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Understanding the Key Parameters

Published in Chemical Solubility Calculation 6 mins read

To calculate the solubility of a sparingly soluble salt from conductivity measurements, you primarily use the relationship between the specific conductance of its saturated solution and its molar conductance at infinite dilution.

The solubility of a sparingly soluble salt (S) can be determined using its specific conductance (K) and molar conductance (ΛM). According to the provided reference, these are related as: S=K×ΛM. The reference also states S=1000×ΛMK. For practical calculations of solubility in moles per liter (mol/L), the second relationship is correctly interpreted as S = (1000 × K) / ΛM.

Understanding the Key Parameters

Before diving into the calculation, it's essential to understand the terms involved:

  • Solubility (S): For a sparingly soluble salt, solubility refers to the maximum concentration of the salt that can dissolve in a given solvent (usually water) at a specific temperature, forming a saturated solution. It is typically expressed in moles per liter (mol/L).
  • Specific Conductance (K or κ): Also known as conductivity, specific conductance is a measure of a solution's ability to conduct electricity. It is the reciprocal of resistivity and is measured in Siemens per centimeter (S/cm) or Siemens per meter (S/m). For solubility calculations, it's crucial to use the specific conductance contributed only by the salt, not the solvent.
  • Molar Conductance (ΛM or Λ): Molar conductance is the conductivity of a solution containing one mole of an electrolyte placed between two electrodes 1 cm apart, where the area of the electrodes is large enough to contain all the solution. For sparingly soluble salts, their saturated solutions are extremely dilute. Therefore, their molar conductance (ΛM) is approximated by the molar conductance at infinite dilution (Λ°M or Λ₀). This value represents the conductivity of one mole of the electrolyte when completely dissociated, with negligible interionic interactions. It is typically expressed in Siemens centimeter squared per mole (S cm²/mol).

The Calculation Formula

While the provided reference includes the relation "S=K×ΛM", this relationship is generally not dimensionally consistent for calculating solubility in common units (mol/L) when K is in S/cm and ΛM is in S cm²/mol.

The accurate and widely accepted formula, which aligns with the latter part of the reference (S=1000×ΛMK, interpreted as S = (1000 × K) / ΛM), for calculating the solubility (S) of a sparingly soluble salt is:

$$S = \frac{1000 \times K}{\Lambda_M}$$

Where:

  • S = Solubility in moles per liter (mol/L)
  • K = Specific conductance of the salt in Siemens per centimeter (S/cm)
  • ΛM = Molar conductance at infinite dilution in Siemens centimeter squared per mole (S cm²/mol)

The factor of 1000 arises from the unit conversion between mol/cm³ (which results from S/cm divided by S cm²/mol) and mol/L (since 1 L = 1000 cm³).

Step-by-Step Calculation Process

To accurately determine solubility from conductivity, follow these steps:

  1. Prepare a Saturated Solution:

    • Add an excess amount of the sparingly soluble salt to distilled or deionized water.
    • Stir the mixture thoroughly for an extended period (e.g., 24-48 hours) at a constant temperature to ensure equilibrium is reached and the solution becomes saturated.
    • Allow any undissolved solid to settle.
    • Carefully filter or decant the clear, saturated solution.
  2. Measure the Specific Conductance of the Solution (K_solution):

    • Use a calibrated conductivity meter and conductivity cell.
    • Ensure the temperature is precisely controlled and recorded, as conductivity is highly temperature-dependent.
    • Measure the specific conductance of the saturated solution (K_solution) in S/cm.
  3. Measure the Specific Conductance of the Solvent (K_water):

    • Measure the specific conductance of the pure water used to prepare the solution (K_water) at the same temperature. This is crucial because even highly purified water has a small inherent conductivity.
  4. Calculate the Specific Conductance of the Salt (K_salt):

    • Subtract the conductivity of the pure water from the conductivity of the saturated solution to obtain the specific conductance contributed solely by the dissolved salt.
      $$K{salt} = K{solution} - K_{water}$$
    • This value ($K_{salt}$) will be used as 'K' in the solubility formula.
  5. Determine the Molar Conductance at Infinite Dilution (Λ°M):

