General Chemistry · 2017. 1. 7. · The most common type of elementary reaction involves the collision of two atoms, ions, or molecules, and is called bimolecular. A termolecular

Chemical Kinetics

박준원 교수(포항공과대학교 화학과)

General Chemistry

• Rate laws

• Reaction mechanisms

• Reaction mechanisms and rate

• Effect of temperature on reaction rates

• Catalysis

이번 시간에는!

Chemical Kinetics (I-1)


General Chemistry

• Why do some chemical reactions proceed with lightning

speed when others require days, months, or even years?

• How do catalysts increase the rates of chemical reactions?

• Why do small changes in temperature often have large

effects on the reaction rate?

• How does a study of the rate of a chemical reaction inform

us about the way in which molecules combine to form

products?

All of these questions is the subject of chemical kinetics.

Rate of chemical reactions

𝑎A + 𝑏B → 𝑐C + 𝑑D

Average reaction rate = change in concentration

chage in time

Average rate = ∆[P]∆𝑡

, [P] = concentration of a product

rate = −1

𝑎

𝑑[A]

𝑑𝑡 = −

1

𝑏

𝑑 B𝑑𝑡

= 1

𝑐

𝑑 C𝑑𝑡

=1

𝑑

𝑑 D𝑑𝑡

[18.1]

1

<Measuring reaction rates>

Rate laws

The relation between the rate of a reaction and concentration is

called a rate law.

Rate law (or empirical rate expression)

zeroth order in A for n = 0; first order in A for n = 1; second

order in A for n = 2; the order of a reaction does not have to be

an integer. Overall reaction order is n + m.

rate = 𝑘[A]𝑛[B]𝑚

2

<Integrated rate laws>

Integrated rate law expresses the concentration

of a species directly as a function of time.

First-order reactions

[ F I G U R E 1 8 . 6 ]

N2O5 g → 2 NO2 g + 1

2 O2(g)

rate = −𝑑 N2O5

𝑑𝑡= 𝑘[N2O5]

𝑐 = 𝑐0𝑒−𝑘𝑡 (𝑐0: initial concentration) [18.2]

𝑡1/2 = ln 2𝑘

= 0.6931

𝑘 [18.3]

Oxtoby, D. W.; Gillis, H. P.; Campion, A., Principles of modern chemistry, 7th ed.; Cengage Learning: Boston, 2012; p 844.

second-order reactions

2 N02 g → 2 NO g + O2(g)

rate = −1

2

𝑑 NO2

𝑑𝑡= 𝑘[NO2]

2

1

𝑐=

1

𝑐0+ 2𝑘𝑡 [18.4]

𝑡1/2 =1

2𝑘[NO2]0

[ F I G U R E 1 8 . 7 ]


Reaction mechanisms

Many reactions do not occur in a single step, but rather proceed

through a sequence of steps to arrive at the products. Each step

is called an elementary reaction and occurs through the collisions.

<Elementary reactions>

A unimolecular elementary reaction involves only a single

reactant molecule.

3

N2O5∗ g → NO2 g + NO3 g

rate = 𝑘[N2O5∗]

The most common type of elementary reaction involves the collision of two atoms, ions, or molecules, and is called bimolecular.

A termolecular reaction step involves the simultaneous collision of three molecules.

Elementary reactions involving collisions of four or more molecules are not observed, and even termolecular collisions are rare if other pathways are possible.

NO g + O3 g → NO2 g + O2(g)

rate = 𝑘 NO [O3]

I g + I g + Ar(g) → I2(g) + Ar(g)

rate = 𝑘[I]2[Ar]

Solvent molecules are present in liquid phase and may affect the reaction,

even though they do not appear in the rate expression because the

solvent concentration cannot be varied appreciably.

<Reaction mechanism>

A reaction mechanism is a detailed sequence of elementary reactions.

eq 1 + eq 2 ->

A reaction intermediate (here, NO3) is a chemical species that is formed

and consumed in the reaction.

