Conductance in Electrolytic Solutions, Molar Conductivities, and Kohlrausch's Law

Introduction to Electrolytic Conductance

When an electric current passes through a conductor, it is called electrical conduction. There are two main types of electrical conductors: metallic conductors and electrolytic conductors. Metallic conductors, like metals and alloys, conduct electricity through the movement of free electrons. This process does not involve any chemical change. Electrolytic conductors, on the other hand, are typically solutions of acids, bases, and salts in polar solvents like water, or molten ionic compounds. In these substances, electrical conduction occurs through the movement of ions. This process is accompanied by chemical reactions at the electrodes, known as electrolysis.

The ability of an electrolytic solution to conduct electricity depends on several factors. Primarily, it relies on the presence of ions, their concentration, and their mobility. The charge carriers in electrolytic solutions are ions, both positive (cations) and negative (anions). When an electric potential difference is applied across the electrodes immersed in the solution, these ions move towards the oppositely charged electrodes, thereby establishing an electric current.

Factors Affecting Electrolytic Conductance

Several factors influence the conductance of an electrolytic solution. Understanding these factors is crucial for comprehending the behavior of electrolytes in solution.

• Nature of the Electrolyte: Strong electrolytes, which dissociate almost completely into ions even at low concentrations (e.g., NaCl, KCl, strong acids like HCl, H₂SO₄), generally exhibit higher conductance than weak electrolytes (e.g., acetic acid, NH₄OH). This is because strong electrolytes produce a greater number of ions in solution.
• Concentration of the Solution: As the concentration of an electrolytic solution increases, the number of ions per unit volume also increases. This generally leads to an increase in conductance. However, the effect of concentration on molar conductivity is more complex and will be discussed later.
• Temperature: For electrolytic solutions, an increase in temperature generally increases conductance. This is because higher temperatures lead to increased kinetic energy of the ions, resulting in greater mobility. Additionally, at higher temperatures, the viscosity of the solvent decreases, further facilitating ion movement.
• Nature of the Solvent: The ability of the solvent to dissolve the electrolyte and its own dielectric constant play a significant role. Polar solvents are better at dissolving ionic compounds and facilitating ion formation than non-polar solvents. The viscosity of the solvent also impacts ion mobility; less viscous solvents allow ions to move more freely.
• Presence of Impurities: Impurities, especially ions, can significantly alter the conductance of a solution.

Measurement of Conductance

The conductance of an electrolytic solution is typically measured using a conductivity cell and a Wheatstone bridge. The conductivity cell is designed to have electrodes of a known area and separation distance. The resistance of the solution is measured, and from this, the conductance is calculated.

Conductivity Cell and Cell Constant

A conductivity cell consists of two platinum electrodes, usually coated with platinum black (a finely divided form of platinum), which are fixed at a certain distance apart. Platinum black is used to minimize polarization effects at the electrodes, which can occur if the electrodes are bare metal.

The resistance (R) of a conductor is directly proportional to its length (l) and inversely proportional to its cross-sectional area (A). For an electrolytic solution, this relationship can be expressed as:

R = ρ * (l/A)

where ρ (rho) is the resistivity or specific resistance of the solution. The term (l/A) is known as the cell constant, denoted by 'G*'.

G* = l/A

The cell constant depends on the geometry of the cell and is usually determined by measuring the resistance of a solution of known conductivity (e.g., a standard KCl solution) at a specific temperature.

Conductance (G)

Conductance is the reciprocal of resistance.

G = 1/R

The unit of conductance is Siemens (S), which is equivalent to mho (ohm⁻¹).

Using the cell constant, the relationship between resistance and conductance becomes:

R = G* / G

And therefore, conductance G = G* / R.

Specific Conductance (κ)

Specific conductance, also known as conductivity (κ, kappa), is the conductance of a solution of unit volume. It is the reciprocal of resistivity.

κ = 1/ρ

Substituting ρ = R * (A/l) into the equation for κ:

κ = 1 / (R * (A/l)) = (1/R) * (l/A)

Since G = 1/R and G* = l/A, we get:

κ = G * G*

Thus, specific conductance is the product of the measured conductance of the solution and the cell constant.

