9.5 temperature-dependence of reaction rate -- arrhenius

§9.5 Temperature-dependence of reaction rate

-- Arrhenius equation

Extensive reading:

Levine, pp. 554-559 Section 17.8

Goals:

1. Describe the effect of temperature on reaction rate;

2. Activation energy: definition, measurement, estimation;

3. Fundamentals for higher level scientific researches

Concentration dependence of reaction rate: study at constant temperature!

Temperature dependence of reaction rate: study at constant concentration?

Differential rate law: r-c

Integrate rate law: c~t.

1

dcr k c

dt= − =

01ln

ck t

c=

Use k which is independent of c instead of r.

§9.5 Arrhenius equation

9.5.1 Types of rate-temperature curves

From the middle 19 century, people began to study the effect of temperature on the

reaction rate.

qualitative

T

k

T

k

T

k

T

k

T

k

Type I Type II Type III

Type IV: Type V


Ludwig Ferdinand Wilhelmy, a German

scientist who published the first quantitative

study in chemical kinetics.

1850, Wilhelmy studied the acid-catalyzed

conversion of a sucrose solution into a 1:1

mixture of fructose and glucose with

a polarimeter. He wrote a differential equation

to describe the reaction, integrated it, and used it

to interpret his experimental results. Wilhelmy

found that the reaction's rate was proportional to

the concentrations of sucrose and of acid present.

9.5.2 The first quantitative study


12 22 11'[C H O ]r k=

k’ is quite low.

k = 0.0043 s-1

It was found that for homogeneous

reaction, an important generalization is

that reaction rate doubles or triples for

every 10 degree increase in temperature.

10 2 ~ 3T

T

k

k

+ =

2

ln AB

d k

dT T= +

in which A and B are experimental / empirical

constants with their physical meaning unclear.

9.5.3 Empirical rules

(1) vant’ Hoff’s Law

Difference between Experimental

reports and Research paper

Semi-quantitative

quantitative

linearization

1884, vant’ Hoff’s equation:

2

ln r m

p

HK

T RT

=


lnd k

dT

2

1

T

In 1889, Arrhenius made theoretical

consideration on the hydrolysis of

sucrose, in which sucrose molecules

were surrounded by water, if all sucrose

molecules could react directly with

water, the reaction should completed

instantly. However, this is not the case.

(2) Arrhenius hypothesis Arrhenius postulated that only a small part of

sucrose molecules with higher energy

(activated molecules) can react with water and,

therefore, the reaction can only proceed at a

low rate.

By taking enough energy, the common sucrose

molecules can become activated molecules.

The energy needed for this conversion was

called activation energy.

[A revolutionary concept!]


Arrhenius, S.A. (1889). "Über die Dissociationswärme und den Einflusß der Temperatur auf den

Dissociationsgrad der Elektrolyte". Z. Phys. Chem. 4: 96–116; 4:226-248


12 22 11'[C H O ]r k= k = 0.0043 s-1

https://en.wikipedia.org/w/index.php?title=Z._Phys._Chem.&action=edit&redlink=1

Arrhenius extended the ideas of vant’ Hoff

and suggested a similar empirical equation.

Is the simplification reasonable?

Arrhenius equation2

d ln

d

aEk

T RT=

2

d lnB

d

k A

T T= + 2

d ln

d

aEk

T RT=

2

ln r m

p

HK

T RT

=

2

ln r m

p

UK

T RT

=

(2) Arrhenius hypothesis



2 d ln=

da

kE RT

T

Definition of activation energy—

experimental activation energy

If Ea is independent on temperature, integration of the equation

yields

A is the pre-exponential factor which has

the same unit as the rate constant.

2

d ln

d

aEk

T RT=

or

(2) Arrhenius hypothesis



ln = +lnaEk A

RT−

= exp aEk A

RT

−

9.5.4 Experimental measurement of activation energy

(1) Experimental measurement:

ART

Ek a lnln +−=

(1) Graphic method

(2) Calculation method

Graphic method:

to plot lnk against 1/T [Arrhenius plot], for the

reaction of Arrhenius type, a straight line may be

obtained, the slope of which equals –Ea/R


ClCOOCH3 + H2O → CO2 + CH3OH + H+ + Cl−

T / K 273.72 278.18 283.18 288.14

104 k / s-1 0.4209 0.7016 1.229 2.087

T / K 198.18 308.16 318.29

104 k / s-1 5.642 14.05 32.65

06.211

9.8515ln +−=T

k

R = 0.99992

T. T. Ching, S. C. Kwong and S. C. Kim, JACS, 2012, 134:

11388-11391

9.5.4 Experimental measurement activation energy

(1) Experimental measurement:


(2) Calculation method:

ART

Ek a lnln

1

1 +−=

ART

Ek a lnln

2

2 +−=

−−=

212

1 11ln

TTR

E

k

k aConstant

Is there any problem in this result?

