chemistry paper no. 10: physical chemistry iii (classical ... · thermodynamics, non-equilibrium...

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Paper No. 10: Physical Chemistry –III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface Chemistry, Fast Kinetics

Module No. 25: Gibbs Adsorption equation, surface activity and surface films

Subject Chemistry

Paper No and Title 10: Physical Chemistry –III (Classical Thermodynamics,

Non-Equilibrium Thermodynamics, Surface Chemistry,

Fast Kinetics

Module No and Title 25: Gibbs Adsorption equation, surface activity and surface

films

Module Tag CHE_P10_M25

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TABLE OF CONTENTS

1. Learning outcomes

2. Adsorption and surface tension: Gibbs adsorption equation

3. Surface active and surface inactive materials

4. Formation of surface films on liquids

5. Formation of electrical double layer at interfaces and electro-kinetic effects

6. Catalytic activity at surface

7. Summary

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1. Learning outcomes

After studying this module, you shall know:

Understand relation between surface tension and adsorption and derive Gibbs

adsorption equation

Know about surface active materials

Understand the formation of surface films and surface pressure

Learn about catalytic activity at surface

2. Adsorption and surface tension: Gibbs adsorption equation

In the last module, we have studied about the phenomenon and applications of surface

tension. We saw that it is due to the presence of unbalanced forces on the surface of a

liquid, the liquid tries to contract its area and attain a minimal value of surface energy.

The latter was numerically and dimensionally equal to the force of surface tension.

It was observed that addition of a solute, of lower surface tension as compared to that of

the liquid, results in the accumulation of the solute on the liquid surface. This is due to

the fact that the solute tends to decrease the surface tension of the liquid and hence its

surface energy per unit surface area. J.W. Gibbs derived a quantitative relationship

between the extent of adsorption of a solute and the respective change in the surface

tension of a liquid due to its addition. This relation is called as the Gibbs adsorption

equation and can be derived as follows:

For a system consisting of a solvent ( number of moles, 𝑛1) and a solute ( number of

moles, 𝑛2) (i.e., a two component system), the Gibbs free energy will be given by:

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(1)

Where, 𝜇1𝑎𝑛𝑑 𝜇2 are the respective chemical potentials of the two components, S is the

surface area and γ is the surface energy per unit area.

‘γS’ deals with the contribution due to the surface free energy towards the Gibbs free

energy of the system.

Therefore the variation in the total Gibbs free energy of the system will be given by:

(2)

Since 𝐺 = 𝑓 (𝑇, 𝑃, 𝑛1, 𝑛2, 𝑆)

Therefore, we have:

(3)

Or,

(4)

Under conditions of constant temperature and pressure, we have

(5)

Equating (2) and (5), we get:

(6)

In the bulk of the liquid, the analogous expression for the Gibbs free energy change

(devoid of surface energy) will be given by:

𝑛1𝑜𝑑𝜇1 + 𝑛2

𝑜𝑑𝜇2 = 0 (7)

where, 𝑛1𝑜 and 𝑛2

𝑜 are the respective amounts of solvent and solute in the bulk phase.

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As per the thermodynamic condition of chemical equilibrium, the chemical potentials of

the components of a system in a state of equilibrium must be same in all its phases. On

slight perturbation, a new state of equilibrium is achieved and the subsequent changes in

the respective chemical potentials of its components, in all the phases, must also be

identical.

Therefore, 𝑑𝜇1𝑎𝑛𝑑 𝑑𝜇2 in equations (6) and (7) must be same.

From equation (7), we have

𝑑𝜇1 = −𝑛2𝑜

𝑛1𝑜 𝑑𝜇2 (8)

Substituting the above equation (8) in equation (6), we get:

(9)

or,

(10)

or,

(11)

Equation (11) involves the difference of two terms on the right hand side, i.e., the amount

of solute associated with the liquid at the surface and in the bulk phase. The numerator of

this equation thus gives the excessive concentration of the solute present at the surface of

the liquid. This when divided by the surface area S, gives the excessive concentration of

the solute per unit surface area (𝚪2).

Thus, we have,

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(12)

Equation (12) is termed as the Gibbs adsorption equation.

Also, the chemical potential of the solute is given by the expression:

(13)

where, 𝜇2∗(l) is the chemical potential of pure solute in the liquid phase.

