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

CHEMISTRY 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

CHEMISTRY 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

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

CHEMISTRY 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

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:

CHEMISTRY Paper No. 10: Physical Chemistry –III (Classical

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

(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:

𝑛₁^𝑜𝑑𝜇₁+ 𝑛₂^𝑜𝑑𝜇₂ = 0 (7) where, 𝑛₁^𝑜 and 𝑛₂^𝑜 are the respective amounts of solvent and solute in the bulk phase.

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

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

𝑛₁^𝑜 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,

CHEMISTRY Paper No. 10: Physical Chemistry –III (Classical

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

(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:

Γ

₂

=

⁻¹

𝑅𝑇

𝑑𝛾 𝑑 ln𝑐₂

𝑐^𝑜

⁄

Γ

₂

=

^−𝑐2

𝑅𝑇 𝑑𝛾

𝑑𝑐₂ (16)

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

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

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 (lim

𝑐₂→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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Module No. 25: Gibbs Adsorption equation, surface activity and surface films

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

(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),

Γ

₂

=

^−𝑐2

𝑅𝑇 𝑑𝛾 𝑑𝑐₂

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or

𝑑𝛾 = −𝑅𝑇 Γ

₂^𝑑𝑐2

𝑐₂

⁽²⁴⁾

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

Γ

₂

∝ 𝑐

₂

or

Γ

₂

= 𝐾 𝑐

₂

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

𝑑𝛾 = −𝑅𝑇 𝐾 𝑑𝑐

₂

(26) Integrating equation (26), we get

𝛾 − 𝛾^∗

= −𝑅𝑇 𝐾 𝑐

₂

(27)

𝛾 − 𝛾^∗

= −𝑅𝑇 Γ

₂

(28)

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

𝐹

= 𝑅𝑇 Γ

₂

(29)

The surface excess concentration (Γ₂) is also equal to number of moles (𝑛₂^𝜎) of the fim forming substance per unit area (𝐴), i.e.,

Γ

₂

=

^𝑛_𝐴^2𝜎

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

𝐹𝐴

=

𝑛₂^𝜎

𝑅𝑇

(31)

or 𝐹𝐴̅

= 𝑅𝑇

(32)

where, 𝐴̅ 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:

2𝑆𝑂_{2 (𝑔)} + 𝑂_{2 (𝑔)} 𝑁𝑂

⇒ 2 𝑆𝑂_{3 (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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𝑁_{2 (𝑔)} + 3𝐻_{2 (𝑔)} 𝑓𝑖𝑛𝑒𝑙𝑦 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑁𝑖

⇒ 2 𝑁𝐻_{3 (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 (|∆𝐻̅̅̅̅̅̅|). If the latter is low, it would mean that the extent _𝑎𝑑𝑠 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:

𝑟

_𝑠

=

^𝐽

𝑆

=

¹

𝐴 1 𝜈_𝐵

𝑑𝑛_𝐵

𝑑𝑡

(33)

where, 𝐽 is conversion rate, 𝑆 is the catalyst surface area and 𝑛_𝐵 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 (𝑟_𝑠) 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 (𝜃_𝐴).

Therefore,

𝑟

_𝑠

= 𝑘𝜃

_𝐴

(34)

CHEMISTRY 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

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.,

𝜃_𝐴 = 𝐾_𝐴 𝑃_𝐴

1 + 𝐾_𝐴 𝑃_𝐴 + 𝐾_𝐶 𝑃_𝐶+ 𝐾_𝐷 𝑃_𝐷

The rate law is then given by: 𝑟_𝑠 = 𝑘𝜃_𝐴

𝑟

_𝑠

= 𝑘

^{𝐾𝐴 𝑃𝐴}

1 + 𝐾_𝐴 𝑃_𝐴+ 𝐾_𝐶 𝑃_𝐶+𝐾_𝐷 𝑃_𝐷

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

𝐾_𝐶 𝑃_𝐶+ 𝐾_𝐷 𝑃_𝐷 ≪ 1 + 𝐾_𝐴 𝑃_𝐴 then the rate law becomes:

𝑟_𝑠 = 𝑘 ^{𝐾𝐴 𝑃𝐴}

1 + 𝐾_𝐴 𝑃_𝐴

(36)

Using above equation (36), we get,

𝑟_𝑠 = 𝑘𝐾_𝐴𝑃_𝐴 at low P

𝑟_𝑠 = 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 700^oC. the reaction follows first order kinetics below 10^-2 torr and zero order above 1 torr.

CHEMISTRY 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

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:

𝑟

_𝑠

= 𝑘𝜃

_𝐴

𝜃

_𝐵

(38)

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

𝑟

_𝑠

= 𝑘

^{𝐾𝐴 𝑃𝐴 𝐾𝐵𝑃𝐵}

(1 + 𝐾_𝐴 𝑃_𝐴+ 𝐾_𝐵 𝑃_𝐵+ 𝐾_𝐶 𝑃_𝐶+𝐾_𝐷 𝑃_𝐷 )²

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

𝐾_𝐵𝑃_𝐵 ≫ 1 + 𝐾_𝐴 𝑃_𝐴+ 𝐾_𝐶 𝑃_𝐶+ 𝐾_𝐷 𝑃_𝐷

Then 𝑟_𝑠 = 𝑘 _𝐾^{𝐾𝐴 𝑃𝐴}

𝐵 𝑃_𝐵

(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, 𝑟_𝑠 = 𝑘𝜃_𝐴 𝜃_𝐵, 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 𝑃_𝐵. Thus, the rate:

𝑟_𝑠 ∞ 𝜃_𝐴 𝑃_𝐵

or

𝑟

_𝑠

= 𝑘 𝜃

_𝐴

𝑃

_𝐵

(41)

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

In the cases, where the reactant molecules undergoes dissociation, appropriate value of 𝜃_𝐴 (discussed in earlier modules) will have to be substituted to get the rate law.

CHEMISTRY 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

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 𝐹𝐴̅

= 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