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1 Electrolyte Solutions Review of Electrical Quantities Charge Charge is a fundamental is a fundamental characteristic of matter and (similar to mass) it is understood better for what it does (it attracts and repels) rather than what it is. Charge is measured with many different units. Modern chemists usually use three units to describe charge SI units In the SI system, the fundamental unit of charge is the Coulomb, itself defined in terms to the Ampere, the unit of electrical current. 1 C = 1 A⋅s When referring to macroscopic amount of charge, the Coulomb is usually most convenient. Atomic units When referring to microscopic amounts of charge, such as the charge of an electron or the charge of a nucleus, atomic units are often convenient (and often taken for granted). e = 1 au = 1.6022 × 10-19 C Faraday The connection between the microscopic and the macroscopic amount of charge is often made using the unit of Faraday which is the charge in Coulombs for one mole of electrons. F = eNA = (1.6022 × 10-19 C)(6.0221 × 1023 mol-1) = 96,485 C/mol = 96485 J/V⋅mol 2 Force (via Coulomb’s Law) The electrical force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. F= 1 Q1Q 2 4πε 0ε r 2 where ε0 is the permittivity of free space = 8.854 × 10-12 C2/J⋅m and is necessary to equate the mechanical definition of force (1 N = 1kg⋅m2/s2) to the electrical definition. ε is the dielectric constant of a material and is different for each material. The dielectric constant is a measure the effective attraction between charges is reduced when the charges are immersed in a material. The dielectric constant of water is very high (ε = 78) compared to that of other solvents. The strongly polar nature of the water “dilutes” the strength of a charge immersed in water. Thus attractions between charges in water are significant over smaller ranges than in other solvents. In a nutshell, water greatly affects the interaction between ions in solution. Electric Field An alternative way of considering the electrical force with the interaction between a charge and an electric field. F = Q1E Comparing this equation with the force equation from Coulomb’s law yields the expression for the electric field of a charge. E= F 1 Q1 = Q 2 4πε 0 ε r 2 The electric field has several sets of equivalent units in the SI system. Usually volts per meter are the most convenient 1N 1J 1V = = C C⋅m m 3 Energy The force between objects is found from the derivative of their potential energy with respect to distance. F=− ∂V ∂x ⇒ V = − ∫ F ⋅ dx Using electrical quantities, the energy of interaction becomes F=− ∂W ∂r ⇒ W = −∫ 1 Q1Q 2 1 Q1Q 2 ⋅ dr = 2 4πε 0ε r 4πε 0 ε r The energy of interaction is inversely proportional to the distance between the charges. (We are usually interested in energy more than force in chemical systems.) Again note how the dielectric constant reduces the energy of interaction between two charges compared to the interaction energy in a vacuum. Potential (Voltage) Just as using the electric field yields an alternative way of considering the electrical force, the potential (electrostatic potential) can be used in an alternative way of viewing the electrical interaction energy. W = Q1φ The potential can be written as φ= W 1 Q1 = Q 2 4πε 0ε r Just as there is a differential relationship between force and energy, F=− ∂W ∂r ⇒ W = − ∫ F ⋅ dr there is a differential relationship between the electric field and the potential. E=− ∂φ ⇒ φ = − ∫ E ⋅ dr ∂r The electrostatic potential can be very convenient for chemists to use when thinking about how charges are arranged with molecules (molecular geometry) and among molecules (intermolecular forces), since positive charges will move to a region with the lowest potential and negative charges will move to a region with the highest potential. 