The unit of specific conductance is Siemens per meter (S m⁻¹) or Siemens per centimeter (S cm⁻¹). If resistance is measured in ohms and the cell constant in cm⁻¹, then κ is in S cm⁻¹.

Example: If a conductivity cell has a resistance of 100 Ω when filled with a solution, and its cell constant is 0.5 cm⁻¹, then the specific conductance (κ) of the solution is:

κ = G * G* = (1/R) * G* = (1/100 Ω) * 0.5 cm⁻¹ = 0.005 S cm⁻¹.

Equivalent Conductance (Λ_eq)

Equivalent conductance is defined as the conductance of a solution containing one gram equivalent of the electrolyte. It is related to specific conductance and the normality (N) of the solution.

Λ_eq = κ / N

where N is the normality of the solution in equivalents per liter.

If κ is in S cm⁻¹ and N is in equivalents per cm³ (1 N = 1 equivalent / 1000 cm³), then Λ_eq will be in S cm² equivalent⁻¹.

The unit of equivalent conductance is typically S cm² equivalent⁻¹.

Molar Conductance (Λ_m)

Molar conductance is defined as the conductance of a solution containing one mole of the electrolyte. It is related to specific conductance and the molar concentration (C) of the solution.

Λ_m = κ / C

where C is the molar concentration of the solution in moles per liter.

If κ is in S cm⁻¹ and C is in moles per cm³ (1 M = 1 mole / 1000 cm³), then Λ_m will be in S cm² mole⁻¹.

The unit of molar conductance is typically S cm² mol⁻¹.

Relationship between Λ_m and Λ_eq:

Λ_m = Λ_eq * n-factor

where n-factor is the valency of the ion. For example, for NaCl, n-factor = 1, so Λ_m = Λ_eq. For H₂SO₄, n-factor = 2, so Λ_m = 2 * Λ_eq.

Variation of Molar Conductance with Concentration

The molar conductivity of an electrolyte solution is not constant but changes with concentration.

• For Strong Electrolytes: The molar conductivity of strong electrolytes decreases linearly with the square root of concentration (√C) according to the Debye-Hückel-Onsager equation.

  Λ_m = Λ_m° - A√C

  where Λ_m° is the molar conductivity at infinite dilution (i.e., at zero concentration), and A is a constant that depends on the solvent, temperature, and the charge of the ions. At infinite dilution, the ions are completely separated, and interionic interactions are negligible, leading to maximum molar conductivity.
• For Weak Electrolytes: The molar conductivity of weak electrolytes increases sharply with dilution (decrease in concentration). This is because the degree of dissociation increases significantly with dilution. As more ions are formed upon dilution, the molar conductivity increases. The relationship between Λ_m and √C is not linear for weak electrolytes.

Molar Conductance at Infinite Dilution (Λ_m°)

Λ_m° represents the molar conductivity of an electrolyte at infinite dilution, where the concentration of the electrolyte is zero. At this point, the electrolyte is completely dissociated into ions, and the interionic attractions are negligible. Each ion contributes independently to the total molar conductivity.

For strong electrolytes, Λ_m° can be obtained by extrapolating the plot of Λ_m versus √C to zero concentration.

For weak electrolytes, the plot of Λ_m versus √C does not yield a straight line and cannot be extrapolated to zero concentration reliably. However, Λ_m° values for weak electrolytes can be determined using Kohlrausch's Law.

Kohlrausch's Law of Independent Migration of Ions

Kohlrausch's law is a fundamental principle that describes the molar conductivity of an electrolyte at infinite dilution. It states that:

At infinite dilution, the molar conductivity of an electrolyte is equal to the sum of the molar conductivities of its constituent ions, each multiplied by the number of ions per formula unit.

Mathematically, for an electrolyte like AX_y which dissociates into x cations (A^y+) and y anions (X^x-):

Λ_m°(AX_y) = x * λ₊° + y * λ_-°

where:

• Λ_m°(AX_y) is the molar conductivity of the electrolyte AX_y at infinite dilution.
• λ₊° is the molar ionic conductivity of the cation A^y+ at infinite dilution.
• λ_-° is the molar ionic conductivity of the anion X^x- at infinite dilution.
• x and y are the stoichiometric coefficients of the cation and anion, respectively.

