Irreversible Electrochemistry Past Paper PDF 2024

Faculty of science Chemistry Department Irreversible Electrochemistry C 352 Collected and Prepared by Dr. Ghada Esmail  Content Chapter 1: Nature of Electrolyte…………………………………….….2 Chapter 2: Structure of Electrical of Double Layer………….31 Chapter 3: Char...

Faculty of science Chemistry Department Irreversible Electrochemistry C 352 Collected and Prepared by Dr. Ghada Esmail  Content Chapter 1: Nature of Electrolyte…………………………………….….2 Chapter 2: Structure of Electrical of Double Layer………….31 Chapter 3: Charge Transfer & Electrode Kinetics………..….54  Physical Chemistry Chapter one Nature of Electrolyte Solutions Introduction Electrochemistry is the branch of physical chemistry which concerned with physical and chemical properties of ionic conductors (electrolyte) as well as with phenomena occurring at the interfaces between ionic conductors and electronic conductors (electrodes). Electrochemistry was divided into different aspects, these are A- Ionics: which concerns with ions in bulk solutions b- Electronics: which concerns with the region between ionic conductors and electronic conductors and the electric charges across it. It’s important to understand the behavior of ions in solution and liquids arising from solids composed into ions before studying of the processes occur at electrode/solution interface. So the nature of electrolyte solution will be discussed in this chapter 2  Physical Chemistry Electrolyte: Electrolyte can be defined as a substance that dissociates into ions in solution and conducts electric current. These solutions conduct electricity due to the mobility of the positive ions (cations) towards cathode and negative ions (anions) towards anode. There are two types of electrolytes 1-Strong electrolyte: It is the substance that completely ionized when dissolved in a polar solvent like water, For example, NaCl, HNO3, HClO3, CaCl2, KCl, NaCl → Na+ + Cal- CaCl2 → Ca2+ + 2Cl- 2-Weak electrolyte: It is the substance that incompletely (partial) ionized when dissolved in a polar solvent like water, For example, NH4OH, CH3COOH, H2CO3 and most organic acids. There is equilibrium between ions and unionized molecules. CH3COOH⇌ CH3COO-+ H+ Electrical conductivity of electrolyte solutions Movement of ions in solution can be studied by using a pair of electrodes into the electrolyte and by introducing a potential difference between the electrodes. Resistance of electrolyte (R) Resistance of conductor refers to the opposition to the flow of current. Like metallic conducting materials, electrolyte solutions follow Ohm’s law 𝑉 𝑅= 𝐼 3  Physical Chemistry Where R is the resistance (Ω, “ohms”), V is the potential difference (V, “Volts”), and I is the current (A, “Amperes”). The resistance “R” for any electrical conductor is directly proportional to its length (L) and inversely proportional to its cross-sectional area (A) 1 RαL and Rα A ρL R= (I. 1) A Where is ρ is proportionality constant called specific resistance or resistivity Specific resistance or resistivity (ρ) It’s defined as the resistance of a conductor has 1 cm length and cross sectional area 1 cm2. Its unit is (Ω.m or Ω.cm). Resistivity it is characteristic of the material. RA ρ= L Conductance (G) The conductance is defined as a measure of the ability of conductor to flow an electric current. It is reciprocal of the resistance. The Unit of conductance is (Ω -1 or mho or Siemens (S)), [as 1 ohm-1 = 1 Siemens] 1 G= R By substituting, R from equation (1) A A G= = ϰ ρL L 1 ϰ= ρ Where: ϰ is the specific conductance or conductivity 4  Physical Chemistry Specific conductance or conductivity (ϰ) It is the conductance of a conductor has 1 cm length and cross sectional area 1 cm2. In other words, it is the conductance of 1 cm3 of a conductor or electrolyte. The specific conductance is the reciprocal of specific resistance and denoted by the symbol ϰ (kappa). GL ϰ= A L ϰ= RA Its unit is (Ω-1.m-1 or Ω-1.cm-1 or S.m-1) Molar Conductance The molar conductance is defined as the conductance of that volume of solution containing one mole of an electrolyte. It is denoted by μ or Ʌm. Consider the solution containing one mole of an electrolyte has volume V cm3, its conductance in this case equal molar conductance μ. Also we know the conductance of a solution occupying 1 cm3 volume called specific conductance ϰ μ …………….. V cm3 ϰ …………….. 1 cm3 So molar conductance equal μ = ϰ ×V The molarity of solution is given by this equation n mol M= × 1000 = V (cm)3 𝑐𝑚3 As n: number of moles, n = 1 Hence: 5  Physical Chemistry 1000 μ= ϰ × M Ω−1. cm−1 μ= × 1000 = mol. cm3 The units of molar conductance are (cm2. Ω-1.mol-1 or m2.S.mol-1) Equivalent conductance It is defined as the conductance of that volume of solution containing one gram equivalent of an electrolyte. It is denoted by Ʌ (lambda). Also, it is related to specific conductance (ϰ) by the following equation 1000 Ʌ= ϰ ×V or Ʌ= ϰ × 𝑁 Where: N is normality of solution. Its units are (cm2. Ω-1. equiv-1 or m2.S. equiv -1) The relation between equivalent conductance and molar conductance can be given by the following equation μ = Ʌ × feq where: feq is equivalent factor of electrolyte. It is usually the total charge on either anions or cations. It may be equal to basicity in case of acids or equal to acidity in case bases. Conductance measurements The conductivity of electrolyte solution is measured by conductance cell. This type of cell is formed by two platinum electrodes have a certain area separated by a fixed distance as figure (1). AC voltage is applied through the cell and the electric current flow through the solution and measure the resistance of electrolyte. The 6  Physical Chemistry applying of alternating voltage (AC voltage) instead of direct voltage (DC voltage) to prevent electrolysis of the water. The conductivity of any electrolyte depends on magnitude called cell constant (c) Cell constant is defined as the ratio of the distance between the electrodes, L, to the electrode area, A and has units cm-1 L c= A It is related to specific conductance by the following equation If GL L ϰ= or ϰ= A RA Hence ϰ 𝑐= 𝑜𝑟 𝑐 = ϰ × R G Fig (1): conductance cell 7  Physical Chemistry Example 1: 1.0 N solution of a salt surrounding two pt electrodes which have area 4.2 cm2 and the distance between them is 2.1 cm, if the resistance of the cell containing this solution is 50 Ω. Calculate the equivalent conductivity of this solution Solution 1- We calculate cell constant L 2.1 c= = = 0.5 𝑐𝑚−1 A 4.2 2- Calculate the specific conductance c= ϰ×R c 0.5 ϰ= = = 0.01 Ω−1. cm−1 R 50 3- Calculate the equivalent conductance 1000 Ʌ= ϰ × N 1000 Ʌ = 0.01 × = 10 cm2 Ω−1. equiv −1 1.0 Example 2: The equivalent conductivity of a solution containing 2.54 g of CuSO4/L is 91.0 cm2. Ω-1. equiv-1. What is the resistance of a cm3 of this solution when placed between two electrodes 1.00 cm apart, each having an area of 1.00 cm2. Solution: 1- Calculate normality of this solution 𝑤𝑡 1 N= × 𝑒𝑞. 