Differentiating equation (13), we get:

(14)

Using equation (12) and equation (14),

(15)

For a dilute solution, we have:

Γ2 = −1

𝑅𝑇

𝑑𝛾

𝑑 ln𝑐2𝑐𝑜⁄

Γ2 = −𝑐2

𝑅𝑇 𝑑𝛾

𝑑𝑐2 (16)

The above equation (16) is termed as Gibbs Adsorption isotherm.

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3. Surface active and surface inactive materials

The Gibbs adsorption equation (16) derived above is useful in classifying certain

substances as surface active and surface inactive materials.

If the addition of a substance to a solvent causes a decrease in the surface tension of the

latter, the substance is termed as a surface active material. Using the Gibbs adsorption

equation, 𝑑𝛾

𝑑𝑐2 is negative which makes Γ2 positive. This implies that the substance has a

relatively higher concentration at the surface of the solvent compared to the bulk of the

solution. Examples of surface active materials include: soaps, detergents, dyes, most

organic compounds (fatty acids, alcohols), etc. The limiting value of the decrease in

surface tension with concentration ( lim𝑐2→0

𝑑𝛾

𝑑𝑐2) is termed as surface activity.

On the other hand, if a substance increases the surface tension of the solvent to which it is

added (𝑑𝛾

𝑑𝑐2 is positive), it is called as a surface inactive material. Such materials have

higher concentration in the bulk of the solution as compared to on their surface (i.e., Γ2

negative). This behavior is termed as negative adsorption and substances like glycerol,

sugars, inorganic salts, etc. belong to this category. These surface inactive materials

produce a very small change in the concentration of the solvent and hence increase the

surface tension of the latter to a small extent only.

The difference in the behavior of these two materials can be explained on the basis of the

intermolecular attractions operating between solute and the solvent. If the solvent-solvent

interactions are stronger than the ones operating between solute and solvent (i.e., solute

exhibits positive deviations from the Raoult’s law), the solute molecules are pushed up

from the bulk of the solvent to the surface, thus making Γ2 positive. This decreases the

attractive forces in the surface layer; hence reducing the surface tension of the solvent.

On the other hand, if solute-solvent interactions are stronger than the solvent-solvent

interactions (i.e., solute exhibits negative deviations from Raoult’s law), the solute

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molecules are retained in the bulk of the solution in order to maximize the interactions

and gain more stability.

3.1 Orientation of surface active materials

The surface active materials have a specific orientation in the solution, especially when

the concentration of the solution is high. Considering the example of a fatty acid in water.

The polar groups (carboxyl or hydroxyl) orient towards the surface of the water and the

non-polar hydrocarbon chain points vertically away from the solution. This was

mathematically proved by B. Szyszkowski who gave the relation showing the variation in

the surface tension of water with the addition of water-soluble fatty acids:

𝛾

𝛾∗= 1 − 𝑋 ln

𝑐

𝑌 (17)

where, 𝛾 𝑎𝑛𝑑 𝛾∗are surface tension of solution of concentration c and pure water,

respectively, and X and Y are constants.

Rearranging equation (17), we get:

(18)

Differentiatig equation (18), we get,

(19)

or, (20)

Using equation (16) in above equation (20), we get:

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(21)

Since the right hand side of the above equation (20) is constant, therefore the surface

excessive concentration also attains a constant value. It can be thus concluded that latter

does not depend on the hydrocarbon chain length of the fatty acid molecule. This is

possible only if the solute (fatty acid) molecules orient themselves in a specific manner,

with their polar heads attracted towards water and non-polar groups pointing vertically

away from it.

4. Fomation of surface films on liquids

Certain substances like long chain fatty acids (oleic, stearic acid, etc.), alcohols, amides,

etc. are insoluble in water and have a tendency to spread and form films on the water

surface. The orientation of these substances is highly specific with their polar ends facing

towards water and hydrocarbon chains facing away, as discussed above (Fig. 1).

Fig. 1: Orientation of surface film molecules

Consider a solution of such a substance, say, stearic acid dissolved in benzene. When a

drop of this dilute solution is added to water contained in a ‘Langmuir tray’, after

sometime the benzene evaporates and stearic acid forms a thin film on the water surface.