Applications of Kohlrausch's Law

Kohlrausch's law has several important applications, particularly in determining the molar conductivity of weak electrolytes at infinite dilution and calculating the solubility of sparingly soluble salts.

1. Determination of Molar Conductance of Weak Electrolytes at Infinite Dilution

Weak electrolytes, such as acetic acid (CH₃COOH), cannot have their Λ_m° determined by simple extrapolation because their degree of dissociation is very low at measurable concentrations. Kohlrausch's law allows us to calculate Λ_m° for weak electrolytes using data from strong electrolytes.

Example: Calculation of Λ_m° for Acetic Acid (CH₃COOH)

We need the molar conductivities at infinite dilution for three strong electrolytes:

• Sodium acetate (CH₃COONa): Λ_m°(CH₃COONa) = λ_CH₃COO⁻° + λ_Na⁺°
• Hydrochloric acid (HCl): Λ_m°(HCl) = λ_H⁺° + λ_Cl⁻°
• Sodium chloride (NaCl): Λ_m°(NaCl) = λ_Na⁺° + λ_Cl⁻°

We want to find Λ_m°(CH₃COOH) = λ_H⁺° + λ_CH₃COO⁻°.

By combining the equations using Kohlrausch's law:

Λ_m°(CH₃COOH) = Λ_m°(CH₃COONa) + Λ_m°(HCl) - Λ_m°(NaCl)

This combination works because:

(λ_CH₃COO⁻° + λ_Na⁺°) + (λ_H⁺° + λ_Cl⁻°) - (λ_Na⁺° + λ_Cl⁻°) = λ_CH₃COO⁻° + λ_H⁺°

Thus, the molar conductivity of a weak electrolyte can be calculated by adding the molar conductivities of a salt of the weak acid/base and the molar conductivity of a strong acid/base, and subtracting the molar conductivity of a common salt.

2. Calculation of Degree of Dissociation (α) of Weak Electrolytes

The degree of dissociation (α) of a weak electrolyte at a given concentration (C) can be calculated using its molar conductivity (Λ_m) and molar conductivity at infinite dilution (Λ_m°):

α = Λ_m / Λ_m°

This relationship is valid for weak electrolytes where the degree of dissociation is not close to 1.

The Ostwald's dilution law for weak electrolytes relates the degree of dissociation to the concentration:

K_a = (Cα²) / (1 - α)

For very weak electrolytes, α is small, so (1 - α) ≈ 1, and K_a ≈ Cα². Substituting α = Λ_m / Λ_m°, we get:

K_a = C * (Λ_m / Λ_m°)²

3. Calculation of Solubility of Sparingly Soluble Salts

For sparingly soluble salts, the concentration of the saturated solution is very low. In such cases, the molar conductivity of the saturated solution (Λ_m) is approximately equal to the molar conductivity at infinite dilution (Λ_m°).

The solubility (s) of the salt in moles per liter can be related to its molar conductivity as follows:

Λ_m° = κ * 1000 / C (where C is molarity)

If we use specific conductivity (κ) in S cm⁻¹ and molar conductivity at infinite dilution (Λ_m°) in S cm² mol⁻¹, then:

Λ_m° = κ / C

where C is the molar concentration.

For a sparingly soluble salt, the molar concentration (C) of the saturated solution is equal to its solubility (s) in moles per liter.

Therefore, s = κ / Λ_m°

If solubility is required in grams per liter, it is calculated as:

Solubility (g/L) = s (mol/L) * Molar Mass (g/mol)

To use this method, one must know the Λ_m° of the sparingly soluble salt, which can be determined using Kohlrausch's law from the data of other strong electrolytes.

Example: Calculate the solubility of AgCl at 25°C if its specific conductivity is 1.12 × 10⁻⁶ S cm⁻¹ and Λ_m°(AgCl) is 138.3 S cm² mol⁻¹.

