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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5708_Библиотеки_им_академика_М_И_Перельмана.pdf
247
Chemical Kinetics
Termolecular reaction
22
22
NO O NO+⎯ →⎯⎯⎯
9.3 Order of Reaction
The sum of the exponents of the concentration terms involved in the rate equa-
tion is known as order of reaction
aA bB
rkAB
ab
+=
=
[][]
product
Order of reaction a b.
It is known that not every molecular collision between reactant molecules leads
to reaction, but some fraction of them will. Certainly, increasing the total number
of molecular collisions will increase the number of successful collisions. Successful
collisions are defined as those that lead to chemical reaction.The dependences of
reaction rates on concentrations are called rate laws. Rate laws are expressions of
rates in terms of concentrations of reactants. Also note that rate laws can be in
differential forms or integrated forms. They are called differential- and integrated-
rate laws. Rate laws apply to homogeneous reactions in which all reactants, and
products are in one phase (solution).
Zero-Order Reaction
A reaction is of zero order when the rate of reaction is independent of the
concentration of materials. The rate of reaction is a constant. When the limiting
reactant is completely consumed, the reaction stops abruptly.
The zero-order rate law for the general reaction
AP⎯→⎯⎯⎯
is written as the equation
d[A]/dt k
If x is the concentration of the reactant changed after time t then the equation
becomes
dd
or d d
xt k
xkt
/
After integrating and solving, the equation for zero-order reaction can be written
as
xkt (5)
Plotting concentration versus t will give a straight line with slope -k.
If a is the initial concetration
At ttxa
05
2
.
,/
(6)
248
Practical Pharmaceutical Analytical Chemistry
then Equation (5) becomes a/2 kt
0.5
t
0.5
a/2k (7)
t
0.5
is known as half-life.
First-Order Reaction
A reaction is said to be first-order if the rate of reaction is proportional to the
first power of the concentration of the reaction. A general unimolecular reaction,
represented as follows,
AP⎯→⎯⎯⎯
where A is a reactant and P is a product, it is called first-order reaction. The rate is
proportional to the concentration of a single reactant raised to the first power.
The decrease in the concentration of A over time can be written as:
(8)
r
A
t
kA
A
A
kt
=
[]
=
[]
=
[]
[]
=
d
d
d
d
(9)
Equation (9) represents the differential form of the rate law. If a is the initial con-
centration of the reactant and (a - x) is concentration of the reactant after time
t, then the above expression can be written as
(10)
Rate of reaction d d
dd
xt a xbx
xt ka xbx
/
/
α−
()
=−
()
(11)
where k is the rate constant. After solving this equation, value for k can be
obtained as
k
t
a
ax
=
−
()
2 303
10
.
log
(12)
Plotting log of (a - x) against time creates a straight line with slope -k/2.303. The
plot should be linear up to about 90% conversion.
Half-life (t
½
): It is the time required to reduce the concentration of A by fraction
(i.e. one half) of its initial concentration. Half-life is a constant and is independ-
ent of the reactants. For example, if a x at t
½
and x a/2, then
(13)
k
t
a
aa
k
t
t
k
=
−
()
=
==
2 303
2
2 303
2
2 303
2
06
12
10
12
12
.
log
/
.
log
.
log
.
/
/
/
993
k
(14)
(15)
249
Chemical Kinetics
Second-order Reaction
The rate of a second-order reaction is proportional to either the concentration
of a reactant squared, or the product of concentrations of two reactants. For the
general case of a reaction between A and B, such that
AB P+⎯ →⎯⎯⎯ (16)
the rate of reaction will be given by
d
d
A
t
kA B
[]
=
[][]
1. When initial concentrations of the two reactants are equal, Equation (16)
can be written as:
d
d
A
t
kA
[]
=
[]
2
(17)
If a is the initial concentration of a reactant and (a - x) is the concentration of
the reactant after time t, then the above expressioncan be written as
(18)
Rate of reaction d d
dd
xt a x
xt ka x
/
/
α−
()
=−
()
2
2
(19)
where k is rate constant. After solving the equation, value for k can be obtained
as
k
t a
x
ax
=×
−
()
1
(20)
A plot of x/(a - x) versus t produces a straight line with slope k and intercept
1/[A]
0
. The plot should be linear up to about 50% conversion.
Half-life (t
½
)
(21)
k
x
ax
t
x
ax
t
a
aa
t a
k a
k a k a
=×
−
()
=×
−
()
=
−
()
1
1
12
2
1
12/
/
/
(22)
(23)
When one of the reactants is present in large excess, the second-order reaction
confirms to the first-order reaction and is known as a pseudounimolecular reaction,
e.g. hydrolysis of methyl acetate by acid.