The film can be confined between a barrier and a float, against which any force can be

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measured with the help of a torsion wire (Fig. 2). The area of the film can be changed by

moving the barrier. If the film is compressed, initially the force required is minimal until

a certain critical stage is reached, where the force required to compress the film rises

steeply. Fig. 3 shows the general trend of how the force required to change the area per

molecule of film forming substance varies. Such a curve is called a F-A isotherm and is a

two-dimensional analogue of three dimensional P-V isotherm. On extrapolating the

graph, the value of the critical area comes to be 0.205 nm2 per molecule. This value of

critical area is a reasonable constant for a number of long chain compounds with polar

heads. This indicates that critical area is independent of the hydrocarbon chain length. At

this critical state, the molecules are closely packed forming a monolayer and oriented in a

specific direction (Fig. 1) on the water surface. This area thus represents the area of

cross-section of the molecule. The latter can also be used to calculate the length of the

molecule.

Fig. 2: Formation of surface film: Langmuir film experiment

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Fig. 3: A typical F-A isotherm

As a result of surface tension on the film-covered surface, a force F acts in the direction

shown in Fig. 2. If the barrier of length ‘x’ moves by a distance ‘dy’, then the area of

clean water surface increases by amount xdy while that of surface film decreases by

amount xdy (Fig. 2).

The net increase in energy = 𝛾∗𝑥𝑑𝑦 − 𝛾 𝑥𝑑𝑦

Where,

𝛾∗and 𝛾 are the surface tension of pure water and film covered surface,

respectively.

The energy is supplied by the movement of the barrier by a distance dy, against a force

‘𝐹 × 𝑥’, such that:

𝐹 × 𝑥𝑑𝑦 = (𝛾∗ − 𝛾) 𝑥𝑑𝑦 (22)

or 𝐹 = 𝛾∗ − 𝛾 (23)

It is to be noted that F acts as the surface pressure and is equal to force per unit length of

the barrier. It can be seen from Fig. 3 and equation (23) that the difference between the

surface tension of the film-covered surface and clean water surface is very small, unless

the film becomes closely packed.

The Gibb’s adsorption isotherm derived above is given by equation (16),

Γ2 = −𝑐2

𝑅𝑇 𝑑𝛾

𝑑𝑐2

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or 𝑑𝛾 = −𝑅𝑇 Γ2𝑑𝑐2

𝑐2 (24)

Now, the surface excess concentration is proportional to bulk concentration, i.e.,

Γ2 ∝ 𝑐2

or Γ2 = 𝐾 𝑐2 (25)

Substituting equation (25) in equation (24), we get:

𝑑𝛾 = −𝑅𝑇 𝐾 𝑑𝑐2 (26)

Integrating equation (26), we get

𝛾 − 𝛾∗ = −𝑅𝑇 𝐾 𝑐2 (27)

𝛾 − 𝛾∗ = −𝑅𝑇 Γ2 (28)

Using equation (23) in the above equation (28), we get:

𝐹 = 𝑅𝑇 Γ2 (29)

The surface excess concentration (Γ2) is also equal to number of moles (𝑛2𝜎) of the fim

forming substance per unit area (𝐴), i.e.,

Γ2 =𝑛2𝜎

𝐴 (30)

Substituting equation (30) in equation (29), we get:

𝐹𝐴 = 𝑛2𝜎𝑅𝑇 (31)

or 𝐹�̅� = 𝑅𝑇 (32)

where, �̅� is the area per mole of the substance.

Equation (32) is applicable to monolayer formation at the surface and is analogous to the

ideal gas law, assuming the gas is two dimensional in nature. Fig. 4 shows the variation

in surface pressure with area of the film-forming molecule. The curves have a very

similar appearance to the P-V isotherms of a real gas. Infact, the uppermost curve follows

the 2-D ideal gas law derived above (equation 32).

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Fig. 4: Variation in surface pressure with area of the film-forming molecule

The formation of surface films of the Langmuir type has diverse many applications. The

above equation (32) can be used to calculate the surface pressure or the area of the

molecule. Langmuir and Blodgett gave a very simple method to determine the size of the

molecules; results of which match well with those obtained from X-ray diffraction

studies. Other applications include measurements like surface diffusion, surface

potentials, film viscosity, chemical reactions in monolayers at the surface, reduction in

water evaporation using long-chain alcohols, etc.

5. Formation of electrical double layer at interface and electro-kinetic

effects

Whenever two phases of dissimilar chemical composition are in contact with each other,

an electric potential difference gets developed resulting in charge separation across the

interface.