NO2 g + NO2 g → NO3 g + NO g (slow) eq 1

NO3(g) + CO(g) → NO2(g) + CO2(g) (fast) eq 2

2 NO2(g) + NO3(g) + CO(g) → NO3(g) + NO(g) + NO2(g) + CO2(g)

NO2(g) + CO(g) → NO(g) + CO2(g)

<Kinetics and chemical equilibrium>

2 NO g + 2 H2 g ⇄ N2 g + 2 H2O(g)

NO g + NO g ⇄ N2O2 g

N2O2 g + H2 g ⇄ N2O g + H2O g

N2O g + H2 g ⇄ N2 g + H2O(g)

𝑘1

𝑘−1 𝑘2

𝑘−2 𝑘3

𝑘−3

𝐾 = 𝐾1𝐾2𝐾3 =[H2O]𝑒𝑞

2 [N2]𝑒𝑞

[NO]𝑒𝑞2 [H2]𝑒𝑞

2

=𝑘1𝑘2𝑘3

𝑘−1𝑘−2𝑘−3=

[N2O2]𝑒𝑞[N2O]𝑒𝑞[H2O]𝑒𝑞[N2]𝑒𝑞[H2O]𝑒𝑞

[NO]𝑒𝑞2 [N2O2]𝑒𝑞[H2]𝑒𝑞[N2O]𝑒𝑞[H2]𝑒𝑞

Chemical Kinetics (I-2)


General Chemistry

Reaction mechanisms and rate

In many reactions mechanisms, one step is significantly slower than all the others; this step is called the rate-determining step.

If the rate-determining step is the first one, the analysis is particularly simple.

A possible mechanism is

The first step is slow and determines the rate (𝒌𝐨𝐛𝐬 = 𝒌𝟏), and the subsequent fast step does not affect the reaction rate.

4

2 NO2 g + F2 g → 2 NO2F(g)

rate = 𝒌𝐨𝐛𝐬 NO𝟐 [F𝟐]

NO2(g) + F2(g)𝑘1→NO2F(g) + F g (slow)

NO2(g) + F(g)𝑘2→NO2F(g) (fast)

Mechanisms in which the rate-determining step occurs after one or more

fast steps are often signaled by a reaction order greater than 2, by a

non-integral reaction order, or by an inverse concentration dependence

on one of the species.

An example is

One possible mechanism would be a single-step termolecular reaction,

but termolecular collisions are rare, and if there is an alternative pathway

it is usually followed.

2 NO g + O2 g → 2 NO2(g)

rate = 𝒌𝐨𝐛𝐬[NO]𝟐[𝐎𝟐]

One such alternative is the two-step mechanism

NO g + NO g ⇄ N2O2 g fast equilibrium

N2O2 g + O2 g ⇄ 2 NO2(g) slow

rate = 𝑘2 N2O2 [O2]

[N2O2]

[NO]2=

𝑘1

𝑘−1= 𝐾1

[N2O2] = 𝐾1[NO]2

rate = 𝑘2𝐾1[NO]2[O2]

𝑘1

𝑘2 𝑘−1

𝑘−2

<The steady state approximation>

In some reaction mechanisms, there is no single step that is much slower

than the others. In such cases we use the steady-state approximation.

𝑘−1

𝑘1

N2O5 g + M g ⇄ N2O5∗(g) + M(g)

N2O5∗ g

𝑘2→ NO3(g) + NO2(g)

NO3 g + NO2 g 𝑘3→ NO g + NO2 g + O2 g (fast)

NO3 g + NO g 𝑘4→ 2 NO2 g (fast)

N2O5(g) → 2 NO2(g) +1

2 O2(g)

The N2O5* is a reactive intermediate. The net rate of change of [N2O5

*] is

then

The approximation consists of the assumption that after the short time

the rate of production and loss of N2O5* becomes equal, and

setting the rate of change of the N2O5* concentration to 0 gives

solving for [N2O5*] gives

𝑑[N2O5

∗]

𝑑𝑡= 𝑘1 N2O5 [M] − 𝑘−1[N2O5

∗][M] − 𝑘2[N2O5∗]

𝑑[N2O5∗]

𝑑𝑡= 0 = 𝑘1 N2O5 [M] − 𝑘−1[N2O5

∗][M] − 𝑘2[N2O5∗]

N2O5∗ 𝑘2 + 𝑘−1 M = 𝑘1 N2O5 [M]

N2O5∗ =

𝑘1 N2O5 [M]

𝑘2 + 𝑘−1[M]

The rate of the overall reaction N2O5 → 2 NO2 +1

2 O2 is

The expression has two limiting cases:

1. Low pressure When [M] is small enough, 𝒌𝟐 ≫ 𝒌−𝟏 M and we can use

the approximation.