250
Practical Pharmaceutical Analytical Chemistry
2. If the starting concentrations of the two reactants, [A] and [B] are different,
the variable x is used.
If the initial concentrations of the two reactants are equal, Equation (16) can be
written as:
−
[]
=
[][]
d
d
A
t
kA B
(24)
If a and b are the initial concentration of the reactants and (a - x) and (b - x) are
concentrations of the reactants after time t, then the above expression can be
written as
Rate of d d
dd
reaction xt axbx
xt ka xb x
/
/
α−
()
−
()
=−
()
−
()
(25)
where k is the rate constant. After solving the equation, value for k can be
obtained as
k
ta b
ba x
ab x
=
−
()
−
()
−
()
2 303
10
.
log
9.4 Factor Affecting Rate of Reaction
The following factors affect the reaction rate: concentrations of reactants, pH, sol-
vent, catalyst, light, pressure, temperature and surface area of solid reactants or catalyst.
Concentration
It is known that not every molecular collision between reactant molecules leads to
reaction, but some fraction of them do. Therefore, with increase in reactant con-
centrations, reaction rate increases because the number of molecules in any given
volume increases and increased number of effective molecular collisions takes place.
Pressure
For solids and liquids, the effect is negligible, because solids and liquids are essen-
tially incompressible, but if the reactants are gases, pressure will have an effect on
reaction rate. If the pressure is increased, volume of gases decreases according to
Boyle’s law; therefore the molecules will collide more often, which means that there
will be a greater number of successful collisions (leading to reaction) in a given
period of time. Therefore, the reaction rate is expected to increase with pressure.
Surface Area
With increase in the surface area, the reaction rate tends to increase as more
reactants will be in contact to each other or catalyst. With a larger surface area,
there will be more molecular collisions that could potentially lead to reaction.
Nature of Reactants
Some substances are by nature more reactive than others. Acid–base reactions,
formation of salts and exchange of ions are some examples of fast reactions.
251
Chemical Kinetics
Reactions in which large molecules are formed or break apart are usually slow.
Reactions breaking strong covalent bonds are also slow.
Temperature
Raising the temperature increases the reaction rate. Temperature affects the rate of
a chemical reaction in two ways. First effect is molecular velocity. Temperature is a
measure of the average kinetic energy of the molecules. The higher the tempera-
ture, the higher the average kinetic energy, therefore higher number of collisions.
Secondly, raising the temperature increases the activation energy. Every chem-
ical reaction has characteristic activation energy. Activation energy is the mini-
mum energy that the reacting molecules must bring into the collision in order
to react. If they collide with less than this amount of energy, they simply bounce
off of each other unchanged. If the average kinetic energy of the molecules is
small in comparison to the activation energy, most of the molecular collisions do
not lead to reaction. Collisions may be occurring frequently, but most of them
are ineffective, so the reaction rate is very low. If the system is heated, the average
kinetic energy of the molecules is increased. When the average kinetic energy
is comparable to the activation energy, a much greater fraction of the molecules
successfully react when they collide. The percentage of successful collisions usu-
ally does not reach 100%, even when the average kinetic energy far exceeds the
activation energy. This is because the molecules must collide in the correct ori-
entation in order to react. Quantitatively, the relationship between the rate—as
reaction proceeds—and its temperature is determined by the Arrhenius equation
kA
Ea/RT
=
−
e
(26)
where E
a
is the activation energy, R is 8.314 J/mol K, T is the absolute tempera-
ture in Kelvins, and A is the frequency factor. The Arrhenius equation is based
on the collision theory, which supposes that particles must collide with both the
correct orientation and sufficient kinetic energy if the reactants are to be con-
verted into products.
9.5 Theories of Rate of Reaction
1. Collision theory: According to this theory, not all collisions of reactants
results in formation of product. For successful colloision, the two mol-
ecules must collide with sufficient energy and in correct orientation. This
critical value of energy is called energy of activation and is denoted by E
a
.
Activation energy is defined as the minimum energy required by reactant
molecules to reach the threshold energy level.
2. Transition state theory: This theory was proposed by Eyring. The reactant mol-
ecules are in chemical equilibrium with the activated complex at the top
of the activation energy barrier A B Æ (AB)* Æ Product. According to
this theory, the reacting molecules must achieve a specific configuration
252
Practical Pharmaceutical Analytical Chemistry
before they can form the product. This specific configuration possessing a
definite energy is known as the transition state.