If one such phase is a metal electrode (say, positively charged, present in the region x≤0)

and other is an electrolytic solution (bearing corresponding negative charge and present

in the region x≥0), different charge distributions are possible corresponding to different

potential fields (Fig. 5):

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(a) Helmholtz double layer: The negative charge of the electrolyte solution

corresponding to the positive charge of metal electrode is present in a plane at a

short distance, δ, from the metal surface. This type of double layer containing

charges at a fixed distance is called Helmholtz double layer (Fig. 5a).

(b) Gouy-Chapman double layer: This layer contains the corresponding negative

charge not at a fixed distance, but in a diffused manner throughout the solution.

The situation can be considered analogous to a diffused atmosphere surrounding

an ion in the solution (Fig. 5b).

(c) Stern double layer: This type of double layer is a combination of fixed and

diffused double layers. A fixed layer containing small negative charge is present

at a distance δ, and the remaining negative charge, required to balance the positive

charge of the metal electrode, is present in the diffused layer lying beyond this

distance. Another possibility is that the fixed layer contains more than

required negative charge and the diffused layer is positively charged (Fig. 5c).

The possibility of specific ions (cations or anions) getting adsorbed on the metal

surface is also included in this theory.

We can also have the other possibility of the metal electrode being negatively charged

and the electrolyte solution bearing positive charge.

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Fig. 5: Various types of double layers. Y-axis displays the potential relative to that in the solution

Another successful model, given by Grahame, distinguishes between the two planes of

ions. It proposes the existence of inner-Helmholtz plane at the distance of closest

approach of the centers of the chemisorbed anions to the metal surface. Beyond this

plane, at a distance of closest approach of the centers of hydrated ions, lies the outer-

Helmholtz plane. The diffused layer begins at the outer-Helmholtz plane (Fig. 6). This

model satisfactorily explains the effects related to double layer.

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Fig. 6: Electric double layer showing inner and outer-Helmholtz plane

The phenomenon of double layer formation leads to charge separation at the solid-liquid

interface. Due to this, the particles in a heterogeneous fluid (fluid containing particles)

get affected in the presence of an electric field, producing ‘electro-kinetic effects’. Some

of them have been discussed below:

(a) Electrophoresis: It is defined as the movement of charged particles, suspended in

a liquid, under the influence of an electric field. These suspended particles carry

charge which the sum of the charge present on the particle and the charge present

in the fixed portion of the double layer. On the other hand, the mobile diffused

part of the double layer, being oppositely charged, moves in the other direction.

(b) Electro-osmosis: It is defined as the movement of the liquid through a

immobilized set of particles, porous membrane or plug, in response to an applied

electric field. It occurs due to the force exerted by the field on the opposite-

charged liquid present inside the charged porous membrane/disk. When these ions

move, they drag along the liquid, which solvates them. The latter moves with a

uniform velocity called ‘electro-osmotic velocity. This can be explained by taking

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a device filled with water and fitted with a fixed porous quartz plug (Fig. 7). The

diffused and hence the mobile part of the double layer is positively charged.

When the electric potential is applied, the positive charge moves to the cathode

compartment, dragging along water with itself.

Fig. 7: The process of electro-osmosis

(c) Electro-osmotic counter-pressure: If we wish to stop the above process of electro-

osmosis, some pressure difference will have to be applied across the system. The

latter is termed as electro-osmotic counter-pressure.

(d) Streaming potential: If we force water to flow through the porous plug, it carries

charge from one side to the other. This flow of counter charges inside the plug

causes the charges to accumulate, creating a potential difference, called streaming

potential, between the electrodes.

(e) Streaming current: When the two electrodes are short-circuited, the current that

flows through the plug is called the streaming potential.

(f) Sedimentation potential (Dorn effect): If we consider a suspension of particles,

the particles upon settling down under the effect of gravity, carry their respective

charge towards the base of the container. The charge on the diffused part of the

layer, on the other hand, remains in the upper part of the vessel. This creates a

potential difference between the upper and lower part of the container, known as

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sedimentation potential. If sedimentation is produced by centrifugal force, the

process is then called as centrifugation potential.

The electro-kinetic effects discussed above come into play due to the reason that the

entire double layer is not fixed; instead some part of it is loosely bound to the solid

surface and hence is mobile in nature. Their magnitude depends on the amount of charges

present in the mobile part of the double layer. The latter, in turn, depends upon the

potential developed at the dividing line (shear surface) between the fixed and the mobile

part of the double layer. This potential is called the zeta (ζ) potential. Thus, it can be

concluded that the magnitude of all the electro-kinetic effects depend on the magnitude of

zeta potential.