4 Molar Conductivity Definitions Ohm’s law Recall Ohm’s law, where the proportionality constant that describes a material’s relationship between voltage and current is the resistance. φ = V = IR ⇒ I = V R Conductance An alternative way of describing the current/voltage is using the conductance, G, in instead of the resistance. 1 G= ⇒ I = GV R Conductivity The conductance is an extensive property. Thus in order to make comparisons between materials, the intensive property, conductivity, κ, should be examined. The conductance of a material is directly proportional of the material’s surface area through which the current flows. Also the conductance is inversely proportional to the length through which the current flows. Thus the definition of conductivity reflects both of these dependencies. G=κ A l ⇒ κ=G l A Molar conductivity The conductivity of water is a constant value (at a given temperature). When solutes are added to the water, the conductivity of the solution may change. When the conductivity changes, the conductivity of the solution may have a concentration dependence. To better examine the nature of the solutes dissolved in a solution, the concentration dependence is controlled by dividing the conductivity by the molar concentration to yield the molar conductivity, Λ. Λ= κ c When a solute does not affect the conductivity of a solution, the solute is called a nonelectrolyte. If the solute affects the conductivity of a solution, it is an electrolyte. 5 Weak Electrolytes Svante Arrhenius in his doctoral dissertation of 1884 examined the nature of solutions by testing their conductivities. He proposed that electrolytes dissociate into ions when they dissolve in water. (He almost failed in his dissertation. Dissociated ions in solution were too controversial!) His results were odd and needed explanation. 1. Some electrolyte solutions had a relatively constant molar conductivity especially at low concentration. 2. The conductivity of other electrolyte solutions changed significantly with concentration especially at low concentration. The first attempt at explanation failed. The argument depended on considering that the movement of the electrical charge through the solution depended on the concentration. Kohlrausch’s law of independent migration of ions (1879) demonstrated that the movement (mobility) of the ions was independent of concentration (especially at low concentration). Thus the simplest explanation for the different molar conductivities was inadequate. Arrhenius Theory To explain how the molar conductivity changes with concentration, Arrhenius assumed that the electrolyte dissociated into ions. 1. All molecules fully dissociate at infinite dilution 2. The degree of dissociation, α, for an electrolyte can be predicted from its conductivity in solution. Λ0 is the molar conductivity at infinite dilution. α= Λ Λ0 0 < α <1 Ostwald’s Dilution Law Ostwald was able to use Arrhenius’ results to find a relationship between the molar conductivity of some electrolytes and their concentrations. ⎛ Λ ⎞ ⎜Λ ⎟ c ⎝ 0⎠ =K ⎛ Λ ⎞ 1− ⎜ ⎟ ⎝ Λ0 ⎠ 2 The relationship seems to be a mess; however, a key feature is that the value K was constant over large ranges of concentration. It is worth noting that Ostwald’s dilution law is useful only for weak electrolytes. It completely fails for strong electrolytes. 