Debye-Hückel Law Relationship for NaCl

The Debye-Hückel Limiting Law (DHLL) provides a way to calculate the mean ionic activity coefficient ($\gamma_\pm$) based on the solution's ionic strength ($I$) and ion charges.

Debye-Hückel Limiting Law Formula

The law is commonly expressed using the base-10 logarithm:

$ \log_{10} \gamma_\pm = -A |z_+ z_-| \sqrt{I} $

• $\gamma_\pm$: Mean ionic activity coefficient
• $A$: Debye-Hückel constant (value depends on temperature and solvent)
• $z_+$: Charge number of the cation
• $z_-$: Charge number of the anion
• $I$: Ionic strength of the solution

NaCl Solution Ionic Strength

For a 1 molal aqueous solution of NaCl:

• NaCl is a 1:1 electrolyte, meaning $z_+ = +1$ and $z_- = -1$.
• The term $|z_+ z_-|$ becomes $|(+1)(-1)| = 1$.
• Ionic strength ($I$) is calculated as $I = \frac{1}{2} \sum m_i z_i^2$. For dilute solutions, we can approximate molality ($m$) to molarity and assume the density of water is 1 kg/L. Thus, for 1 molal NaCl:
  $ I \approx \frac{1}{2} (1 \times (1)^2 + 1 \times (-1)^2) = \frac{1}{2}(1 + 1) = 1 $

Activity Coefficient Calculation

Substitute the values for NaCl and the calculated ionic strength into the DHLL:

$ \log_{10} \gamma_\pm = -A (1) \sqrt{1} $

This simplifies to:

$ \log_{10} \gamma_\pm = -A $

To express the mean ionic activity coefficient ($\gamma_\pm$) directly, we convert this logarithmic relationship to its exponential form:

$ \gamma_\pm = 10^{-A} $

Activity Coefficient Result

The relationship derived from the Debye-Hückel Limiting Law for a 1 molal aqueous NaCl solution is $\gamma_\pm = 10^{-A}$.