9.5.5 Ea and energy change of reaction

Raa UUE −=+,

Paa UUE −=−,

reactant, product, activated molecule,

reaction path.

−+ −= ,, aa EEU

When Ea,->Ea,+, U < 0, the reaction is

exothermic.

UUUEE RPaa =−=− −+ ,,


principle of micro-reversibility

−+ −= ,, aa EEU

When Ea,-< Ea,+, U > 0, the reaction is a

endothermic one.

For a strong endothermic reaction, the

activation energy for backward reaction is very

small.

What about Ea, +?

Problems with this explanation:

(1) Do the molecules possess the same

energy?

(2) What kind of energy is the activation

energy?

9.5.5 Ea and energy change of reaction


9.5.6 Tolman’s definition of Ea

The minimum energy that the molecules

must absorb before the reaction can take

place is known as the activation energy.

EEEa −= *

Boltzmann distribution

According to Tolman, the activation energy

of elementary reaction is the difference

between the average energy of the activated

molecules and the average energy of total

molecules:

EEEa −= *


Does the Tolman activation energy

depend on temperature?

Activation energy for elementary reaction

Activation energies for overall reaction:

A combination of the activation energy of elementary reactions composing of the

overall reaction. apparent activation energy

9.5.7 Activation energy of a overall reaction?

1,aE

2,aE

3,aE 4,aE

5,aE6,aE

The activation energy of ammonia synthesis.


9.5.8 Theoretical evaluation of Ea:

The activation energy can be related to the

energy change of the reaction. The energy

change can be calculated using dissociation

energy of bond (NOT BOND ENERGY). To

do this, some empirical rules may be used:

1) Dissociation reaction:

Ea will not be less than and need not be

larger than the dissociation energy of the

bond, i.e., Ea = DCl-Cl

Cl−Cl → 2 Cl


2) Combination reaction of radicals

2CH3· → CH3−CH3

Ea = 0

3) Radicals react with molecules:

A + B−C →A−B + C

If the reaction is a exothermal one, Ea 5% DB-C;

4) Molecules react with molecules:

A−B + C−D →A−C + B−D

If the reaction is exothermal, Ea = 30% (DAB +

DCD)

Half-life of first-order reaction with

different activation energy

9.5.9 Ea on reaction rate

Ea /

kJmol-140 60 80 100 120

t1/2 210-5 s 0.066 s 5.6 h 11.6 d 68.7 y

Ea ranges between 40 ~ 400 kJ mol-1.

For first-order reaction, when Ea increases by

4 kJ mol-1, k decreases by 80%.

Ea < 80 kJ mol-1: fast reactions

Ea > 100 kJ mol-1 : slow reactions


Ea,1

T1

T2

Ea,2

9.5.10 Temperature on reaction rate

T2 > T1

What about the fraction of activated

molecule increases?

Problem: Can your find another way to

increase the fraction of activated molecules?

9.5.11 modification of Arrhenius equation

The Arrhenius plots for some reactions are

curved, which suggests that the activation

energy of these reactions is a function of

temperature. At this situation, the temperature

dependence of k can be usually expressed as:

−=

RT

EATk cm exp

2

lnB

d k A

dT T= +

Problem:

Discussion the relationship

between this equation and

vant’ Hoff empirical equationDeduce the relationship between Ea and Ec.

exp cEk A

RT

= −


m, usually be 0, 1, 2, 1/2, etc., is not

very large.

In a relatively small temperature

range, Ea seems independent on

temperature.

ca EmRTE +=However, for some reaction such as:

CCl3COOH → CHCl3 + CO2, m = -10.7

CH3Br + H2O → CH3OH + H+ + Br−,

m = -34.3

The effect of temperature on the activation

energy of these reactions is too large to ignore.

Temperature-dependence of Ea

RT= 2.44 kJ mol-1



To measure activation energy of the

reaction over a large span of temperature

would result in exceptional difficulties.

ART

Ek a lnln +−=

When T →, A = k. Is this correct?

How can we measure the experimental

activation energy of an reaction?



9.5.12 A on reaction rate

−=

RT

EAk aexp

9.5.13 Application of Arrhenius equation

1) make explanation for some

experimental results;

2) calculate the reaction rate at different

temperature;

3) determine the optimum

temperature for reaction.

ART

Ek a lnln +−=


胡适：

大胆假设、小心求证！

Bold hypothesis, careful verification

Bring up hypothesis boldly while prove it

conscientiously and carefully

Summary: Pathway for scientific researches


9.5 temperature-dependence of reaction rate -- arrhenius

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