𝑤𝑡 V(L) 2.54 1 N= × ×2 159.5 1 8  Physical Chemistry N = 0.0318 𝑁 2- Calculate the specific conductance 1000 Ʌ= ϰ × N Ʌ×N ϰ= 1000 91.0 × 0.0318 ϰ= = 2.89 × 10−3 Ω−1. cm−1 1000 3- Calculate the resistance L R= ϰA 1 R= = 2.89 × 10−3 × 1 R = 1/ 2.89×10-3×1= 346 Ω Example 3: A certain solution of NaCl salt has molar conductance of 123.2 cm2. Ω-1. mol-1. If this solution placed on a cell which consisted of two platinum electrodes which area 5.0 cm2 and the distance between them is 2.5 cm, give a resistance 202.9 Ω. calculate the specific conductance of this solution then calculate the concentration of this solution. Solution Firstly, calculate cell constant L 2.5 c= = = 0.5 𝑐𝑚−1 A 5 Then, calculate the specific conductance c= ϰ×R 9  Physical Chemistry c 0.5 ϰ= = = 0.0025 Ω−1. cm−1 R 202.9 Finally, calculate the concentration from this equation 1000 μ= ϰ × M 1000 so M= ϰ × μ 1000 M = 0.0025 × = 0.02 𝑀 123.2 Factors Affecting the Electrolytic Conductance In general, conductance of an electrolyte depends upon the following factors, (1) Nature of electrolyte: The conductance of an electrolyte depends upon the number of ions present in the solution. Therefore, the strong electrolytes dissociate almost completely into ions in solutions so their solutions have high conductance. On the other hand, weak electrolytes dissociate incompletely and give less number of ions. Therefore, the solutions of weak electrolytes have low conductance. (2)Temperature: The conductivity of an electrolyte depends upon the temperature. With increase in temperature, the conductivity of an electrolyte increases. (3) Ionic size & mobility: The ionic mobility decreases with increase in its size and hence conductivity also decreases. E.g. In molten state, the conductivities of lithium salts are greater than those of cesium salts since the size of Li+ ion is smaller than that of Cs+ ion. 10  Physical Chemistry However, in aqueous solutions the extent of hydration affects the mobility of the ion, which in turn affects the conductivity. Heavily hydrated ions show low conductance values due to larger size. E.g. In aqueous solutions Li+ ion with high charge density is heavily hydrated than Cs+ ion with low charge density. Hence hydrated Li+ bigger than hydrated Cs+. As a result, lithium salts show lower conductivities compared to those of cesium salts in water. Greater the size of the ions or greater is the solvation of the ions, less the conductance. (4) Nature of the solvent: The conductance of electrolytes depend on properties of solvents such as viscosity and dielectric constant 11  Physical Chemistry a- Viscosity of solvent The ionic mobility is reduced in more viscous solvents. Hence the conductivity decreases. b- Dielectric constant Generally, the dielectric constant of the solvent provides a rough measure of a solvent's polarity. The strong polarity of water is indicated by its high dielectric constant, the exact value at 25 °C is 78.6. It is denoted by D or ε According to Coulomb's law, the force between two charged particles is directly proportional to the product of the two charges, and inversely proportional to the square of the distance between them: 𝑞1 𝑞2 𝐹𝛼 𝑟2 𝑞1 𝑞2 𝐹= 𝜀𝑟 2 Where: F is attraction force between 2 charges q1, q2 are charges in coulombs r distance between 2 charges ε is the proportionality constant which called dielectric constant of medium Solvents have high dielectric constant increase the ionization of electrolyte hence; increase the conductance as it decreases the attraction force between different ions while solvents with low dielectric constant increase the attraction force between anions and cations and formed unionized molecules hence, decrease the conductance 12  Physical Chemistry (5) Concentration The specific conductance and equivalent conductance influenced by the concentration of electrolyte a- Effect of concentration on specific conductance the specific conductance (ϰ) depends on the number of ions per unit volume of the solution. Since on dilution the degree of dissociation increases but the number of ions per unit volume decreases, therefore it is expected that the specific conductance of solution decrease upon dilution (decreasing in concentration) b- Effect of concentration on equivalent conductance Both the equivalent conductivity and molar conductance increase upon dilution since the extent of ionization increases. This increase is due to the fact that molar conductance is the product of specific conductance (ϰ) and the volume (V) of the solution containing 1 mole of electrolyte. As the decreasing value of specific conductance is compensated by the increasing value of (V) thus, the value of molar conductance (Ʌ𝑚 ) will increase with dilution. Figure 2 shows that as the concentration decrease or dilution increase, the equivalent conductance increases and reaches to maximum value at certain dilution and does not change upon further dilution. This concentration is termed as infinite dilution and the equivalent conductance at infinite dilution is known as the limiting equivalent conductance Λ∞. At this infinite dilution the ionization of even weak electrolyte is complete. 13  Physical Chemistry Fig 2: variation of equivalent conductance with dilution Theories of ionization The first quantitative theory of electrolytes was suggested by Arrhenius in period 1883-1887. Arrhenius supposed the following postulates 1. the molecules of an electrolyte in solution breaks up into two types of charged particles called ions, this process is known as ionization. The positively charged ion is called cation while negatively charged ion is called anion. For example: When NaCl is dissolved in water, it breaks up into Na+ and Cl– ion. 𝐼𝑜𝑛𝑖𝑧𝑎𝑡𝑖𝑜𝑛 NaCl → Na+ + Cl- 2. The total charge on the cation is equal but opposite to that on the anion so that the solution as a whole is electrically neutral. 3. The process of ionization is reversible. The ions present in a solution are constantly reuniting and the molecules are dissociating. Thus, there is a state of 14  Physical Chemistry equilibrium between the dissociated ions and unionized molecules. Such equilibrium is called ionic equilibrium. For example: AB ⇌ A+ + B- 4. Electrical conductivity of electrolytes is due to the migration of ions towards oppositely charged electrodes. i.e cations move towards the cathode and anions move towards the anode. 5.The properties of an electrolyte in solution are the properties of ions produced. 6. The extent of ionization is expressed in terms of the degree of ionization (α). The degree of ionization is defined as the ratio of no. of moles of ions to no. of moles of undissociated molecules. Mathematically, the degree of ionization can be expressed as: no of molecules which dissociate into ions α= Total no of molecules 𝑛 α= 𝑁 And 0 < α Br-> I- >SCN- While in case of polar aprotic solvents, solvation of anions decreases slightly in the reverse order SCN-> I- >Br->Cl-> F- 3-Ion- ion interactions Actually, the original Debye-Huckel theory (Ionic Atmosphere Theory) was a first theory which treated the ion-ion interaction theoretical Debye-Huckel theory (Ionic Atmosphere Theory) Debye-Huckel assumed that each ion is surrounded more closely by ions of opposite charge than by ions of like charge and possessing asymmetrical columbic field. Such a sphere surrounding the central ion is called ionic atmosphere or cloud as shown in the figure (5.a). Also, when the external electric field is applied 23  Physical Chemistry the ionic atmosphere is distorted away from spherical symmetry (asymmetrical ionic atmosphere) as shown in fig (5.b). Fig 5.a Fig 5.b Debye and Huckel derived an equation based on the quantitative treatment of ion- ion attraction. This equation was later on modified by Onsagar and is known as Debye-Huckel-Onsagar (DHO) equation for strong electrolyte. It shows how the value of equivalent conductance at infinite dilution of strong electrolytes is much less than unity due to following effects: (i) Relaxation effect When an EMF is applied for a central ion of negative charge, it migrates towards the anode where the ionic atmosphere of positive ions is left behind to disperse, at this time a new ionic atmosphere is under formation. The rate of formation of new ionic atmosphere is not the same at which the previous ionic atmosphere disperses and the later takes more time. This time is called the ‘relaxation time’. This effect will decrease the mobility of the ions and is known as ‘relaxation effect (ii) Electrophoretic effect The solvent molecules attach themselves to ionic atmosphere and the ions move in the direction opposite to that of central ion. It produces friction due to which the mobility of the central ion is retarded. This effect is called the electrophoretic effect. 