6 Law of Mass Action Two decades earlier (1864), Norwegian chemists, Cato Guldberg and Peter Waage, examined the concentration dependence of reversible reactions and found an expression very similar to our modern formulation of an equilibrium constant. For the reaction, A + B U C + D, such as an esterification, Guldberg and Waage put forth the following relationship (in simplified form) known as the law of mass action. (a 0 − x)(b 0 − x) = K ( c0 + x ) (d 0 + x) Note that the expression relates the initial concentrations of the species with a change away from initial conditions, “x”. Arrhenius’ ideas of ionic dissociation fit very nicely with the law of mass action. For his theory of ionic dissociation, Arrhenius won the third Nobel Prize in Chemistry in 1903. Combined with Ostwald’s dilution law, we get a view as to how examining the conductivity of solutions yields the evidence to need to confirm our modern view of the behavior of weak electrolytes in solution as a reversible reaction between dissociation and association. ⎛ Λ ⎞ ⎛ Λ ⎞ c2 ⎜ 2 ⎜Λ ⎟ ( cα )2 Λ 0 ⎟⎠ c2 ( α ) 0 ⎠ ⎝ ⎝ K=c = = = ⎛ Λ ⎞ ⎡ ⎛ Λ ⎞ ⎤ c [1 − ( α )] c (1 − α ) 1− ⎜ ⎟ c ⎢1 − ⎜ Λ ⎟ ⎥ ⎝ Λ0 ⎠ ⎣ ⎝ 0 ⎠⎦ 2 2 In the modern view, the constant that Ostwald found was the equilibrium constant (acid-dissociation constant). Initial Change Equilibrium HA (aq) U H+ (aq) + A- (aq) c 0 0 -cα cα cα c(1 – α) cα cα Arrhenius’s theory, together with van’t Hoff’s work on osmosis (remember the van’t Hoff factor) was solid evidence that electrolytes split into ions when dissolved in water. 7 Strong Electrolytes The Arrhenius theory of ionic dissociation did an excellent job of explaining the conductivity of solutions like acetic acid solutions. However for solutions like hydrochloric acid solutions, the theory didn’t fit at all since no constant (for Ostwald’s dilution law) could be found. For dilute strong electrolyte solutions, the molar conductivity increases linearly with concentration (conductivity is directly proportional to the number of ions in solution). However, for more concentrated solutions (above 0.001 M) the conductivity of the solution increases less than expected. The interaction of the solvated ions with other solvated ions affects the conductivity especially at high ion concentration. Thus, understanding the conductivity of a solution begins with understanding the environment surrounding the ion. To address these discrepancies, Peter Debye and Erich Hückel in 1923 constructed a model of electrolyte solutions Debye – Hückel Theory Finding the potential of an ion in solution Debye and Hückel wanted to find the potential that an ion experiences in solution due to the other ions in solution. Finding the potential would be simple if one knew the charge density. However to find the charge density, one needs the potential. Thus, our situation is like a dog chasing its tail! We start by considering the fundamental relationship between charge density, ρ, and potential, φ, that comes from Coulomb’s law. ∇2φ = ρ ε0 The ∇2 operator, called the Laplacian, is a three-dimensional second derivative. In Cartesian coordinates the operator is ∂2 ∂2 ∂2 ∇2 = 2 + 2 + 2 ∂x ∂y ∂z Because of the nature of the electrical force, the spherical polar coordinates are easier. (Truly!) 