physical-chemistry ii chapter-2-the rates of chemical reactions · physical-chemistry ii...

Physical-Chemistry II

Chapter-2-The rates of chemical reactions

الكيميائيةسرعات التفاعالت

Dr. El Hassane ANOUAR

Chemistry Department, College of Sciences and Humanities, Prince Sattam bin

Abdulaziz University, P.O. Box 83, Al-Kharij 11942, Saudi Arabia.

2/28/2017 College of Science and humanities, Al-kharj 1

Physical

Chemistry II

(Chem 3320)

Important :

These slides are prepared in reference to chapter 21 in Physical Chemistry, Ninth

Edition, Peter Atkins, and Julio De Paula

Introduction


This chapter begins with: o Reaction rates depend on the

concentration of reactants (and

products) Outlines the techniques for its

measurement

Show that

o Rates can be expressed in terms

of differential equations (DEs)

known as rate laws. Predict the concentrations of species

at any time after the start of the

reaction Solution of DEs

Provides insight into the

series of elementary

steps by which a

reaction takes place.

Simple rate

laws

Combination

of rate Laws.

Approximations

i. Concept of the rate determining stage of a reaction

ii. The steady-state concentration of a reaction intermediate

iii. The existence of a pre-equilibrium.

The definition of reaction rate

Introduction


Examples of reaction mechanisms

1. Polymerization reactions

2. Photochemistry reactions (the reactions are initiated by light).

The principles of chemical kinetics.

The study of reaction rates

Develop the rate reactions

in more details

Rates of reactions may be measured and

interpreted.

showing how

More complicated or more specialized

cases.

Application

Rate of a chemical reaction might

depend on variables (P, T and catalyst)

Optimize the rate by the

appropriate choice of conditions

Ability

2/28/2017 College of Science and humanities, Al-kharj

4

1. Empirical chemical kinetics (الكيمياء الحركية التجريبية)

The first steps in the kinetic analysis

of reactions

Stoichiometry of the

reaction and identify any

side reactions.

Establish

The basic data of chemical

kinetics reaction

Concentrations of the

reactants and products at

different times after a

reaction has been initiated.

The rates of most chemical reactions

are sensitive to the temperature

In experiments, the

temperature of the reaction

mixture must be held constant

throughout the course of the

reaction.



1.1 Experimental techniques

1.1.1 Monitoring the progress of a reaction

A reaction in which at

least one component is a

gas might

Overall change in pressure in a

system of constant volume.

The progress may be followed by

recording the variation of pressure with

time.

Result in

Methods used to monitor concentrations

Species involved

Rapidity with which their

concentrations change

Depends on

Example 2.1 (Solution see word file)

Predict how the total pressure varies during the gas-phase decomposition in a

constant-volume container of N2O5?

2 N2O5(g) → 4 NO2(g) + O2(g)



(UV/vis spectra)

the progress of the reaction

H2(g) + Br2(g) → 2 HBr(g)

can be followed by measuring the absorption of visible light

by bromine.

Electrical conductivity

measurements of the solution

In case of reaction that changes the number

or type of ions present in a solution

pH measurements of the solution In case of reactions, in which hydrogen

ions are produced or consumed

Other methods

Emission spectroscopy

Mass spectrometry Gas chromatography

Nuclear magnetic resonance

electron paramagnetic resonance




1.1.2 Application of the techniques

In a real-time analysis the composition of the system is analysed, while the reaction is in

progress.

Flow method

The reactants are injected into

the mixing chamber with a

Steady rate

Its location corresponds to

different times after initiation.

Outlet

tube

The disadvantage of conventional flow techniques is

that a large volume of reactant solution is necessary





The stopped-flow technique: In which the reagents are mixed very quickly in a small

chamber fitted with a syringe instead of an outlet tube.

The reactants are injected into

the mixing chamber with a

Steady rate

A and B mixed very quickly

such as ultraviolet–visible absorption, circular dichroism, and

fluorescence emission, are made on the sample as a function of time.





Flash photolysis technique

The sample is

exposed to a brief

flash of light that

initiates the

reaction and then

the contents of the

reaction chamber

are monitored.

A configuration used for time-resolved absorption

spectroscopy (ultrafast chemical reactions)

The laser pulse can

initiate the reaction

by forming a

reactive species,

such as an excited

electronic state of a

molecule, a radical,

or an ion.

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Flash photolysis technique

The sample is exposed to

a brief flash of light that

initiates the reaction and

then the contents of the

reaction chamber are

monitored.