    • For sparingly soluble salts, their solutions are so dilute that their molar conductance (ΛM) can be assumed to be equal to their molar conductance at infinite dilution (Λ°M).
    • This value is usually obtained from literature resources (handbooks, databases) or calculated using Kohlrausch's Law of Independent Migration of Ions. This law states that at infinite dilution, the molar conductivity of an electrolyte is the sum of the limiting molar conductivities of its constituent ions:
      $$Λ^°M = ν^+λ^°+ + ν^-λ^°_-$$
      Where:
      • $ν^+$ and $ν^-$ are the number of cations and anions, respectively, produced per formula unit of the electrolyte.
      • $λ^°+$ and $λ^°-$ are the limiting ionic molar conductivities of the cation and anion, respectively, at the given temperature.
  6. Apply the Solubility Formula:

    • Substitute the calculated specific conductance of the salt ($K{salt}$) and the determined molar conductance at infinite dilution (Λ°M) into the formula:
      $$S = \frac{1000 \times K
      {salt}}{Λ^°_M}$$
    • The resulting value 'S' will be the solubility of the sparingly soluble salt in mol/L.

Important Considerations for Accuracy

  • Temperature Control: Conductivity is highly sensitive to temperature. All measurements (solution and solvent) must be performed at the exact same, stable temperature. The Λ°M value used must also correspond to this temperature.
  • Purity of Water: Use highly purified water (e.g., deionized or distilled water with very low conductivity) to minimize the background conductivity and ensure accurate subtraction.
  • Equilibrium: Ensure the saturated solution has truly reached equilibrium with the undissolved solid to get an accurate representation of maximum solubility.
  • Accuracy of Λ°M: The accuracy of the calculated solubility largely depends on the reliability of the Λ°M value used.
  • Salt Type: This method is most accurate for 1:1 electrolytes (e.g., AgCl, BaSO₄) that dissociate completely into two ions. For more complex electrolytes, ionic interactions at low concentrations might still introduce minor deviations.

Example (Conceptual)

Let's say you want to find the solubility of Silver Chloride (AgCl).

  1. You prepare a saturated solution of AgCl at 25°C.
  2. You measure the specific conductance of the saturated AgCl solution ($K_{solution}$) as $3.40 \times 10^{-6}$ S/cm.
  3. You measure the specific conductance of the pure water ($K_{water}$) at 25°C as $1.60 \times 10^{-6}$ S/cm.
  4. Calculate the specific conductance of AgCl:
    $K_{AgCl} = (3.40 \times 10^{-6}) - (1.60 \times 10^{-6}) = 1.80 \times 10^{-6}$ S/cm.
  5. Look up the limiting molar conductivities for Ag⁺ and Cl⁻ ions at 25°C:
    $λ^°{Ag^+} = 61.9$ S cm²/mol
    $λ^°
    {Cl^-} = 76.3$ S cm²/mol
    Calculate Λ°M for AgCl:
    $Λ^°{AgCl} = λ^°{Ag^+} + λ^°_{Cl^-} = 61.9 + 76.3 = 138.2$ S cm²/mol.
  6. Calculate the solubility of AgCl:
    $S = \frac{1000 \times (1.80 \times 10^{-6} \text{ S/cm})}{138.2 \text{ S cm²/mol}}$
    $S \approx 1.30 \times 10^{-5}$ mol/L

Summary Table

Parameter Symbol Typical Units Description
Solubility S mol/L Concentration of dissolved salt in a saturated solution
Specific Conductance K or κ S/cm Conductivity contribution from the dissolved salt only
Molar Conductance at ΛM or Λ°M S cm²/mol Conductivity of one mole of electrolyte at infinite dilution
Infinite Dilution (approximated for sparingly soluble salts)
Conversion Factor 1000 (dimensionless) Converts volume from cm³ to L in the calculation

By carefully measuring the specific conductance of the saturated solution and the solvent, and by utilizing the known molar conductance at infinite dilution, the solubility of sparingly soluble salts can be accurately determined through conductivity measurements.