6. Catalytic activity at surface

A catalyst is a substance that affects the rate of a chemical reaction, without itself getting

used up for the same. It does not undergo any change in its mass or its composition. It

only helps in the faster attainment of the reaction equilibrium. It does so by changing the

activation energy pathway leading to the reaction. In general, the process of catalysis can

be classified as follows:

(a) Homogenous catalysis: When the catalyst is present in the same phase (physical

state) as that of the reactants, it is called homogenous catalysis. For example:

2𝑆𝑂2 (𝑔) + 𝑂2 (𝑔) 𝑁𝑂 ⇒ 2 𝑆𝑂3 (𝑔)

(b) Heterogenous catalysis: When the catalyst is present in a different phase as that of

the reactants, the process is called heterogenous catalysis. For example:

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𝑁2 (𝑔) + 3𝐻2 (𝑔) 𝑓𝑖𝑛𝑒𝑙𝑦 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑁𝑖 ⇒ 2 𝑁𝐻3 (𝑔)

(c) Auto-catalysis: When one of the reaction products acts as a catalyst for the

reaction itself, the process is called auto-catalysis. For example, the

decomposition of KMnO4 gives MnO2, which catalyzes its decomposition further.

Comparing homogenous and heterogeneous catalysts. The former are often selective

towards the formation of a desired product and operate at low temperature and pressure

conditions. They, however, are difficult to separate from the reaction mixture. On the

other hand, the heterogeneous catalysts are often robust in nature and offer a major

advantage of being capable of easy removal from the reaction mixture.

In industries, most of the chemical reactions make use of solid catalysts. For example, the

Born Haber’s process involves the synthesis of ammonia from hydrogen and nitrogen

gases, in the presence of finely divided iron as a catalyst, Contact process for the

synthesis of sulphuric acid, involves the oxidation of SO2 to SO3, in the presence of

platinized asbestos or V2O5, polymerization of alkenes using phosphoric acid distributed

on diatomaceous earth, etc. The most common examples of heterogenous catalysts used

are transition metals associated with partially vacant d orbitals (Cr, W, Mn, Fe, Co, Ni,

Pd, Pt, Cu, Ag, Au), metal oxides (Al2O3, Cr2O3, V2O5, ZnO, NiO, Fe2O3), or acids

(H2SO4, H3PO4). There are a number of commercially important reactions that make use

of solid catalysts. For latter to be effective, a very important requirement is that one or

more reactants should be chemisorbed on their solid surface. These catalysts are known

to substantially decrease the activation energy required for the reaction. For example,

consider the decomposition of hydrogen iodide to give hydrogen gas and iodine. This

reaction involves and activation energy of about 44 kcal/mol. The latter decreases to a

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value of 25 kcal/mol and 14 kcal/mol in the presence of gold and Platinum, as catalysts,

respectively.

Some important features of solid catalysts are:

Activity: The activity of a catalyst depends on the strength of chemisorption of

reactant(s) on the solid catalyst surface. In other words, it depends on the value of

enthalpy of adsorption (|∆𝐻𝑎𝑑𝑠̅̅ ̅̅ ̅̅ |). If the latter is low, it would mean that the extent

of adsorption is low, leading to a slow reaction. On the other hand, if it is large,

the reactants will be held very tightly to their adsorption sites, thereby becoming

immobile. This inhibits their tendency to react with one another and/or leaves no

space for others to get adsorbed, again leading to slow or no reaction at all.

Hence, a good catalyst must have moderate values for the enthalpies of adsorption

of reactants on the solid catalyst surface.

Selectivity: A good catalyst should be highly selective in nature, so as to give the

desired product in a given reaction. Shape-selective reactions involve those

reactions that depend upon the size of the reactant molecules and catalyst pore

structure. For example, use of zeolites for cracking hydrocarbons

Surface area: A good catalyst must have large surface area for the reactants to get

adsorbed on to it. To increase the exposed surface area, the catalyst is finely

divided or is commonly distributed on the surface of a porous carrier like alumina

(Al2O3), silica gel (SiO2), charcoal, etc. These carriers may be inert or may

contribute towards the catalytic activity.

There are certain substances that increase the activity of a catalyst and its lifetime. These

are called promoters. For example, iron used in Born Haber’s process contains small

amount of Al2O3, that prevents the sintering process (i.e., joining together of small iron

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crystals); formation of large crystals decreases the surface area and hence the catalytic

activity.

