.: An Introduction to Medicinal Chemistry, 5th Ed., Oxford University Press, Oxford, UK, 2013.

• REMKO, Milan: Molekulové modelovanie. Princípy a aplikácie, Slovak Academic Press, Bratislava, SR, 2000.

• REMKO, Milan: Základy medicínskej a farmaceutickej chémie, Slovak Academic Press, Bratislava, SR, 2005.

• ZATHURECKÝ, Ladislav a kol.: Biofarmácia a farmakokinetika, Osveta, Martin, SR, 1989.