Solubility (s) = κ / Λ_m° s = (1.12 × 10⁻⁶ S cm⁻¹) / (138.3 S cm² mol⁻¹) s ≈ 8.10 × 10⁻⁹ mol cm⁻³

To convert to mol/L:

s = 8.10 × 10⁻⁹ mol cm⁻³ * (1000 cm³/L) = 8.10 × 10⁻⁶ mol/L

Factors Affecting Molar Ionic Conductivity (λ°₊ and λ°_-)

The molar ionic conductivity of an ion at infinite dilution (λ°₊ for cation, λ°_- for anion) depends on:

• Size and Charge of the Ion: Smaller ions with higher charges generally have higher mobility and thus higher ionic conductivity. However, in aqueous solutions, smaller ions are more heavily hydrated, which increases their effective size and reduces their mobility. For example, Li⁺ is smaller than Cs⁺, but Cs⁺ is more mobile in water because it is less hydrated.
• Viscosity of the Solvent: Higher viscosity leads to lower ion mobility and hence lower ionic conductivity.
• Temperature: Increased temperature leads to increased kinetic energy of ions, greater mobility, and higher ionic conductivity.
• Presence of Solvent Molecules (e.g., Water): Ion-solvent interactions, such as hydration, affect the effective size of the ion and its mobility.

Transport Number (t)

When an electric field is applied to an electrolytic solution, both cations and anions move towards their respective electrodes. The transport number of an ion is the fraction of the total current carried by that ion.

The transport number of the cation (t₊) is the fraction of current carried by cations:

t₊ = (Conductivity due to cations) / (Total conductivity) = (Λ_m⁺) / (Λ_m)

Similarly, the transport number of the anion (t_-) is:

t_- = (Conductivity due to anions) / (Total conductivity) = (Λ_m^-) / (Λ_m)

where Λ_m⁺ and Λ_m^- are the molar ionic conductivities of cations and anions, respectively.

The sum of the transport numbers of the cation and anion is always equal to 1:

t₊ + t_- = 1

The transport number is independent of concentration for strong electrolytes.

Relationship with Molar Ionic Conductivity:

t₊ = λ₊ / (λ₊ + λ_-)

t_- = λ_- / (λ₊ + λ_-)

where λ₊ and λ_- are the molar ionic conductivities of the cation and anion at the given concentration. At infinite dilution, these become λ₊° and λ_-°.

Applications of Transport Number

Transport numbers are important in several electrochemical applications:

• Hittorf's Method: Used to determine changes in concentration near electrodes during electrolysis, which allows for the calculation of transport numbers.
• Potentiometry: Used in the calculation of liquid junction potentials in electrochemical cells.
• Electrolytic Refining and Electroplating: Understanding ion migration is crucial for efficient processes.

Summary Table of Conductance Terms

Here is a quick reference for the key terms related to conductance:

Term	Symbol	Definition	Unit (SI)	Unit (Common)
Conductance	G	Reciprocal of Resistance (1/R)	Siemens (S)	Siemens (S) or mho
Resistance	R	Opposition to current flow	Ohm (Ω)	Ohm (Ω)
Resistivity (Specific Resistance)	ρ	Resistance of a unit cube of material (R * A/l)	Ohm-meter (Ω·m)	Ohm-centimeter (Ω·cm)
Specific Conductance (Conductivity)	κ	Reciprocal of Resistivity (1/ρ)	Siemens per meter (S/m)	Siemens per centimeter (S/cm)
Cell Constant	G*	Ratio of electrode separation to area (l/A)	Meter (m)	Centimeter (cm)
Molar Conductance	Λm	Conductance of a solution containing 1 mole of electrolyte (κ/C)	S m² mol⁻¹	S cm² mol⁻¹
Equivalent Conductance	Λeq	Conductance of a solution containing 1 gram equivalent of electrolyte (κ/N)	S m² eq⁻¹	S cm² eq⁻¹
Molar Ionic Conductivity	λ+°, λ-°	Molar conductivity of a single ion at infinite dilution	S m² mol⁻¹	S cm² mol⁻¹
Transport Number	t+, t-	Fraction of total current carried by an ion	Dimensionless	Dimensionless