24  Physical Chemistry Debye-Huckel Onsager (DHO) was able to derive a theoretical expression to account for the empirical relation known as Kohlrausch's equation Ʌc= Ʌ∞- S √C (4) Where: S is Onsager slope, S = (A+B Ʌ∞) Where: A represents the electrophoretic effect while B represents relaxation effect, and two constants depend on temperature and nature of solvent (di-electric constant) Limitation of Debye-Huckel-Onsagar Equation Since the plotting of Ʌc against C1/2 from equation (4) give linear with negative slope and positive intercept for strong electrolyte. However, it has observed that the equation (4) is followed only at low and moderately concentration as shown in figure (6). It can be clearly seen that the theoretical and experimental move away as the concentration increases. This is simply because some approximation used to derive are DHO equation is not valid. 25  Physical Chemistry Fig (6): comparison between theorical and experimental equivalent conductance for some electrolyte Activity and Activity coefficient The practical approach to the problem of non-ideality of electrolytes, Debye- Huckel introduce a quantity known as the activity (a) which used to measure the "effective concentration" of a species in electrolyte solutions. The activity (a) is related to the concentration of electrolyte such as molarity (M) or molality (m), given by a=γC Where: γ is the activity coefficient which responsible for deviation from ideal behavior. This factor takes into account the ion-ion interaction. At infinite dilution γ →1 and a→ m, the solution approaches to ideality The activity of cation and anion are given by general equations a+ = γ+ C+ a- = γ- C- Where: a+and a- are the activities of the cation and anion respectively. γ+ and γ- are the activity coefficient for cation and anion respectively. C+ and C- are the concentration for cation and anion respectively Because the effects of the positive and negative ions are difficult to separate, we often define the mean ionic activity a± of electrolyte, which equal to a± = γ ± m± As m± mean ionic molality and γ ± is mean ionic activity coefficient For a 1:1 electrolyte, such as NaCl it is given by the following: a± =(a+ a-)1/2 26  Physical Chemistry m± = (m+ m-)1/2 the mean activity coefficient can be expressed theoretical by the following equation γ ± =( γ+ γ-) 1/2 More generally, the mean activity coefficient of an electrolyte of formula AxBy is expressed theoretically by m ± =( m+x m-y) 1/x+y γ ± =( γ+x γ-y) 1/x+y example: the mean activity coefficient for FeCl3 can be expressed as following γ ± =[(γFe+) (γCl-)3] ¼ Ionic strength Ionic strength is the total ion concentration in solution. Knowing ionic strength is important to chemists because ions have an electrical charge that attract or repel against each other. This attraction and repulsion causes ions to behave in certain ways. Basically ionic strength represents interactions between the ions in water and the ions of a solution and it is defined as half the sum of the terms obtained by multiplying of concentration of each ion present in the solution by the square of its valence. I = ½ ƩCizi2 Where: I is ionic strength, can be molar (mol/L) or molal (mol/kg) Ci is the molar concentration of ion i (mol/L or mol/kg) zi is the charge number of that ion For 1,1 electrolyte such as NaCl or KCl, if the concentration is m, the ionic strength equal I = ½ [m×(+1)2+ m×(-1)2] = m 27  Physical Chemistry For 1,2 electrolyte such as CaCl2, the ionic strength equal I = ½ [m×(+2)2+2 m×(-1)2] = 3m For 2,2 electrolyte such as MgSO4, the ionic strength equal I = ½ [m×(+2)2+ m×(-2)2] = 4m Generally multivalent ions contribute strongly to the ionic strength. Debye-Huckel was able to determine the mean activity coefficient theoretical, as he related the mean activity coefficient of electrolyte to its ionic strength as following log γ ± = −𝐴 |z+ z− |√𝐼 (D.H.L.L) Where: A=0.509, it is constant depend on nature of solvent and temperature This equation called Debye-Huckel limiting low (D.H.L.L). Debye-Huckel limiting low is excellent agreement with experimental value of up to an ionic strength of about 0.1 M, but there is a large deviation from experimental for electrolyte solution at high ionic strength. Debye-Huckel can be formulating the following expression for appreciable concentration more than 0.1M −𝐴 |z+ z− |√𝐼 log γ ± = (D.H.E.L) 1+𝐵 √𝐼 Where: B is constant and equal to 1.6 at 25 °C This equation called Debye-Huckel Extended low (D.H.E.L) Problems 1-Calculate the ionic strength for the following A- 0.05 M CdSO4 B- a solution containing of 0.01M NaCl, 0.01M Na2SO4 and 0.01M CuSO4 28  Physical Chemistry Solution A- CdSO4 is (2:2) electrolyte CdSO4 → Cd2+ + SO42- C C C I = ½ [C×(+2)2+ C×(-2)2] = 4 C= 4× 0.05= 0.20 M B- As three electrolytes have the same concentration NaCl → Na+ + Cl- C C C 3C Na2SO4 → 2 Na+ + SO42- C 2C C CuSO4 → Cu2+ + SO42- 2C C C C I = ½ [CNa+×(+1)2+ CCl-×(-1)2+ CCu2+×(+2)2 + CSO42-+×(-2)2] = I = ½ [3C×1+ C×1+ C×4 +2C×4] I = ½ [3C+ C+ 4C +8C]=8C I = 8× 0.01= 0.08 M 2-Using (D.H.L.L) and (D.H.E.L) find mean activity coefficient for 0.1 M KCl, if the constant A= 0.512, B=1.18 at 25 °C then comment on the results Solution Since KCl is (1,1) electrolyte, then I =C= 0.1 M Using D.H.L.L log γ ± = −𝐴 |z+ z− |√𝐼 log γ ± = −0.512 |(+1)(−1)|√0.1 log γ ± = −0.162 29  Physical Chemistry γ± = 0.689 −𝐴 |z+ z− |√𝐼 (D.H.E.L) log γ ± = 1+𝐵 √𝐼 −0.162 log γ ± = 1 + 1.18 √0.1 log γ± = -0.1180 γ± = 0.762 Compare between two values 3- calculate the mean ionic activity coefficient and mean activity for 0.05 m solution of CaCl2 Solution CaCl2 → Ca2+ + 2 Cl- m m 2m I = ½ [m×(+2)2+ 2m×(-1)2] = 3 m= 3× 0.05= 0.15 m Using D.H.L.L log γ ± = −𝐴 |z+ z− |√𝐼 log γ ± = −0.512 |(+2)(−1)|√0.1 γ± = 0.40 m± =[ (m+)x (m-)y] 1/x+y m± =[ (0.05) (0.1)2] 1/3 m ± =0.079 a± = m± γ± a± = 0.079×0.40 a± =0.0316 30  Physical Chemistry CHAPTER II Structure of Electrical Double Layer Electrical double layer, EDL is a structure that appears at the electrode/electrolyte interface when it is exposed to an electrolyte, the ions will be distributed near the metal surface as shown in figure (5). Fig 5: formation of electric double layer There are different cases for the formation of electric double layer First case When the metal immersed in a solution containing its ions. E.g., Ag+/ AgNO3. The structure of EDL depends on chemical potential of ions at solid state µim and chemical potential of ions at in solution µis: a- If µim > µis (µAg > µAgNO3) Oxidation of Ag metal occurs and the metal dissolves in solution and become carrying excess –ve charges, then attract Ag+ ions from solution Figure (6). Ag → Ag++ e- 31  Physical Chemistry Fig (6) b- If µis > µim (µAgNO3 > µAg) Reduction of Ag+ ion occurs and the ions in solution tend to deposit on the metal surface, hence the metal will have excess +ve charges, and then attract NO3- ions from solution Figure (7). Ag+ + e- → Ag Fig (7) c- If µis = µim (µAgNO3 = µAg) No ionic transfer and the surface charge equal zero, hence no EDL formed Note: Oxidation or reduction of metal depends on its position in the electrochemical series 32  Physical Chemistry Second case When a metal such as mercury in contact with electrolyte under applied external source of potential E.g. Hg/ deaerated KCl, by appling a potential within a certain range of