A configuration used for time-resolved absorption spectroscopy

Reactions are monitored

by using electronic

absorption or emission,

infrared absorption, or

Raman scattering.





A strong and short laser pulse, the pump, promotes

a molecule A to an excited electronic state A* that

can either emit a photon (as fluorescence or

phosphorescence) or react with another species B

to yield a product C.

A + hν → A* (absorption)

A* → A (emission)

A* + B → [AB] → C (reaction)

Intermediate or an activated complex


Quenching methods: Are based on stopping, or quenching, the reaction after it has been

allowed to proceed for a certain time.




These methods are suitable only for reactions that are slow enough for there to be little

reaction during the time it takes to quench the mixture.


1.2 The rates of reactions


1.2.1 The definition of the rates

Reaction rates depend on the composition and the temperature of the reaction mixture.

Consider a reaction of the form

A + 2 B → 3 C + D

Rate is a positive quantity

It follows from the stoichiometry for the

above reaction that

𝑟𝑎𝑡𝑒 =d D

dt=1

3

d C

dt= −

d A

dt= −

1

2

d B

dt






A + 2 B → 3 C + D

𝑟𝑎𝑡𝑒 =d D

dt=1

3

d C

dt= −

d A

dt= −

1

2

d B

dt

The undesirability of having different rates to describe

the same reaction is avoided by using the extent of

reaction, ξ

𝛏 =𝐧𝐉 − 𝐧𝐉,𝟎

𝛎𝐉 𝒗 =

𝟏

𝐕

𝒅𝛏

𝐝𝐭=𝟏

𝛎𝐉×𝟏

𝐕

𝐝𝐧𝐉

𝐝𝐭

Stoichiometric

number of species J Volume of the system

unique rate of reaction, v, as the rate of change of the

extent of reaction

is negative for reactants and

positive for products





𝒗 =𝟏

𝛎𝐉×𝟏

𝐕

𝐝𝐧𝐉

𝐝𝐭

Homogeneous reaction in a

constant-volume system

𝒗 =𝟏

𝛎𝐉

𝐝 𝐉

𝐝𝐭

nJV= J

Heterogeneous reaction

σJ =nJA

𝒗 =𝟏

𝛎𝐉

𝐝𝛔𝐉

𝐝𝐭

mol.dm-3

mol dm−3 s−1

mol m−2 s−1

For gas-phase reactions:

Concentrations are often expressed in molecules cm−3

Rates in molecules cm−3 s−1

𝒗 =𝟏

𝛎𝐉×𝟏

𝑨

𝐝𝐧𝐉

𝐝𝐭





Example:

Consider the reaction

2 NOBr(g) → 2 NO(g) + Br2(g)

If the rate of formation of NO is reported as 0.16 mmol dm−3 s−1, we use νNO = +2 to report that

𝑣 = 0.080 mmol dm−3 s−1

Because νNOBr = −2 it follows that

d[NOBr]/dt = − 0.16 mmol dm−3 s−1

The rate of consumption of NOBr is therefore

0.16 mmol dm−3 s−1, or 9.6 × 1016 molecules cm−3 s−1.




1.2.2 Rate laws and rate constants


A + 2 B → 3 C + D 𝒗 = 𝐤𝐫 𝐀 𝐁

Rate constant for the reaction: Independent of

concentration, but depend on T (dm3 mol−1 s−1)

Each concentration raised to

the first power (mol dm−3 )

The rate law of the reaction (s-1)

This equation

Is determined

experimentally In an overall chemical equation for a chemical reaction

𝒗 = 𝒇( 𝐀 , 𝐁 ,… )

𝒗 = 𝒇(𝐩𝐀, 𝐩𝐁, … ) For homogeneous gas-phase reactions

In gas-phase studies:

Concentrations are commonly expressed in

molecules cm−3

Rate constant for the reaction above would be

expressed in cm3 molecule−1 s−1

The units of the rate

constants can be

determined from rate

laws of any form




1.2.2 Rate laws and rate constants

The rate law of a reaction is determined

experimentally, and cannot in general be

inferred from the stoichiometry of the

balanced chemical equation for the reaction.

𝐇𝟐(𝐠) + 𝐁𝐫𝟐(𝐠) → 𝟐 𝐇𝐁𝐫(𝐠)

𝒗 =𝐤𝐚 𝐇𝟐 𝐁𝐫𝟐

𝟑/𝟐

𝐁𝐫𝟐 + 𝐤𝐛 𝐇𝐁𝐫

Once we know the law and the value of the rate constant:

Predict the rate of reaction from the composition of the mixture.

Predict the composition of the reaction mixture at a later stage of the reaction.