1 ∂⎛ ∂ ⎞ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ∇2 = 2 ⎜ r2 + sin θ + ⎟ ⎜ ⎟ ∂θ ⎠ r 2 sin 2 θ ∂Φ 2 r ∂r ⎝ ∂r ⎠ r 2 sin θ ∂θ ⎝ The potential is spherically symmetric (no θ, Φ dependence); therefore the second and third terms become zero for ∇2φ. 1 ∂⎛ 2 ∂ ⎜r r 2 ∂r ⎝ ∂r 4πρ ⎞ ⎟φ = ε0ε ⎠ 8 Again this equation is not useful to find the potential until we have the charge density. Let us consider the electrical environment surrounding an ion labeled “j”. If the charge of this ion is Zje, then the total charge for the rest of the ions in solution is –Zje. Also the electric potential that emanates from the charge can be given the subscript “j” as φj. To calculate the potential, we will attempt to find the charge density surrounding ion “j” by the finding the distribution of charges surrounding the ion. The distribution of charges will depend on the electrical interaction between the charges (which, on average, have the opposite charge of ion “j”) and the potential of ion “j” which imparts to the system the tendency to arrange ions in specific places as well as the thermal energy which imparts the tendency of the ion to spread. This distribution of ions can be approximated using the Boltzmann distribution (remember the MaxwellBoltzmann distribution of gas velocities). N i′ = N i e − Zi eφ j kT Note the distinction being made between Zi and φj. The above expression gives the probability of one ion. To find the charge density surrounding ion “j”, we need to sum of all of the other charges in the solution. ρ j = ∑ Zi eN i′ = ∑ Zi eN i e i − Zi eφ j kT i Using a Taylor series to expand the exponential function yields, 1 2 1 3 x − x +" 2! 3! 2 3 Zi eφ j Z e φ Z e φ Z e φ − ⎛ ⎞ ⎛ ⎞ 1 1 i j i j i j + ⎜ e kT = 1 − ⎟ − ⎜ ⎟ +" kT 2! ⎝ kT ⎠ 3! ⎝ kT ⎠ 2 ⎡ Zi eφ j 1 ⎛ Zi eφ j ⎞ 2 ⎤ Zi eφ j Zi eN i ⎛ Zi eφ j ⎞ ρ j = ∑ Zi eN i ⎢1 − + ⎜ +∑ ⎟ + "⎥ = ∑ Zi eN i − ∑ Zi eN i ⎜ ⎟ kT 2! ⎝ kT ⎠ kT 2! ⎝ kT ⎠ ⎣ ⎦ i i i i e− x = 1 − x + Summing over all charges “i”, the first summation is zero because of the conservation of charge. If Coulombic interaction is much less than the thermal energy, i. e., Zi eφ j kT , then the third (and other) summations can be neglected. Thus the charge density becomes ρ j = −∑ N i Zi2 e 2 φ j i kT =− e2φ j kT ∑N Z i 2 i i Using this charge density, we substitute into the earlier relationship between potential and charge density. 1 ∂⎛ 2 ∂ ⎜r r 2 ∂r ⎝ ∂r 4πe 2 φ j 4πρ ⎞ φ = = − N i Zi2 ∑ ⎟ j ε0ε ε 0εkT i ⎠ 9 To simplify the equation (and to help our later discussion), let us define the constant κ as κ2 = 4πe 2 N i Zi2 ∑ ε 0εkT i We will also define the ionic strength, I, as I= 1 N i Zi2 ∑ 2 i In calculating the ionic strength of a solution, all of the ions in the solution must be included in the sum. Thus the differential equation that needs to be solved to find the potential surrounding the ion “j” is 1 ∂⎛ 2 ∂ ⎞ 2 ⎜r ⎟ φj = κ φj r 2 ∂r ⎝ ∂r ⎠ The solution to this equation (the electric potential surrounding ion “j” is φj = Zi e e κa e−κr ε 0 ε 1 + κa r where a is the “distance of closest approach” between ion “j” and any other ion. “a” is of the order of 25 nm to 100 nm. (Higher ionic charge yields higher “a”.) The constant κ has some physical content as the inverse of the “thickness of the ionic atmosphere”. That is, ε εkT 1 = 0 2 8πe I κ The thickness of the ionic atmosphere can be considered as the maximum distance