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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.