2. High pressure When [M] is large enough, 𝒌−𝟏 M ≫ 𝒌𝟐 and we can use

the approximation.

𝑟𝑎𝑡𝑒 =1

2

𝑑[NO2]

𝑑𝑡= 𝑘2 N2O5

∗ =𝑘1𝑘2 N2O5 [M]

𝑘2 + 𝑘−1[M]

rate = 𝑘1 N2O5 M (second order)

rate =𝑘1

𝑘−1 𝑘2 N2O5 = 𝐾1𝑘1[N2O5] (first order)

Chemical Kinetics (II-1)


General Chemistry

Effect of temperature on reaction rates

<Gas-phase reaction rate constants>

Some reactions do proceed at each collision event; rate constant of 𝟏 ×𝟏𝟎𝟏𝟎 𝐋𝐦𝐨𝐥−𝟏𝐬−𝟏.

However, reaction rates that are much lower – by factors of 1012 or more – are common.

The rate of many chemical reactions increase extremely rapidly at temperature increases, typically a 10℃ rise in temperature may double the rate. In 1889 Svante Arrhenius suggested

Where 𝐸a is a constant with dimensions of energy and A is a constant with the same dimensions as k. Taking the natural logarithm of this equation gives

5

𝑘 = 𝐴𝑒−𝐸a/𝑅𝑇 [18.5]

ln 𝑘 = ln𝐴 −Ea

RT [18.6]

𝑬𝒂 is known as the activation energy. The fraction of molecules having

more than the critical energy 𝐸a increases exponentially as exp (-𝐸a/RT).

𝑇1

𝑇2

𝑇2 > 𝑇1

Frac

tio

n o

f m

ole

cule

s

𝑢 = 2𝜀/𝑚

Speed [ F I G U R E 1 8 . 1 0 ]


<The reaction coordinate and the activated complex>

Activated complex or transition state

The activation energy for an elementary reaction is always positive, but some overall reactions manifest “negative activation energies”.


[ F I G U R E 1 8 . 1 1 ]

Reaction in solution

Reactions in solution are much more complicated to describe than their gas-

phase counterparts because the transport mechanism in solutions is much

different. Collisions between reactants in solution occur much less frequently

than for those in the gas phase, but the reactants may be bound together by

a “cage” of solvent molecules for sufficiently long periods that they can

acquire enough energy through collisions

k1 << k2, if k1 is limited by the rate at which the reactants encounter

one another by diffusion we call the reaction diffusion-controlled.

k2 << k1 and k2, reactions are often called activation-energy-controlled.

𝑘1

𝑘−1

6

with the solvent to react.

AB𝑘2→products

A + B ⇄ AB

Chemical Kinetics (II-2)


General Chemistry

Catalysis

A catalyst is a substrate that takes part in a chemical reaction and

speeds up the rate but undergoes no permanent chemical

change itself. Catalysts therefore

do not appear in the overall

balanced chemical equation.

7


[ F I G U R E 1 8 . 1 9 ]

In homogeneous catalysis the catalyst is present in the same

phase as the reactants.

In heterogeneous catalysis the catalyst is present as a phase

distinct from the reaction mixture.

C𝟐H𝟒(g) + H𝟐(g) → C𝟐H𝟔(g) solid oxide of vanadium (V2O5) as a

catalyst.

TI+ 𝑎𝑞 + 2 Ce4+ 𝑎𝑞Ag+(𝑎𝑞)

TI3+ 𝑎𝑞 + 2 Ce3+(𝑎𝑞)

<Enzyme catalysis>

The vast majority of chemical reactions in living organisms are

catalyzed by enzymes, proteins that have evolved to enhance

reaction rates by many orders of magnitude with exquisite

selectivity.