potential to avoid any electrochemical reactions at electrode surface. Supposed Hg electrode has –ve charges, the K+ ions electrostatically attracted to its surface. Third case When a metallic surface has no charge in contact with electrolyte. Under the influence of binding force (covalent or Vander Waal) some ions make specific adsorption on the metal surface. e.g. Hg/ KI, I- make adsorption on Hg metal by Vander Waal forces then attracted K+ ions form solution and EDL figure (8). Fig(8) Note: -Electrical double layer may be also formed due to adsorption of dipolar molecules on metal surface such as hexyl alcohol 33  Physical Chemistry -All these formation of EDL are accompanied by changes in concentration of solution in electrode region. Structure of Electrical Double Layer, EDL Electrocapillary phenomena: This technique used for studying the electrical double layer at mercury- solution interface. For liquid metals such as Hg, the interfacial tension is a property which easily measured and related to potential difference across the interface and the surface excess of various species in solution. Fig 9: schematic apparatus for the measurement of the surface tension of mercury as a function of cell potential (Lippmann apparatus) Lippmann apparatus consisted of four essential parts. These are 1- Polarizable electrode which defined as the electrode that its potential changes with changing in the electro motive force (e.m.f) such as Hg electrode 34  Physical Chemistry 2- Non Polarizable electrode which defined as the electrode that its potential does not change upon connecting by external source of (e.m.f) such as SHE standard H2 electrode or SCE calomal electrode 3- An external source of variable potential 4- Capillary tube containing Hg and contact with HCl solution to determine the surface tension of Hg as a function of of potential difference across the interface Figure (10). Fig(10) Hg column in the capillary When the Hg electrode coupled with H2 electrode, so the change in the applied potential equal to the change in potential only at the polarizable interface (Hg/HCl), The surface tension relates to surface excess of species at the interface that represent the composition or the structure of the interface. There are three probabilities for the interface Figure (11) 1- If the Hg surface charged with (+ve) so the Cl- will be attracted towards it and forming EDL. By decreasing the potential, the repulsive ions on the surface of electrolyte decrease causing increase in the surface tension. 2- Decreasing the potential, Hg surface neutralize and become having no charge so the surface tension reach to the electrocapillary maximum. 35  Physical Chemistry 3- By decreasing the potential, Hg surface charged with (-ve) hence the H+ ions attracted to surface which cause decreasing in surface tension again Fig (11) There are 2 opposing forces affect on mercury column 1-force affecting downward by weight force of Hg column = πr2h ρ g 2- force affecting upward by surface tension force = 2 πr γ Cos θ Where: g: gravity of acceleration ρ: density of mercury r: radius of capillary γ: surface tension of Hg The contact angle, θ≈ 0 Cos θ = 1 The potential is adjusted, then the mercury,s reservior is moved upward or downward until the interface between Hg and electrolyte is fixed At equilibrium force downward = force upward πr2 h ρ g = 2 πr γ γ = h g ρ r/2 g, ρ and r are constants so γαh the surface tension γ is directly proportional to the height of Hg column. 36  Physical Chemistry Symmetric parabola Asymmetric parabola Fig (12): Electrocapillary curves γ vs V There are two types of electrocapillary curve as shown in Figure (12), these types are symmetric parabola (perfect parabolic) and asymmetric parabola (non-perfect parabolic curves). Most of solutions have asymmetric parabola electrocapillary curves except a few of solutions. The potential at which the surface tension is maximum (electrocapillary maximum e.c.m) is called the point of zero charge. Some Basic Facts about Electrocapillary Curves The measurement of interfacial tension as a function of applied potential cross the interface related by basic electrocapillary equation dγ = - qM dV- RT гd ln C (II.1) Where: qM: charge density on electrode dV: potential difference R: universal gas constant T: absolute temperature C: concentration of electrolyte Г: surface excess which refers to the concentration of ions at electrode surface 37  Physical Chemistry Determination of the charge density on the metal surface: For one electrolyte solution of fixed composition, d ln C=0 Hence the equation (II.1) becomes dγ = - qM dV (II.2) Lippmann - qM = dγ/ dV (II.3) equation The excess of electric charges on the Hg surface can be determine from the slope of the electrocapillary curves, figure (12) at various value of cell potential (differentiate this curve) and plot these values of the slope as a function of cell potential. If the electrocapillary curve was symmetric parabola, then the charge on the electrode would vary linearly with the potential however the charge for asymmetric parabola vary nonlinear, figure (13) Fig (13): Electrocapillary curves qM vs V 38  Physical Chemistry Determination of the capacitance of EDL at the metal /solution surface: at the metal/solution surface, the charges are accumulated or depleted relative to the bulk of the electrolyte; it can be considered a system capable of storing charge. But the ability to store charge is the characteristic property of an electric capacitor. Hence, one can discuss the capacitance of an electrified interface in way similar to that with a condenser. the capacitance of a condenser is given by the total charge (q) required to rise the potential difference across the condenser by 1.0 volt. cdl = dq/dV Where cdl: capacity of electrical capacitors which is constant and independent of the potential. As - qM = dγ/ dV So cdl = - d2γ/ dV2 The slope of electrocapillary curves qM vs cell potential V as shown in figure (13) equal to the value of the differential capacity of double layer. For the symmetric parabolic γ versus V curve, which yields a linear q versus V curve, one obtains a constant capacitance (Fig. 14). 39  Physical Chemistry Fig (14): Electrocapillary curves cdl vs V The Potential of Zero Charge (Epzc) It is the potential difference at which the charge on the electrode is zero and it corresponds to electrocapillary maximum (e.c.m). Its symbol is Epzc. qM = -(dγmax/ dV) = zero Hence, the pzc is the potential at which the e.c.m occurs and it is determined for liquid metals by making electrocapillary measurements (Fig.12). For solid metals Epzc cannot be determined from electrocapillary measurements but by measuring the capacitance of EDL from impedance measurements at different potential and Epzc is taken as the potential at which there minimum cdl The potential of zero charge has several applications in electrochemistry process like electroplating and corrosion process. 40  Physical Chemistry Theories of Electrical Double Layer The Helmholtz- Perrin Theory (The Parallel-Plate Condenser Model) Helmholtz and Perrin consider that the electrified interface consists of two sheets of charge, one on the electrode and the other in the solution. Hence, the term double layer. The charge densities on the two sheets are equal in magnitude but opposite in sign, exactly as in a parallel-plate capacitor (Figure 15). Fig (15): Helmholtz model As this postulate, the EDL as a simple capacitor consists of two parallel planes of charges separated by a medium with dielectric constant. Hence the electrostatic theory of a capacitor can be used for EDL V= (4πd/ε) q q= (ε/4πd) V Where: d is the distance between 2 plates of a capacitor ε is the dielectric constant of medium between the plates. 