Guide to the mechanism of the reaction, for any proposed mechanism must be

consistent with the observed rate law.




1.2.3 Reaction order

Rate laws of many reaction have the form

𝒗 = 𝐤𝐫 𝐀𝒂 𝐁 𝒃…

Is the order of the

reaction with respect to

the species A in the

reaction (A can be

reactant or product)

Is the order of the

reaction with respect

to the species B in the

reaction (B can be

reactant or product)

The overall order of a reaction with a rate

law like that in this is the sum of the

individual orders Overall order = a + b + · · ·

A reaction need not have an

integral order many gas-phase

reactions do not 𝒗 = 𝐤𝐫 𝐀

𝟏/𝟐 𝐁

Some reactions obey a

zero-order rate law

Catalytic decomposition of

phosphine (PH3) on hot tungsten

at high pressures

𝒗 = 𝐤𝐫




1.2.3 Reaction order

When a rate law is not of the form 𝒗 = 𝐤𝐫 𝐀𝒂 𝐁 𝒃…

Reaction does not

have an overall

order

not have a definite

order with respect to

each participant

Important questions:

How do we identify the rate law and obtain the rate constant from the experimental

data?

How do we construct reaction mechanisms that are consistent with the rate law?

How do we account for the values of the rate constants and their temperature

dependence?




1.2.4 The determination of the rate law

Isolation method

In which, the concentrations of all the reactants

except one are in large excess.

The concentrations of excess reactants are

constant throughout the reaction

Approximation

𝑣 = kr[A][B]

Example

If B is in large excess, we can approximate [B] by [B]0

⇒ 𝒗 = 𝒌𝒓′ 𝐀 ; 𝒌𝒓

′ = 𝐤𝐫 𝐁 𝟎

Has the form of a first-order rate law Pseudo-first-order rate law





Method of

initial rates The rate is measured at the beginning of the

reaction for several different initial

concentrations of reactants..

Often used in conjunction with the isolation

method

Suppose that the rate law for a reaction with A isolated is

𝒗 = 𝒌𝒓′ 𝐀 𝒂 𝒗𝟎 = 𝒌𝒓

′ 𝐀 𝟎𝒂 then its initial rate

Use logarithm in each side

𝐥𝐨𝐠𝒗𝟎 = 𝐥𝐨𝐠𝐤𝐫′ + 𝐚 𝐥𝐨𝐠 𝐀 𝟎

Plot 𝐥𝐨𝐠 𝒗𝟎 against 𝐥𝐨𝐠 𝐀 𝟎 Straight line

with slope a


Example 2.2 (Solution in word document)

Consider the reaction: 2 I(g) + Ar(g) → I2(g) + Ar(g)

The recombination of iodine atoms in the gas phase in the presence of argon was investigated

and the order of the reaction was determined by the method of initial rates. The initial rates of

reaction were as follows:

[I0]/(10-5 mol dm-3) 1.0 2.0 4.0 6.0

𝑣0/(mol dm-3s-1) (a) 8.70 × 10− 4 3.48 × 10−3 1.39 × 10−2 3.13 × 10−2

(b) 4.35 × 10−3 1.74 × 10−2 6.96 × 10−2 1.57 × 10−1

(c) 8.69 × 10−3 3.47 × 10−2 1.38 × 10−1 3.13 × 10−1




The Ar concentrations are (a) 1.0 mmol dm−3, (b) 5.0 mmol dm−3, and (c) 10.0 mmol dm−3.

Determine the orders of reaction with respect to the I and Ar atom concentrations and the rate

constant.


1.3 Integrated rate laws


Rate laws are differential equations

Concentration

s as a function

of time.

Integration

1.3.1 First-order reactions

𝐝 𝐀

𝐝𝐭= −𝐤𝐫 𝐀

Integration

𝐥𝐧𝐀

𝐀 𝟎= −𝐤𝐫𝐭 ⇒ 𝐀 = 𝐀 𝟎 𝒆

−𝐤𝐫𝐭

Rate constant of reaction Concentration at an instant t

If ln([A]/[A]0) is plotted

against t, then a first-order

reaction will give a straight

line of slope −kr.