9.6 Determination of Order of Reaction
Several different methods can be employed to determine the order of reaction:
1. Use of characteristic equations
2. Half-life method
3. By variation in rate of reaction
4. Addition of excess of reactants
5. Graphical method
1. Use of characteristic equations: The values of initial concentration a and con-
centration after time t (a x) are determined for a number of values of t
and then substituted into the different integrated rate equations. The order
of reaction is given by the equation that gives a constant value for velocity
constant k.
2. Half-life method: By calculating the half-change time, the order of reaction
can be determined.
In general for nth order reaction
t
½
Į 1/a
n1
(27)
To determine the order of a reaction the half-change time t
½
is measured for
different concentrations. The experiment is started with two initial concen-
trations say a
1
and a
2
, respectively.
(t
½
)
1
/( t
½
)
2
(a
2
/a
1
)
n1
(28)
where (t
½
)
1
and (t
½
)
2
are the corresponding half-change times. If it is independ-
ent of the initial concentration, the reaction is of first order; if at
½
is constant,
the reaction is of second order; if a
2
t
½
is constant, the reaction is of third order.
3. By variation in rate of reaction: When more than one reactants are involved,
the order of reaction can be studied by changing the concentration of one
reactant keeping the concentration of other constant. For example,
dx/dt k[A]
a
[B]
b
(29)
At a certain concentration of A and B, determined ¨x
1
/¨t for a short interval of
time ¨t. Now double the concentration of A and keep concentration of B same,
and determine ¨x
2
/¨t.
(¨x
2
/¨t)/( ¨x
1
/¨t) (k [2A]
a
[B]
b
)/(k [A]
a
[B]
b
) 2
a
(30)
The value of a is the order of reaction with respect to A. By repeating the
experiment with respect to B, the order of reaction b with respect to B can be
determined. The overall order of reaction is the sum of a + b.
253
Chemical Kinetics
4. Addition of excess of reactants (Ostwald isolation method): All the reactants but
one are taken in excess and the order of reaction is determined. This gives
order of reaction with respect to that reactant. The process is repeated with
other reactants one by one, and the order of reaction is determined with
respect to each. The overall order of reaction will be the sum of all these
orders.
5. Graphical method: Graphical method is used to study the order of reac-
tion by the following ways:
(i) For zero-order reaction: The x versus t plot is straight line passing
through the origin with slope equal to -k.
(ii) For first-order reaction: The log of (a - x) versus t plot is straight line
with slope equal to -k/2.303.
(iii) For second-order reaction: The x/(a - x) versus t plot is straight line
passing through the origin with slope equal to ak.
(iv) In case of two reactants: The log of b(a - x)/a(b - x) versus t plot is
straight line passing through the origin. A plot of 1/(a - x ) versus t is
linear graph with slope k.
Summary
Order Units
of K
Integrated rate equation Half
life
(t½)
Characteristics
Zero mol/Ls
kt x a/2k
Concentration of the product changes
linearly with time
The x vs. t plot is straight line passing
through the origin with slope equal
to -k.
First L/s
k
t
a
ax
=
−
()
2 303
10
.
log
0.693/k
Time required to complete
any fraction of the reaction is
independent of initial concentration
The log of (
a - x) vs. t plot is straight
line with slope equal to -k/2.303
Second L/mol.s
k
t a
x
ax
=×
−
()
1
Or for two reactants
k
ta b
ba x
ab x
=
−
()
−
()
−
()
2 303
10
.
log
1/ka
Time required to complete
any fraction of the reaction
varies inversely with the initial
concentration
The x/(
a - x) vs. t plot is straight line
passing through the origin with slope
equal to ak
In case of two reactants, log of
b(
a - x)/a(b-x) vs. t plot is straight line
passing through the origin. A plot of
1/(
a - x) vs. t is linear graph with slope k
254
Practical Pharmaceutical Analytical Chemistry
Experiment 1 Determine the rate constant of the hydrolysis of methyl
acetate.
Requirements
Methyl acetate, 0.5 N HCl, 0.1 N NaOH, thermostat, conical flask, pipette, ther-
mometer, burette, phenolphthalein, methyl orange
Procedure
1. Prepare 0.5 N HCl by taking accurately measured 42.5 mL of conc. HCl in
a 100 mL volumetric flask and add water to make up the volume.
2. Standardize the 0.5 N HCl solution by sodium carbonate using methyl
orange as indicator.
3. Standardize the sodium hydroxide solution prepared by standardized HCl
solution using phenolphthalein as indicator.
4. Place 50 mL of 0.5 N HCl and 10 mL of methyl acetate separately in ther-
mostat for about 30 min.
5. Take a 100-mL conical flask containing 25 mL ice-cold water.
6. Mix 20 mL methyl acetate and 80 mL of 0.5 N HCl solution and imme-
diately pipette out 5 mL of the mixture into the conical flask containing
ice-cold water. Titrate against 0.1 N sodium hydroxide solution using phe-
nolphthalein as indicator.