The major classes and the reactions they catalyze include (1)

oxidoreductase – redox reactions; (2) transferase – functional

group transfer reactions; (3) hydrolase – hydrolysis reactions; (4)

lyase – addition to carbon-carbon double bonds; (5) isomerase –

isomerization reactions; (6) ligase – bond-forming reactions.

<Enzyme kinetics>

The kinetics of enzyme-catalyzed reactions is generally treated using the

steady-state approximation in a form that has become known as

Michaelis-Menten kinetics.

In which E represents the free enzyme, S is the substrate, ES represents

the complex formed, and P is the final product.

The concentration of

𝑘1

𝑘−1

[ES] reaches a steady state.

E + S ⇄ ES

ES 𝑘2→ E + P

𝑑[ES]

𝑑𝑡= 0 = 𝑘1 E S − 𝑘−1 ES − 𝑘2[ES]

The concentration of the free enzyme ([E]) is [ET] – [ES], where [ET] is the

total concentration of enzyme present (i.e., the concentration of enzyme

initially added [Eo].).

𝑑[ES]

𝑑𝑡= 0 = 𝑘1 ET S − 𝑘1[ES][S] − 𝑘−1 ES − 𝑘2[ES]

ES =𝑘1 ET [S]

𝑘−1 + 𝑘2 + 𝑘1[S]

We define the Michaelis constant Km

(in units of molarity) as

𝐾m =𝑘−1 + 𝑘2

𝑘1


[ F I G U R E 1 8 . 2 2 ]

Substrate molecule

which allows us to express the steady state concentration of [ES] as

ES =ET S

𝐾m + [S]

𝑑[P]𝑑𝑡

= 𝑘2 ES =𝑘2 ET [S]𝐾m+[S]

[18.15] Michaelis-Menten equation

The rate is linear (first-order) with respect to [S] at low substrate

concentrations but rolls over to become independent of [S] at higher

concentrations, saturating at a maximum rate 𝑉max .

At [S] >> 𝐾m, rate = 𝑉max = 𝑘2[ET] from eq [18.15]


[ F I G U R E 1 8 . 2 3 ]

Rate ≅ 𝑘2

𝐾mET S

at [S] << 𝐾m

An alternative form of Michaelis-Menten equation is

When 𝑑[𝑃]/𝑑𝑡 = 𝑉𝑚𝑎𝑥 /2, 𝐾𝑚 =[S], so we simply use [S] to find 𝐾m

experimentally.

A small value of 𝐾m indicates that the formation of ES complex is faster

than the dissociation of ES complex to the starting materials and the

product.

𝑑[P]

𝑑𝑡= 𝑘2 ES =

Vmax [S]𝐾m+[S]

𝐾m = [S]Vmax

𝑑[𝑃]/𝑑𝑡− 1

Turnover number is defined as the number of substrate molecules

converted into product per enzyme molecule per second, and is easily

calculated from 𝒌𝐜𝐚𝐭.

at saturation,

The turnover numbers for different enzymes vary over an enormously

wide range; each catalase molecule can decompose 40 million molecules

of hydrogen peroxide per second, whereas the turnover number of

chymotrypsin is about 100 and that of lysozymes is about 0.5.

[S] >> 𝐾m, [ES] = [ET], rate = 𝑉max = k2[ET], 𝑘cat ≡ 𝑘2 = 𝑉max /[ET]

A linear version of

Plot the inverse of the rate versus 1/[S], the slope is 𝐾m / 𝑉max, and the

intercept is 1/ 𝑉max .

From the plot, the key parameters (𝑘cat , 𝐾m)

Michaelis-Menten equation is

can be obtained from the rates at a few

concentrations of the substrate.

1

𝑑 P /𝑑𝑡= (

𝐾m

𝑉max)(

1

S) +

1

𝑉max


[ F I G U R E 1 8 . 2 4 ] 𝐾m =𝑘−1 + 𝑘2

𝑘1

𝐾cat = 𝑉max /[ET]

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General Chemistry · 2017. 1. 7. · The most common type of elementary reaction involves the collision of two atoms, ions, or molecules, and is called bimolecular. A termolecular