Initial concentration




1.3.1 First-order reactions

Example 2.3 Analysing a first-order reaction

The variation in the partial pressure of azomethane with time was followed at 600 K, with the

results given below. Confirm that the decomposition

CH3N2CH3(g) → CH3CH3(g) + N2(g)

is first-order in azomethane, and find the rate constant at 600 K.

t/s 0 1000 2000 3000 4000

p/Pa 10.9 7.63 5.32 3.71 2.59

Solution (see file word)




1.3.2 Half-lives and time constants for a first-order reaction

The half-life, t1/2, of a substance, the time taken for the concentration of a reactant to fall to

half its initial value. The time for [A] to decrease from [A]0 to 1

2[A]0 in a first-order reaction

is given:

𝐭𝟏/𝟐 =𝐥𝐧𝟐

𝐤𝐫

Is independent of its initial concentration

The time constant, τ (tau), the time required for the concentration of a reactant to fall to 1/e

of its initial value.

τ =𝟏

𝐤𝐫




1.3.2 Second-order reactions

Consider the chemical reaction equation of second order:

𝐀 → 𝐏𝐫𝐨𝐝𝐜𝐮𝐭𝐬

The rate law Integrated form of the

second-order rate

𝒗 = −𝐝 𝐀

𝐝𝐭= 𝐤𝐫 𝐀

𝟐

𝟐

𝐀 𝟎−𝟏

𝐀 𝟎= 𝐤𝐫t1/2

Or

𝐀 =𝐀 𝟎

𝟏 + 𝐤𝐫𝐭 𝐀 𝟎

2nd Order 1st Order

At t = t1/2 => [A] = 1

2[A]0

𝐭𝟏/𝟐 =𝟏

𝐤𝐫 𝐀 𝟎

Plot

𝟏

𝐀−𝟏

𝐀 𝟎= 𝐤𝐫t

varies with the initial concentration




1.3.2 Second-order reactions

In general,

for an nth-order reaction (n > 1) of the form A → products:

𝐭𝟏𝟐=

𝟐𝒏−𝟏

𝐧 − 𝟏 𝐤𝐫 𝐀 𝟎𝒏−𝟏

Consider the chemical reaction equation of second order:

𝐀 + 𝐁 → 𝐏𝐫𝐨𝐝𝐜𝐮𝐭𝐬

The rate law

Integrated form of the

second-order rate

𝒗 = −𝐝 𝐀

𝐝𝐭= 𝐤𝐫 𝐀 𝑩

𝐋𝐧

𝐁𝐁 𝟎

𝐀𝐀 𝟎

= 𝐁 𝟎 − 𝐀 𝟎 𝐤𝐫𝐭


1.4 Reactions approaching equilibrium


1.4.1 First-order reactions close to equilibrium

Consider the reaction 𝐀 ⇌ 𝐁

The variation of the composition with time close to chemical equilibrium by considering the

reaction in which A forms B and both forward and reverse reactions are first-order

𝐀 → 𝐁

𝐁 → 𝐀

Forward reaction

Forward reaction

𝒗 = 𝒌𝒓 𝑨

𝒗 = 𝒌𝒓′ 𝑩

The concentration of A is reduced by the forward

reaction (at a rate kr[A]) but it is increased by the

reverse reaction (at a rate 𝒌𝒓′ [B]). The net rate of

change is therefore

𝐝 𝐀

𝐝𝐭= −𝐤𝐫 𝐀 + 𝐤𝐫

′ 𝐁

𝐀 ⇌ 𝐁

𝐤𝐫

𝐤𝐫′




1.4.1 First-order reactions close to equilibrium 𝐝 𝐀


′ 𝐁 𝐀 ⇌ 𝐁 𝐤𝐫′

𝐤𝐫

If the initial concentration of A is [A]0, and no B is

present initially, then at all times [A] + [B] = [A]0

𝐝 𝐀


′ 𝐁 = − 𝐤𝐫 + 𝐤𝐫′ 𝐀 + 𝐤𝐫

′ 𝐀 𝟎

The solution of this

first-order differential

equation 𝐀 =

𝐤𝐫′ + 𝐤𝐫𝒆

−(𝐤𝐫+𝐤𝐫′)𝐭

𝐤𝐫 + 𝐤𝐫′ 𝐀 𝟎

Plot



𝐀 =𝐤𝐫′ + 𝐤𝐫𝒆

−(𝐤𝐫+𝐤𝐫′)𝐭

𝐤𝐫 + 𝐤𝐫′ 𝐀 𝟎

As t → ∞, the concentrations

reach their equilibrium values

𝐀 𝒆𝒒 =𝐤𝐫′ 𝐀 𝟎𝐤𝐫 + 𝐤𝐫

′

𝐁 𝒆𝒒 = 𝐀 𝟎 − 𝐀 𝒆𝒒 = 𝐤𝐫 𝐀 𝟎𝐤𝐫 + 𝐤𝐫

′

The equilibrium constant of the reaction

𝐀 ⇌ 𝐁 ⇌ 𝐂 ⇌ ⋯ 𝐤𝒂′

𝐤𝒂

𝑲 =𝐁 𝒆𝒒

𝐀 𝒆𝒒= 𝐤𝐫𝐤𝐫′ 𝐤𝐫 𝐀 𝐞𝐪 = 𝐤𝐫

′ 𝐁 𝐞𝐪

At equilibrium

For a more general reaction, the overall equilibrium

constant can be expressed in terms of the rate constants

for all the intermediate stages of the reaction mechanism:

𝑲 = 𝐤𝐚𝐤𝐚′ ×𝐤𝐛𝐤𝐛′ ×⋯ .