that ion “j” can attract other ion or the effective range for the potential of ion “j”. The total potential surrounding ion “j” comes from the potential of the ionic environment, φ″, as well as the potential from the ion itself, φ′. φ = φ′ + φ′′ The potential of the ion itself has the simple form of φj = 1 Zi e ε0ε r 10 Thus to find the potential of the ionic environment, we need to subtract the potential of the ion itself from the total potential. φ′′ = φ − φ′ = Zi e e κa e−κr 1 Zi e Zi e ⎛ e κa −κr ⎞ − = e − 1⎟ ⎜ ε 0 ε 1 + κa r ε0ε r ε 0εr ⎝ 1 + κa ⎠ The above potential is valid for all distances r > a. The potential when r = a (that is , when solvent cages of the ions interacts) becomes, φ′′ = = ⎞ Ze ⎛ 1 Zi e ⎛ e κa −κr ⎞ Z e ⎛ e κa e−κa ⎞ − 1⎟ = i ⎜ − 1⎟ e − 1⎟ = i ⎜ ⎜ ε0 εr ⎝ 1 + κa ⎠ r =a ε 0εa ⎝ 1 + κa ⎠ ε 0εa ⎝ 1 + κa ⎠ Zi e ⎛ 1 − (1 + κa ) ⎞ − Zi e ⎛ κa ⎞ − Zi e ⎛ κ ⎞ ⎜ ⎟= ⎜ ⎟= ⎜ ⎟ ε 0 εa ⎝ 1 + κa ⎠ ε 0εa ⎝ 1 + κa ⎠ ε 0ε ⎝ 1 + κa ⎠ φ′′ = − Zi e ⎛ κ ⎞ ⎜ ⎟ ε0 ε ⎝ 1 + κa ⎠ 1 , that is, κ the distance of closest approach of the ion plus the thickness of its ionic atmosphere. This potential is the same as the Coulomb potential, φ′, at a distance r = a + 1/κ – thickness of the ionic atmosphere (The thickness of the ionic atmosphere is on the order of a few nanometers.) Nonideality in electrolytic solutions (calculation of the activity coefficient) Consider the Gibbs free energy of an ion in solution ΔG i = ΔG *i + RT ln a i By redefining the standard state, ΔG*i , we can relate activity to the molar concentration rather the mole fraction. Such a redefinition of the standard state implies a redefinition of the activity coefficient. a i = γ i ci Thus the Gibbs free energy of an ion is solution becomes ΔG i = ΔG*i + RT ln ci + RT ln γ i The nonideal behavior of the ion in solution is expressed through the activity coefficient. 11 ΔG inonideal − ΔG iideal = ΔG*i + RT ln ci + RT ln γ i − ( ΔG*i + RT ln ci ) = RT ln γ i The major contribution to the nonideal behavior is the interaction between the ion and the ionic environment. The interaction is described simplest by considering the work needed to bring the ion from infinity to within the “distance of closest approach”. a RT ln γ i = w = ∫ φ dQ ∞ Recall the potential we found surrounding ion “j” that was due to the ionic environment. − Zi e ⎛ κ ⎞ ⎜ ⎟ ε 0 ε ⎝ 1 + κa ⎠ φ′′ = The charge, Q, is Zi⋅e. Thus an expression for the activity coefficient can be written as −1 ⎛ κ ⎞ −Q 2 ⎛ κ ⎞ Q dQ = ⎜ ⎟ ⎜ ⎟ 2ε 0εRT ⎝ 1 + κa ⎠ ε0 εRT ⎝ 1 + κa ⎠ ∫0 Q Q RT ln γ i = w = ∫ φ dQ ⇒ ln γ j = 0 ln γ j = − Z2j e2 ⎛ κ ⎞ ⎜ ⎟ 2ε0 εRT ⎝ 1 + κa ⎠ Often, the activity coefficient is written in terms of common (base 10) logarithm rather than natural logarithm. log b x = ln x ln b ⇒ log10 x = ln x ln10 ⎛ ⎞⎛ κ ⎞ e2 log γ j = − Z ⎜ ⎟ ⎟⎜ ⎝ 2ε 0 εRT ln10 ⎠ ⎝ 1 + κa ⎠ 2 j Recall that κ = 8πe2 I = B′ I ε0 εkT ⎛ B′ I ⎞ ⎛ ⎞ I 2 log γ j = − Z2j B′′ ⎜ ⎟ = −Z j B ⎜ ⎟ ⎝ 1 + B′a I ⎠ ⎝ 1 + B′a I ⎠ where B = 0.5091 M-½ at 25 °C and B′ = 0.0328 M-½⋅nm-1. This expression for the activity coefficient is known as the extended Debye-Hückel law. For 1:1 electrolytes, the extended Debye-Hückel law can be used solutions up to 0.1 M. 12 -3 If a solution is dilute (<10 M for a 1:1 electrolyte); then on average, r >> a. Thus the potential of the ion can be simplified to φ′′ = − Zi e ⎛ κ ⎞ − Zi eκ ⎜ ⎟≈ ε0 ε ⎝ 1 + κa ⎠ ε0ε Using the simplified potential yields a simplified expression of the activity coefficient known