41  Physical Chemistry Based on this parallel-plate model of the double layer, one has dV= (4πd/ε) dqM (II.4) From Lippmann equation - qM = dγ/ dV ∫ dγ= -∫ qM dV Inserting this expression for dV in equation (II.4), ∫ dγ= -4πd/ε ∫qM dqM γ+ const = (-4πd/ε) ½ qM2 Now, when qM = 0 at the pzc, γ= γmax const. = - γmax Hence γ = γmax – (4πd/ε) ½ qM2 γ = γmax – (ε/4πd) ½ V2 (II.5) The equation II.5 is equation for a parabola symmetrical about γmax, i.e., about the e.c.m. It appears that the Helmholtz-Perrin model would be quite satisfactory for electrocapillary curves which are perfect parabolas. Deviation from parabola symmetrical The deviation from parabola symmetrical is greater than with some solution than others. Electrocapillary curves show, for instance, a marked sensitivity to the nature of the anions present in the electrolyte (Figure 16). In contrast, the curves do not seem to be affected significantly by the cations present, unless they are large organic cations, e.g., tetra alkyl ammonium ions. 42  Physical Chemistry A parallel-plate condenser has a specific capacity (i.e., per unit area of plates) which is given by rearranging Eq. (II.5) in the form cdl = dq/dV cdl =ε/4πd II.6 Thus, if ε and d are taken as constant, it means that the parallel-plate model predicts a constant capacity, i.e., one which does not change with potential but this is not what is observed. Fig 16: Electrocapillary curves γ Fig 17: Electrocapillary curves cdl vs V, for different anions vs V, at different concentrations Defects of Helmholtz- Perrin Theory The defects of this model can be summarized as follows 1-It’s clear from equations II.5 and II.6, that cdl and γ are independent on the concentration of electrolyte, but experimental data showed that on increasing of concentration of electrolyte, γ decrease while cdl increase, (figure 17) 43  Physical Chemistry 2- This model predicts a constant capacity with change in potential but it is not observed in experimental results. 3- It is expected that the colloidal particles as a whole be electrically neutral but experimental results showed that colloidal particles are electrically charged respect to the dispersion medium. It appears that an electrified interface does not behave like a simple double layer. The Ionic Cloud: The Gouy-Chapman Diffuse-Charge Model of the Double Layer There must be an equal amount of charge of opposite sign in the solution. Net charge = zero, however due to the presence of electrode with a certain charge. The charges with opposite sign with respect to electrode will be attracted to it while charges with similar charges will repulsive with it which known (columbic interaction) so there will be different number of (+ ve) and (-ve) ions in bulk solution, Net charge ≠ zero. There are two interactions controlling the motion of ions in solution columbic interaction and thermal energy forces. Due to these interactions Gouy-Chapman assumed that the double layer not fixed as Helmholtz was suggested but it is a diffuse double layer figure (18) 44  Physical Chemistry figure18: Gouy-Chapman Diffuse Model Ions under Thermal and Electric Forces near an Electrode Consider a lamina (in the electrolyte) parallel to the electrode and at a distance x from it (Figure 19). The charge density can be expressed in two ways, 1- According to Poisson equation, for the x dimension in rectangular coordinates ρx= (-ε /4π) (d2ψx/ dx2) II.7 Where: ψx is the outer potential difference between the lamina and the bulk of the solution (taken as " ρx = 0 at x=∞) ρx: the charge density on the unit area of lamina 2- According to Boltzmann distribution ρx =∑i ni Zi e0=∑ ni0 Zi e0 exp (- Zi e0 ψx /kT) II.8 where: ni and nio are the concentrations of the i species in the lamina and in the bulk of the solution, respectively. Zi is the valence of the species i 45  Physical Chemistry e0 is the electronic charge (- Zi e0 ψx /Kt): represents the ratio of the electrical and thermal energies of an ion at the distance x from the electrode. Fig 19: Lemina in the solution. From eqn (II.7) into eqn (II.8) we obtain the Poisson Boltzmann equation d2ψx/dx2 = - 4π/ ε =∑ ni0 Zi e0 exp (- Zi e0 ψx /kT) II.9 A mathematical treatment of equation II.9 leads to dψx/ dx = - (32π k T ni0 / ε)1/2 sinh (Zi e0 ψx /2kT) II.10 Where: dψx/dx: potential gradient at a distance x from the electrode according to the diffuse charge model of Gouy-Chapman. It is preferred to have an expression for total diffuse charge in the solution in terms of the potential. 46  Physical Chemistry The charge “q” contained in a closed surface, or Gaussian box, is equal to ε/4π times the area of the closed surface (taken here as unity) times the component of the field normal to the surface. q = ( ε /4π) (d ψ/d x ) II.11 From equation II.11, we can determine the charge enclosed q within the closed volume at surface of which of the field is d ψ/dx To compute the total diffuse-charge density ρx, the Gaussian box chosen in (Figure 20) is a rectangular box with one unit area side deep in the solution at x =∞, where " d ψ and d ψ/dx = 0, and with the other side "very close to the electrode Fig 20: Gaussian box containing the entire diffuse charge, qd If the model considered is that of point-charge ions (and this is the Gouy-Chapman model), then the box must be brought up to x = 0.0 The total diffuse-charge density scattered in the solution under the interplay of thermal and electrical forces, qd, is given from equations (II.10) and (II.11) qd = -2 (ε n0 K T/2π) 1/2 sinh (Zi e0 ψ0 /2kT) Where: II.12 47  Physical Chemistry ψ0 is the potential at x = 0 relative to the bulk of the solution when the potential is taken as zero (ψ∞ = 0). Equation (II.12) shows that the total diffuse charge varies (according to the present model) with the total potential drop in the solution. The differentiation of qd with respect to the potential ψM gives the capacity of double layer. cdl = (dqM/ dψM)= ( dqd /dψM ) = (ε Z2 e02 n0/2π kT)1/2 cosh (Z e0 ψM /kT) cosh function gives inverted parabolas. An Experimental Test of the Gouy-Chapman Model: The experimental capacity-potential curves of different concentrations of an electrolyte are not the inverted parabolas (Figure 22) which the Guoy-Chapman diffuse model predicts (Figure 21) especially at high concentration. However, at low concentration and near to pzc, the experimental capacity value is almost near to the calculated capacity from Guoy-Chapman model Finally, the Gouy-Chapman theory might be the best model which described the electrified interface. 1- It represents an important contribution to a true description of the double layer 2-it successes in explanation of the electro kinetic phenomena of colloids particles This model has some defects 1- The predicted shape of the capacity-potential curves wrong, especially at high concentration 2-Gouy Chapman theory predicts that the capacity depends on the concentration to an extent that is not at all observed. 