7. Record the time as t
0
and note down the volume of 0.1 N sodium hydrox-
ide consumed as V
0.
8. After 10 min, again pipette out 5 mL of the mixture into the conical flask
containing ice-cold water and titrate against 0.1 N sodium hydroxide solu-
tion using phenolphthalein as indicator.
9. Record the time as t
10
and note down the volume of 0.1 N sodium hydrox-
ide consumed as V
10.
10. Take out 10 mL of reaction mixture and place in thermostat or water bath
at 60°C for 1 h.
11. Repeat the steps 8 and 9 after every 10 min till t
50
is recorded.
12. After 1 h of placing reaction mixture in thermostat or water bath at 60°C
pipette out 5 mL of the mixture and titrate with 0.1 N sodium hydroxide
solution using phenolphthalein as indicator. The line will be t
f
and vol-
ume of sodium hydroxide consumed is V
f
.
13. Record all the readings in the following table and calculate the velocity
and t
½
of the reaction.
Observations and calculations
Room temperature °C
k
t
VV
VV
t
=
−
−
∞
∞
2 303
10
0
.
log
255
Chemical Kinetics
Observations
S. no. Time
(min)
Volume of sample
withdrawn (mL)
Burette reading
(mL)
(V
f
V
t
)K
t
0
5 V
0
t
10
5 V
10
t
20
5 V
20
t
30
5 V
30
t
40
5 V
40
t
50
5 V
50
t
f
5
V
f
V
0
is the volume of NaOH consumed by reaction mixture at t
0
and will be due
to HCl as no hydrolysis of methyl acetate has taken place. V
f
is the total volume
consumed, which will be due to HCl and acetic acid due to hydrolysis of ethyl
acetate. V
f
- V
t
will give the amount of acetic acid produced due to hydrolysis
at any time interval during the course of hydrolysis.
Plot a graph between concentration on Y-axis and time on X-axis. A curved
plot will be obtained while plotting log (a - x) on Y-axis and time on X-axis. A
straight plot is obtained indicating that the rate of reaction depends on concen-
tration of reactants.
Calculate t
½
by equation t
½
0.693/k.
Result The hydrolysis of methyl acetate was studied under room tempera-
ture and it was observed that the hydrolysis of methyl acetate follows first-
order rate equation. The graph of log of V
- V
0
/V
- V
t
versus t was found
to be and t
½
was found to be .
Experiment 2 Determine the relative strengths of two acids by studying the
hydrolysis of methyl acetate.
Requirements
Methyl acetate, 0.1 N HCl, 0.1 N H
2
SO
4
, 0.1 N NaOH, thermostat, conical
flask, pipette, thermometer, burette, phenolphthalein, methyl orange
Procedure
1. Determine the rate constant of the hydrolysis of methyl acetate by 0.1 N
HCl at room temperature as described in Experiment 1.
256
Practical Pharmaceutical Analytical Chemistry
2. Determine the rate constant of the hydrolysis of methyl acetate by 0.1 N
H
2
SO
4
at room temperature as described in Experiment 1.
3. Calculate the rate constant k as k
1
for 0.1 N HCl and k
2
for 0.1 N H
2
SO
4
.
Observations and calculations
Room temperature = °C
Volume of HCl added = mL
Volume of H
2
SO
4
added = mL
Volume of methyl acetate used = mL
k
t
VV
VV
t
=
−
−
∞
∞
2 303
10
0
.
log
Observations
S. No. Time
(min)
Volume of sample
withdrawn (mL)
Burette reading
(mL)
(V
f
V
t
)K
1
K
2
0.1 N
HCl
0.1 N
H
2
SO
4
0.1 N
HCl
0.1 N
H
2
SO
4
1 t
0
5 V
0
2 t
10
5 V
10
3 t
20
5 V
20
4 t
30
5 V
30
5 t
40
5 V
40
6 t
50
5 V
50
7
t
f
5
V
f
The relative strength of acids is determined by the following equation:
Relative strength
Strength of acid
Srength of acid
===
1
2
1
2
1
2
α
α
k
k
Result The strength of 0.5 N HCl and 0.5 N H
2
SO
4
was studied under room
temperature and it was observed that the hydrolysis followed order
rate equation. It was found that acid is times stronger than
acid.
Experiment 3 To determine the rate law and the rate constant for the iodina-
tion of acetone.
Requirements
Acetone, iodine, H
2
SO
4
, sodium acetate, sodium thiosulphate, starch, thermostat,
conical flask, pipette, thermometer, burette