as the Debye- Hückel limiting law ⎛ B′ I ⎞ 2 log γ j = − Z2j B′′ ⎜ ⎟ ≈ −Z j B I ⎝ 1 + B′a I ⎠ The extending Debye-Huckel law can be improved by adding a correction factor. ⎛ ⎞ I log γ j = − Z2j B ⎜ ⎟ + cI ⎝ 1 + B′a I ⎠ The coefficient for the correction factor is found empirically (that is, from experimental data) rather than theoretically. If a is assumed to be the inverse of B′ such that a × B′ = 1, then equation becomes known as the Davies equation. ⎛ I ⎞ log γ j = − Z2j B ⎜ ⎟ + cI ⎝ 1+ I ⎠ The Davies equation can be used for some electrolytes up to 1 M concentration. 13 Thermodynamics of Ions It is impossible to measure absolute thermodynamic quantities for a single ion since we never isolated cations from anions. On the other hand because ions are always paired in solution, finding relative values for the thermodynamic quantities of ions is perfectly acceptable. To find relative values, we need to choose a standard. The standard chosen is 1 M (H+) at 25 °C such that, ΔH 0f ( H + ) = 0 ΔG 0f ( H + ) = 0 ΔS0f ( H + ) = 0 With our standard in place, we can consider the thermodynamics of other ions by examining the thermodynamics of solution with the hydrogen ion. For example, consider an experimentally determined value for the enthalpy of formation for hydrochloric acid (at 1 M and 25 °C). ΔH 0f ( HCl ( aq ) ) = ΔH 0f ( H + ( aq ) ) + ΔH 0f ( Cl− ( aq ) ) ⇒ ΔH 0f ( Cl− ( aq ) ) = ΔH 0f ( HCl ( aq ) ) − ΔH 0f ( H + ( aq ) ) = ΔH 0f ( HCl ( aq ) ) Next we could determine the enthalpy of formation of the sodium ion from the experimentally determined quantity, ΔH 0f ( NaCl ( aq ) ) ΔH 0f ( NaCl ( aq ) ) = ΔH 0f ( Na + ( aq ) ) + ΔH 0f ( Cl− ( aq ) ) ⇒ ΔH 0f ( Na + ( aq ) ) = ΔH 0f ( NaCl ( aq ) ) − ΔH 0f ( Cl− ( aq ) ) Next we could determine the enthalpy of formation of the bromide ion from the experimentally determined quantity, ΔH 0f ( NaBr ( aq ) ) ΔH 0f ( NaBr ( aq ) ) = ΔH 0f ( Na + ( aq ) ) + ΔH 0f ( Br − ( aq ) ) ⇒ ΔH 0f ( Br − ( aq ) ) = ΔH 0f ( NaBr ( aq ) ) − ΔH 0f ( Na + ( aq ) ) Next we could determine the enthalpy of formation of the potassium ion from the experimentally determined quantity, ΔH 0f ( KBr ( aq ) ) ΔH 0f ( KBr ( aq ) ) = ΔH 0f ( K + ( aq ) ) + ΔH f0 ( Br − ( aq ) ) ⇒ ΔH 0f ( K + ( aq ) ) = ΔH 0f ( KBr ( aq ) ) − ΔH 0f ( Br − ( aq ) ) Et cetera, et cetera, et cetera … 14 Concentration dependence of chemical potential As with other solutions, the chemical potential of a solution component depends on its activity. In examining the chemical potential of the solute, we choose a new standard state of c = 1 M. (rather than a mole fraction of 1). With a new standard state , concentration dependence of the chemical potential of an ion is expressed as μi ( aq ) = μi0 ( aq ) + RT ln a i = μi0 ( aq ) + RT ln γ i ci Because ions cannot be separated in solution, the activity coefficient for a single ion, γi, is not sensible. Thus an average activity coefficient (an average for both ions) is used. The average used is a geometric average rather than arithmetic average. Thus for a 1:1 electrolyte the mean ionic activity coefficient is γ± = γ+ γ− In general, for an electrolyte with “a” cations and “b” anions, the mean ionic activity coefficient is γ ± = a + b γ a+ γ −b For example, for a solution of aluminum sulfate 3 2 3 5 γ 3+ γ 2− γ ± = 2+3 γ 2Al3+ γ SO 2− = Al SO 4 4 or 3 γ 5± = γ 2Al3+ γ SO 2− 4 0 0 μ Al2 (SO4 ) = μ 0Al2 (SO4 ) = 2μ 0Al3+ + 3μSO 2− = RT ln a i = μ i ( aq ) + RT ln γ i ci 3 3 4