48  Physical Chemistry 3-At high concentrations (e.g., 1M) the predicted capacity is almost an order of magnitude higher than the observed value. Fig 21: theoretical capacity-potential curves Gouy-Chapman prediction Fig 22: experimental capacity-potential curves Where: a < b < c < d in concentration 49  Physical Chemistry Stern Model: Stern combined Gouy-Chapman model and Helmholtz model and he suggests that the electrified interface consists of a) Some ions stuck at the electrode surface as compact layer (Helmholtz model) b) Scattering ions in solution due to electric and thermal forces as diffused layer (Gouy-Chapman model) as shown in Figure (23) Fig (23): Stern model Where: qM: charge density on metal surface qs: charge density on solution qH: charge density on Helmholtz layer qG: charge density on Gouy layer The interface as a whole (the electrode side taken along with the electrolyte side) is electrically neutral-the net charge density qM on the electrode must be equal in magnitude and opposite in sign to the net charge density qs on the solution side. 50  Physical Chemistry -qM = qs According to the Stern model Figur (23), the charge qs on the solution is partially stuck (the Helmholtz-Perrin charge qH) to the electrode and the remainder qG is diffusely spread out (in Gouy-Chapman style) in the solution. qs= qH + qG There are therefore two regions of charge separation. The first region is from the electrode to the Helmholtz plane (the plane defined by the locus of centers of the stuck ions) The second region is from Helmholtz plane of fixed charges into the bulk solution (Gouy-Chapman). When charges are separated, potential drops result versus distance from electrode. The Stern model implies, therefore, two potential drops, fig (24). Fig (24): the potential variation according to Stern model As shown in figure (24) the potential variation across an interface as consisting of two regions, 51  Physical Chemistry 1- A linear variation region corresponding to the ions stuck on the electrode, since the first layer contain the greatest number of charges so sharp drop in potential will be occur. 2- Exponential variation region corresponding to the ions in bulk solution which are under the combined influence of the ordering electrostatic and the disordering thermal forces. Two potential drops across electrified interface are ψM-ψB =(ψM – ψH) + (ψH – ψB) Where: ψM is the inner potential at the metal ψH is the inner potential at the Helmholtz plane ψB is the potential in the bulk of the solution. By differentiate the potential across electrified interface with respect to the charge on metal surface d(ψM-ψB)/dqM = d (ψM – ψH) /dqM + d(ψH – ψB)/dqM (II.13) In the last term of equation (II.13), we replace dqM with dqd because the total charge on the electrode is equal to the total diffuse charge d(ψM-ψB)/dqM = d (ψM – ψH) /dqM + d(ψH – ψB)/dqd (II.14) Each term is the reciprocal of a different capacity which has form (c=dq/dψ), Hence, eqn (II.14) can be rewritten as following. 1/ct = (1/cH )+ (1/cG) Where: 52  Physical Chemistry ct: is total capacity of the interface. cH: is Helmholtz- perrin capacity (the capacity of the region between the metal and Helmholtz plane to store charge). cG: is Gouy-Chapman capacity or diffuse-charge capacity. Notes: For solutions with high concentrations, cG becomes large while cH does not change and hence: cG >> cH So (1/ct) ≈ (1/cH) and ct= cH For solutions with low concentrations, cG becomes small with respect to cH Hence: cG > cH, hence ct= cH At high temperature, some ions leave surface and diffuse in bulk solution, cH >> cG, hence ct= cG 53  Physical Chemistry Chapter III Charge Transfer and Electrode Kinetics Reversible and Irreversible process. In reversible cell such as Daniel cell, Zn(s)/ZnSO4 (1 M) || CuSO4 (1 M)/Cu(s) Oxidation of zinc occurs at anode while reduction of copper occurs at cathode. At equilibrium (Rate of oxidation= rate of reduction at equilibrium) So, there is no net current (no charge transfer) The cell reaction: Zn + Cu2+ ⇌ Zn2+ + Cu (reversible reaction) Fig (25): Daniel cell If the cell connected to an outside source of E.M.F exactly equal to Ecell, no chemical reaction occurs as we in equilibrium Ecell = Ee.m.f While if Ecell > Ee.m.f by very small amount, current will flow from Daniel cell and the normal cell reaction occurs However if Ecell < Ee.m.f by very small amount, the current will flow but it flows in the reverse direction and the reverse chemical reaction occurs. 54  Physical Chemistry Zn2+ + Cu ⇌ Zn + Cu2+ These conditions required for any reversible galvanic cell so we can defined the reversible process as it is the process at which the current pass through the cell in reverse direction due to reverse chemical reaction only without much disturb the equilibrium and this happened when Ecell < Ee.m.f by very small amount. However if Ecell < Ee.m.f by large amount, the current will flow across an electrode/solution interface and the potential of the electrode will be change from reversible value to another value, so we can defined the irreversible process as it is the process at which the current flow across working electrode at any potential different than the equilibrium potential. The degree of irreversibility is measured by the deviation of electrode potential from the reversible process value under the same conditions of temperature, pressure and concentration and the irreversible (working) electrode is said to be polarized or exhibit polarization and quantified in terms of over-potential. Over-potential or polarization It is defined as “it is the change of electrode potential form reversibility to the irreversibility” OR “it is the difference between irreversible and reversible potential of electrode” and it is represented by the following equation. η = Ew - Er Where: Ew: irreversible (working) electrode potential Er : reversible electrode potential 55  Physical Chemistry η: over-potential Note: The polarization is said to be Anodic polarization: when the working potential Ew is more positive than reversible potential Er Cathodic polarization: when the working potential Ew is more negative than reversible potential Er e.g. For Zn electrode, Er = -0.7 V If Ew = -0.8 V cathodic polarization If Ew = -0.6 V anodic polarization Types of over-potential Over-potential can be divided into three different types that are not all well- defined. 1- Ohmic or Resistance over-potential (ηo): It originates in any electrochemical cell as a result of formation of an oxide film or other substances such as insoluble salts, sulfides, grease, gas bubbles or oil on the electrode surface that sets up a resistance (R), which resist the passage of current (I), the ohmic over-potential is determined from this relation. ηo= IR The ohmic over-potential depends on current density, nature and concentration of electrolyte and distance between two electrodes. To minimize to ohmic over-potential, we must be using strong electrolyte with high concentration, solvent of high dielectric constant. 56  Physical Chemistry 2- Ohmic Pseudo over-potential (ηo pseudo): This type occurs when the distance between the luggin capillary tip of the reference electrode (calomel electrode) and the electrode surface is large. Similar to ohmic over-potential, the ohmic Pseudo over-potential has high value at high current density, low concentration or large distance from capillary tip. 3- Concentration over-potential (ηc): It occurs when electrochemical reaction is sufficiently rapid to lower the concentration of ions at electrode/solution interface than the bulk solution. Therefore the electrode is effectively in a solution of different concentration from the bulk solution and thus there is difference between the potential of working electrode and corresponding reversible reference electrode. The rate of reaction is dependent on the ability of ions in bulk solution to reach the electrode surface (mass transfer). The mass transfer from bulk solution to electrode surface occurs by (diffusion, ionic migration and convection) however diffusion is usually of the greatest significance. Note: Concentration over-potential is higher in dilute solution and can be eliminated by agitation heating or stirring. 4- Activation over-potential (ηa): It is the excess potential required for electrode to reduce the activation energy of the redox reaction and making the reactants overcome the energy barrier and transfer to transition state (T.S) of the reaction. 