Contrary to promoters, there are substances, termed as poisons, which decrease the

catalytic activity by binding strongly to the catalyst. These may be present as impurities

in the reactants or may be formed as one of the by-products in the reaction. Examples of

such substances generally include compounds containing species with lone pair of

electrons (like H2S, CO, CS2, etc) or heavy metals (like lead, mercury, etc.). A very

important example is the reaction occurring in the catalytic converters of the automobiles.

This involves the use of finely divided platinum to convert carbon monoxide from the

exhaust gases to carbon dioxide. Lead free gasoline must be supplied to the cars equipped

with catalytic converters as lead acts a catalytic poison decreasing the efficiency of the

latter.

The general mechanism of fluid-phase reactions catalyzed by solid (heterogeneous)

catalysts involves the following steps:

(a) Diffusion of the reactant molecules in to the solid surface of the catalyst

(b) Chemisorption of atleast one of the reactant molecules on the solid surface

(c) Chemical reaction between the adsorbed molecules present on adjacent sites of

the solid catalyst surface, or a chemical reaction between an adsorbed molecule

and fluid-phase molecules colliding with the surface

(d) Desorption of the products from the catalyst surface

(e) Diffusion of reactant products away from the solid catalyst surface into the bulk

fluid

If the reaction involves two reactant molecules that are adsorbed on the solid catalyst

surface, migration of the adsorbed molecules on the surface may occur between steps (b)

and (c). The overall reaction mechanism is often complicated and, it is more likely that

one of the above steps is slower as compared to other. In this case, the rate of this slowest

step is considered.

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Let us consider only solid catalyzed reaction of gases, where step (c) is the slowest.

When two chemisorbed species are reacting on the solid catalyst surface in this step, the

reaction is said to occur by a Langmuir-Hinshelwood mechanism. On the other hand, if

in this step, a chemisorbed species reacts with fluid-phase species, the mechanism is

called Rideal-Eley mechanism. The former are said to be more common that the latter.

The Langmuir-Hinshelwood mechanism:

For the sake of simplicity, let us consider that step (c) consists of a single slow

(unimolecular or bimolecular) elementary reaction, followed by one or more steps. The

rate of adsorption and desorption is higher than the rate of chemical reaction of all

species; therefore the state of equilibrium between adsorption and desorption is

maintained for all species in a reaction. Under these conditions and the assumption that

the surface of solid catalyst is uniform (which actually is not true), we can use the

Langmuir isotherm.

The conversion rate of a heterogeneous catalyst per unit surface area is given by:

𝑟𝑠 = 𝐽

𝑆=

1

𝐴

1

𝜈𝐵 𝑑𝑛𝐵

𝑑𝑡 (33)

where, 𝐽 is conversion rate, 𝑆 is the catalyst surface area and 𝑛𝐵 is the stoichiometric

number of any species B in the reaction.

If the elementary reaction on the catalyst surface is unimolecular in nature:

𝐴 → 𝐶 + 𝐷

The conversion rate of a heterogeneous catalyst per unit surface area (𝑟𝑠) is proportional

to the number of adsorbed A molecules per unit surface area (𝑛𝐴

𝑆), which in turn is

proportional to the fraction of sites covered by A molecules (𝜃𝐴).

Therefore, 𝑟𝑠 = 𝑘𝜃𝐴 (34)

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Where, 𝑘 is the rate constant.

If the products C and D also compete with the adsorption sites on the solid catalyst

surface, we use a form of the Langmuir adsorption equation where more than one gas is

adsorbed on the adsorbent surface, i.e.,

𝜃𝐴 =𝐾𝐴 𝑃𝐴

1 + 𝐾𝐴 𝑃𝐴 + 𝐾𝐶 𝑃𝐶 + 𝐾𝐷 𝑃𝐷

The rate law is then given by: 𝑟𝑠 = 𝑘𝜃𝐴

𝑟𝑠 = 𝑘𝐾𝐴 𝑃𝐴

1 + 𝐾𝐴 𝑃𝐴+ 𝐾𝐶 𝑃𝐶+𝐾𝐷 𝑃𝐷 (35)

If the products are weakly adsorbed on the solid surface, i.e.,

𝐾𝐶 𝑃𝐶 + 𝐾𝐷 𝑃𝐷 ≪ 1 + 𝐾𝐴 𝑃𝐴

then the rate law becomes:

𝑟𝑠 = 𝑘𝐾𝐴 𝑃𝐴

1 + 𝐾𝐴 𝑃𝐴 (36)

Using above equation (36), we get,

𝑟𝑠 = 𝑘𝐾𝐴𝑃𝐴 at low P

𝑟𝑠 = 𝑘 at high P (37)

Thus, we find that the reaction becomes of first order at low pressure and of zero order at

high pressure. The latter can be explained by the fact at high pressure, the surface gets

completely covered by a monolayer of gas molecules, showing no further increase in the

rate with increase in pressure. Example is the decomposition of phosphine (PH3) gas in

the presence of tungsten as the catalyst, at 700oC. the reaction follows first order kinetics

below 10-2 torr and zero order above 1 torr.

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If the elementary reaction on the catalyst surface is biimolecular in nature:

𝐴 + 𝐵 → 𝐶 + 𝐷

and both the reactants are adsorbed on the catalyst surface, the rate law is given by:

𝑟𝑠 = 𝑘𝜃𝐴 𝜃𝐵 (38)

Again, using the Langmuir adsorption equation for non-dissociative adsorbed species, we

get,

𝑟𝑠 = 𝑘𝐾𝐴 𝑃𝐴 𝐾𝐵𝑃𝐵

(1 + 𝐾𝐴 𝑃𝐴+ 𝐾𝐵 𝑃𝐵+ 𝐾𝐶 𝑃𝐶+𝐾𝐷 𝑃𝐷 )2 (39)

If the reactant B is strongly adsorbed as compared to other species, then

𝐾𝐵𝑃𝐵 ≫ 1 + 𝐾𝐴 𝑃𝐴 + 𝐾𝐶 𝑃𝐶 + 𝐾𝐷 𝑃𝐷

Then 𝑟𝑠 = 𝑘𝐾𝐴 𝑃𝐴 𝐾𝐵 𝑃𝐵

(40)

In this case, the reactant B inhibits the reaction, as it is more strongly adsorbed than the

reactant A. The fraction of surface covered by molecules of A goes to zero; therefore rate

of the reaction, 𝑟𝑠 = 𝑘𝜃𝐴 𝜃𝐵 , goes to zero.

The rate of the reaction will be maximum when the two reactants are adsorbed at equal

rates.

In case of Rideal-Eley mechanism, let us consider the following bimolecular reaction:

𝐴 (𝑎𝑑𝑠) + 𝐵(𝑔) → 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠

Since one of the reactants is in the gas phase, the rate of collision of B with the catalyst

surface will be proportional to its partial pressure 𝑃𝐵. Thus, the rate:

𝑟𝑠 ∞ 𝜃𝐴 𝑃𝐵

or 𝑟𝑠 = 𝑘 𝜃𝐴 𝑃𝐵 (41)

The value of 𝜃𝐴 can be substituted to get the desired rate law.

In the cases, where the reactant molecules undergoes dissociation, appropriate value of

𝜃𝐴 (discussed in earlier modules) will have to be substituted to get the rate law.

CHEMISTRY



7. Summary

In this module, we have learnt that:

J.W. Gibbs derived a quantitative relationship between the extent of adsorption of

a solute and the respective change in the surface tension of a liquid due to its

addition

Gibbs adsorption isotherm is given by: Γ2 = −𝑐2

𝑅𝑇

𝑑𝛾

𝑑𝑐2

Based on this equation, certain substances are classified as surface active and

surface inactive materials

The surface active materials have a specific orientation in the water: polar heads

face towards towards water whereas non-polar groups pointing vertically away

from it

The 2-D ideal gas law, applicable to surface film (monolayer formation) at the

surface is 𝐹�̅� = 𝑅𝑇

Whenever two phases of dissimilar chemical composition are in contact with each

other, an electric potential difference gets developed resulting in the formation of

double layer across the interface

The phenomenon of double layer formation at the solid-liquid interface produces

several ‘electro-kinetic effects’

Many commercially important reactions make use of solid catalysts

For catalyst to be effective, an important requirement is that one or more reactants

should be chemisorbed on their solid surface

For heterogeneous catalysis, Langmuir-Hinshelwood mechanism is commonly

followed as compared to Rideal-Eley mechanism

chemistry paper no. 10: physical chemistry iii (classical ... · thermodynamics, non-equilibrium...

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