𝐀 ⇌ 𝐁 𝐤𝐫′

𝐤𝐫

𝐤𝒃′

𝐤𝒃




1.4.2 Relaxation methods

Relaxation term denotes the return of a system to equilibrium.

In chemical kinetics, an external applied influence or a perturbation

(e.g; sudden change in T, P…) shifted the equilibrium reaction

position, and the equilibrium is adjusting to the equilibrium

composition characteristic of the new conditions => This is called

relaxation.

When a sudden temperature increase is applied to a simple

A ⇌ B equilibrium that is first-order in each direction, the

composition relaxes exponentially to the new equilibrium

composition: 𝒙 = 𝒙𝟎 𝒆

−𝒕/𝝉 𝟏

𝝉= 𝐤𝐫 + 𝐤𝐫

′

The departure from equilibrium

immediately after the temperature jump

the departure from quilibrium

at the new temperature after

a time t




1.4.2 Relaxation methods

Example 2.4 Analysing a temperature-jump experiment

The equilibrium constant for the autoprotolysis of water, H2O(l) ⇌ H+(aq) + OH−(aq), is

Kw = a(H+)a(OH−) = 1.008 × 10−14 at 298 K. After a temperature-jump, the reaction returns to

equilibrium with a relaxation time of 37 μs at 298 K and pH ≈ 7. Given that the forward

reaction is first-order and the reverse is second-order overall, calculate the rate constants for the

forward and reverse reactions.

Solution (see page. 28 word file chapter 2)


1.5 The temperature dependence of reaction rates


1.5.1 The Arrhenius parameters

The rate constants of most reactions increase as the

temperature is raised.

For many chemical reactions, it is found that a plot

of ln kr against 1/T gives a straight line of an called

Arrhenius equation:

𝐥𝐧 𝐤𝐫 = 𝐥𝐧𝐀 −𝐄𝐚𝐑𝐓

Pre-exponential factor or

the ‘frequency factor’

The activation energy

Intercept

The two quantities A and Ea are

called the Arrhenius parameters




Example 2.5 Determining the Arrhenius parameters

The rate of the second-order decomposition of acetaldehyde (ethanal, CH3CHO) was measured

over the temperature range 700–1000 K, and the rate constants are reported below.

Find Ea and A.

T/K 700 730 760 790 810 840 910 1000

kr/(dm3 mol−1 s −1) 0.011 0.035 0.105 0.343 0.789 2.17 20.0 145

Solution (see page 30 word file, chapter 2)

The higher the activation energy, the stronger the temperature

dependence of the rate constant (that is, the steeper the slope).

In other word, a high activation energy signifies that the rate

constant depends strongly on temperature.





If a reaction has Ea= 0 Rate of reaction is independent of temperature.

In some cases Ea <0 Reaction has a

complex mechanism.

indicates

The rate decreases as the

temperature is raised.

Signal

In some cases, the temperature

dependence of some reactions is

non-Arrhenius (i.e., Ln K = f(1/T)

is not a straight line).

The activation

energy at any

temperature

However

𝐄𝐚 = 𝐑𝐓𝟐𝐝𝐥𝐧𝐤𝐫𝐝𝐓





1.5.2 The interpretation of the Arrhenuis parameters 𝐤𝐫 = 𝐀𝒆

−𝐄𝐚𝑹𝑻

Potential energy of chemical reaction rises to a

maximum, and the cluster of atoms that

corresponds to the region close to the

maximum is called the activated complex.

The crucial configuration at the maximum of

potential is called the transition state of the

reaction.

For a reaction involving the collision of two molecules, the activation energy (Ea) is the

minimum kinetic energy that reactants must have in order to form products.

The exponential factor, A in Arrhenius equation can be an interpret as the fraction of

collisions that have enough kinetic energy to lead to reaction.

physical-chemistry ii chapter-2-the rates of chemical reactions · physical-chemistry ii...

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