57  Physical Chemistry It may be observed for most electrode processes. Activation over-potential is a condition occurring when the reaction rate is controlled by the slowest "step" in a series of reaction steps. For example, evolution of H2 occurs in many steps. One of these steps is slow and is the rate determining step and requires energy of activation. The rate determining step depends on the electrode potential and can be accelerated by displacing the electrode potential towards more positive or more negative value for anodic or cathodic process respectively. It is called activation over-potential because it increases the velocity of the rate determining step The total over-potential at an electrode is equals to the sum of all types of over- potential. η= ηo + ηc +ηa Measurement of Over-potential: To measure the over-potential of a polarized electrode whether cathodic or anodic polarization, two methods are used: 1- Direct method: The apparatus used shown in figure (26), there are three electrodes experimental that are used for determining the over-potential. Briefly, three electrodes are referred to as working, counter or auxiliary, and reference electrodes. When calomel electrode (reference electrode) inserted through Luggin capillary “the simple device was first suggested by Luggin” in this way more accurate potential are obtained and connected as cathode. Then Luggin capillary tip must be close to working electrode to minimize the pseudo ohmic over-potential and 58  Physical Chemistry measure the potential of the working electrode with respect to the reference electrode, reversible potential (Er) Then when the polarizing current circuit passes between working electrode and counter electrode, the potential of working electrode will be change. The potentiometer is used to measure the potential of working electrode, irreversible potential, (Ew) We can measure η = Ew - Er At high current density, pseudo ohmic over-potential (IR drop) becomes appreciable even with the use of Luggin capillary. Under these conditions, the following method is used. Fig 26: Measurement of polarized electrode potential 59  Physical Chemistry Indirect method: The tip of Luggin capillary is place at several known distance from the working electrode and series of direct potential measurements are made with current passing in the cell. As the distance between Luggin capillary tip and working electrode increase, The over-potential increase due to large value of pseudo ohmic over-potential however at zero distance, the pseudo ohmic over-potential can be neglected. This can be obtained by linear extrapolation method to zero distance as shown in figure (27) Fig (27): over-potential versus distance between 2 electrodes Decomposition potential of an electrolyte: It is defined as the minimum voltage must be applied to cause continuous electrolysis of the solution or to beginning of current flow in electrolytic cell 60  Physical Chemistry To determine the decomposition of any electrolyte, we use the electrolytic cell as shown in fig (28) which consists of two smooth inert metal electrodes such as platinum electrode immersed in an electrolyte. Fig (28): electrolytic cell used to determine decomposition potential A gradually increase in the applied potential from power supply leads to electrolyze the solution. The current passed in the cell is measured by the ammeter while the voltmeter is introduced to measure the potential difference between the two electrodes. The Plot of the current against the applied potential gives a curve as shown in fig (29). This curve shows that at beginning of applied potential, small current (residual current) is usually observed until point b. This serves to build up (charging) the electrical double layer (edl) at the electrode solution interface and reduction of impurities. After point b, the current undergoes vertical increase due to oxidation-reduction process at two electrodes surface unit reach to point c, after this point, there is no increase in current by increase of potential, so this current called (limiting current). 61  Physical Chemistry Fig (29): curve of determination of decomposition potential For most aqueous solutions of acids and bases, the decomposition potential is almost the same with approximate value of 1.7 volt. This may attributed to the fact that in these solutions, the two main electrochemical reactions are: At the cathode, evolution of hydrogen gas In acidic solutions: 2H+ + e- ⇌ H2 (g) In basic solution: 2H2O + 2e- ⇌ 2OH- + H2 (g) At the anode, evolution of oxygen gas. in acidic solutions 2H2O ⇌ O2(g) +4H+ + 4e- In basic solution: 2OH- ⇌ H2O + ½ O2 (g) + 2e- Formation of H2 or O2 not occur until the reversible hydrogen or oxygen electrode potential are exceed in the +ve direction. 62  Physical Chemistry The minimum potential necessary to start electrolysis is called reversible decomposition potential D0, However in practice the applied potential Di required maintaining certain decomposition current in electrolytic cell can be written: Di= D0+ η Di= (Eanode+ Ecathode )+ (ηa)cathode+ (ηa)anode+ IR Where: Ea, Ec: reversible anode and cathode potential (ηa)cathode, (ηa)anode: activation over-potential for anode and cathode IR : ohmic over-potential It is assumed that the electrolyte is sufficiently well stirred so the concentration over-potential is neglected. From Nernest equation, Ec (H2) = 0.0 V and Ea (O2) = 1.23 V Therefore, the potential difference about (0.47V) is over-potential which required causing polarization. Note: In the previous curve, the current from (b-c) is due diffusion only, as rate of mass transfer > rate of charge transfer, the process from (b-c) is controlled by activation. However, the current from (c-d) limiting current, the rate of charge transfer > rate of mass transfer, hence the process from (b-c) is controlled by concentration (diffusion). 63  Physical Chemistry Energy Barrier and Electrode Kinetics: Reversible electrode potential: The equilibrium at an electrode is dynamic and its ions are produced and discharged simultaneously at its surface. Taking the dissolution of a metal as an example: M⇌ M z+ + z e– The rate of ionization of metal equal to the rate of discharge of its ions. Not all metal atoms are ionized and only those metal which got a minimum free energy of activation (ΔG) to ionize. The rate of ionization depend on the height of energy barrier which must be exceeded so as to bring the atoms in the activated state, hence each metal can be ionized or not according to its energy barrier Fig (30): energy-barrier diagram for metal electrode 64  Physical Chemistry From studying the energy-barrier diagram for metal electrode figure (30), it found that, there is characteristic free energy of activation ΔG in either the forward or reverse process. The free energy of activation for the reaction equal the difference between two values ΔG= ΔG1- ΔG2 Where: ΔG1: free energy of activation for the forward reaction ΔG2: free energy of activation for the backward reaction ΔG: free energy of activation for all reaction which related to cell potential according to this equation. ΔG= - zEF (III.1) Where: F: Faraday constant z: number of electron involved in the reaction According to Maxwell’s distribution law: n = nt exp (- ΔG/RT) Where: n: number of particles with excess free energy nt: total number of particles R: gas constant T: absolute temperature By applying the Maxwell’s distribution law on the reversible reaction Rate of forward reaction = r1= k1 exp (- ΔG1/RT) Rate of backward reaction = r2= K2 exp (- ΔG2/RT) As k1, k2: rate constant of forward and backward reactions respectively at equilibrium, r1 = r2 k1 exp (- ΔG1/RT) = K2 exp (- ΔG2/RT) (III.2) By rearrangement of equation (III.2) and taking ln 65  Physical Chemistry ln (K2/ k1)= -ΔG/RT (III.3) From equation (III.1) into equation (III.3) ln (K2/ k1)= zEF /RT (III.3) Rearrange of the equation (III.3) E= RT/ zF ln (K2/ k1) This equation shows that reversible process has a specific potential at constant activity and temperature. Irreversible electrode potential: For a polarized electrode under steady-state current flow, a net reaction takes place at electrode surface. For example in an anodic polarization: Rate of ionization of metal > Rate of discharge of metal ions, hence the electrode potential deviated from the irreversible value by over-potential η. Fig (31): energy-barrier diagram for polarized electrode 66  Physical Chemistry This over-potential η has two effects as shown in figure (31). 1- Part of it helps in forward reaction (dissolution of metal) by decreasing the free energy of activation of it by amount (αηF) ΔG1ʹ = ΔG1 - αηF 2- Remainder part diminishes backward reaction (the rate of discharge of metal ions) by increasing the free energy of activation of it by amount (1-α)ηF ΔG2ʹ = ΔG2 + (1-α)ηF Where: α: fraction of over-potential assisting the dissolution of metal reaction 1- α: fraction of over-potential assisting the discharge of metal ions Hence the rate of forward equal to r1ʹ= k1 exp (- ΔG1ʹ/RT) r1ʹ= k1 exp [-(ΔG1 – αηF) /RT] r1ʹ= k1 exp(-ΔG1/RT)× exp (αηF/RT) r1ʹ= r1× exp (αηF/RT) the same for backward reaction r2ʹ= k2 exp (- ΔG2ʹ/RT) r2ʹ= k2 exp [-(ΔG2 + (1-α)ηF)/RT] r2ʹ= k2 exp(-ΔG2/RT)× exp -[(1-α)ηF/RT] r2ʹ= r2× exp – [(1-α)ηF/RT] if the rates of electrode are expressed by gram equivalent/ cm2. Sec. so we may be replace the by (i/F) terms. As i is current density i/F= (Amp/cm2)/coulmb]/ gram equivalent i/F= Amp. gram equivalent/cm2coulmb 67  Physical Chemistry rαi The rate of forward and backward will be written r1 ʹ α i 1 r2ʹ α i2 At equilibrium, r1ʹ= r1 , r2ʹ= r2 and r1= r2 Hence, i0= i1= i2 Where i0: is exchange current which defined as the current flowing a cross unit area of the electrode in each direction at reversible process (η=0). It is considered to be the start point for any process (anodic or cathodic) and it is specific value for each electrode While in case irreversible i1= i0 exp (αηF/RT) i2 = i0 exp - [(1-α)ηF/RT] For anodic polarization The rate of metal dissolution or the anodic current density equal to ia = i1 – i2 = i0 exp (αηF/RT) - i0 exp - [(1-α)ηF/RT] (III.4) The previous equation is the general equation for an anodic or cathodic process Under condition of low irreversibility, with η < 0.02 V by making approximation exp (x) ≈ (1+x) exp (-x) ≈ (1-x) eqn (III.4) becomes ia = i0 [1+(αηF/RT)] - i0 [1-((1-α)ηF/RT)] ia = i0+ i0 (αηF/RT) - i0 + i0 (ηF/RT) - i0 (αηF/RT) ia = i0 (ηF/RT) iaα η 68  Physical Chemistry Hence, the net anodic or cathodic currents are linearly related to over-potential and the linear polarization method is an electrochemical technique used for determination of the rate of corrosion of metals or alloys at low over-potential, Fig (32) Fig (32): Linear Polarization- Resistance Diagram Under condition of high irreversibility, with η > 0.05 V The rate of backward reaction is very small and can be neglected the magnitude i0 exp - [(1-α)ηF/RT] as, i1>>> i2 and the equation (III.4) becomes: ia = i0 exp (αηF/RT) By rearrangement η = (RT/αF) ln (ia/i0) η= (-2.303RT/αF) log i0 +(2.303RT/αF) log ia Generally η= a ± b log i (III.5) This equation (III.5) is straight line equation and called Tafel equation 69  Physical Chemistry Where: a=(-2.303RT/αF) log i0 b= (2.303RT/αF) The values of a and b are characteristic for each metal Tafel equation is a considered to be the principle of electro-kinetics for electrodes. As it is used to determine the rate of the corrosion of metals and alloys and concerned with the reaction mechanism of metal. Figure (32) shows Evans diagram (polarization diagram) which allows the determination of the corrosion point where both the hydrogen cathodic and the metal anodic line intercept. Tafel equation can be applied at η > 0.05 V, by simply extrapolating the linear portions of both anodic and cathodic curves until the intercept as indicated in Fig (33). Fig (33): Evan Diagram 70  Physical Chemistry Consequently, from slope value of Tafel line give information about reaction mechanism Study of Mechanism of electrode reaction: The Tafel slope value give information about RDS of the reaction mechanism. For example, for cathodic polarization, when a metal dissolves in acid solution, the cathodic reaction is represented by: Overall reaction 2H+ + 2e- ⇌ H2 It is occurred through four steps: 1- Diffusion of a proton H+ from bulk solution towards electrode. 2- Electron transfer from electrode to discharge H+ and producing Hads on the surface electrode. H+ + e- → Hads 3- Combination of Hads to give H2 gas. Hads + Hads ⇌ H2 4- Evolution of H2 gas. From calculation of Tafel slope (b), we can determine the R.D.S of reaction 1- If b = 0.12 V, the proton discharges (step 2) is R.D.S, this occurs for some metals 2- If b = 0.03 V, the Combination of Hads (step 3) is R.D.S Where: rate =k θH2, θH: surface coverage of metal by Hads This means the rate depend on adsorption of Hads on metal surface. 2- If b > 0.12 V, adsorption of organic material or sulfides on electrode surface. 71  Physical Chemistry Question (1) 1- If you know the potential of zero charge for Hg electrode at 0.05 mol/L K 2SO4 solution is -0.47 volts. Which of the following statement is true? a- γ of Hg at -0.74 V > γ of Hg at -0.47 b- γmax of Hg at 0.03 mol/L < γmax of Hg at 0.05 mol/L c- at potential -0.1 V, the metal becomes carrying –ve charge and K+ cation attracted to it d- the charge density (qM) at -0.32 V < the charge density (qM) at -0.22 V 2- The ionic strength (I) for a solution containing both 0.01 M (2:2) and 0.01 M (1:3) electrolytes a- 0.04 b- 0.07 c- 0.1 d- 0.2 3- If the electrolyte (1:3) in the previous example is (AlCl3), the mean activity coefficient for this electrolyte equal to ………………………….. 4- If you know that Er= 0.34 V for copper electrode, if you applied potential 0.21 V for this electrode. So this electrode exhibit a- anodic polarization b- cathodic polarization c- non polarized 5- For cathodic polarization of metal in acidic solution, if the Tafel slope b= 0.12 V, the reaction will be controlled by a) mass transfer b) charge transfer c) adsorption of Hads 6- For Daniel cell, If applied potential increases slightly than Ecell, the …….. occurs a) normal cell reaction a) reverse cell reaction c) no chemical reaction 7- The concentration over-potential can be eliminated by a) stirring b) dilution c) using solvent with high ε d) a and c 72  Physical Chemistry Problems 1- Calculate the mean ionic activity coefficient for solution that containing 0.01mol. kg-1 CaCl2 and 0.03 mol. kg-1 NaF. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………………..……………..……… ………………………………………………………………..…………………….. ………………………………………………………………..…………………………….. ………………………………………………………………………..…………………….. ………………………………………………………………..…………………………….. 2- Calculate γ±, and a± for a 0.03 m solution of K4 Fe (CN)6. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………………..……………..……… ………………………………………………………………..…………………….. ………………………………………………………………..…………………………….. ………………………………………………………………………..…………………….. 3- If ionic molar conductivity at infinite dilution for Ba2+ and Cl- are 127.3, 76.3 S m2 mol−1. Calculate limiting equivalent conductance of this solution. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. ………………………………………………………………………..…………………….. 73