Understanding the Debye-Hückel Limiting Law

The Debye-Hückel limiting law is a fundamental concept in physical chemistry, particularly in the study of electrolyte solutions. It helps predict the activity coefficients of ions in dilute solutions. The activity coefficient ($\gamma$) measures how much the behavior of an ion deviates from ideality. The limiting law provides a theoretical basis for calculating these coefficients under specific conditions (very dilute solutions).

Debye-Hückel Limiting Law Formula Breakdown

The law is expressed mathematically to relate the mean activity coefficient ($\gamma_{\pm}$) to the properties of the ions in the solution. The key components of the formula are:

• Activity Coefficient ($\gamma_{\pm}$): Represents the deviation of ion behavior from ideal conditions. It's often expressed on a molality basis.
• Ionic Charge ($z_{+}$, $z_{-}$): The absolute values of the charges on the cation (+) and anion (-) respectively. The term $|z_{+}Z_{-}|$ represents the product of these absolute charges.
• Ionic Strength ($I$): A measure of the total concentration of ions in a solution, taking into account their charges. It is defined as $I = \frac{1}{2} \sum_{i} m_i z_i^2$, where $m_i$ is the molality of ion $i$ and $z_i$ is its charge.
• Constant (0.509): This constant depends on the solvent and temperature. For water at 25°C, the value is approximately 0.509.

Identifying the Correct Debye-Hückel Limiting Law Equation

The Debye-Hückel limiting law specifically applies to very dilute solutions where interactions are minimal and can be approximated. Under these conditions, the relationship between the logarithm of the mean activity coefficient and the ionic strength is given by:

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

Where:

• $A$ is the Debye-Hückel constant (approximately 0.509 for water at 25°C).
• $|z_{+}z_{-}|$ is the absolute value of the product of the charges of the cation and anion.
• $\sqrt{I}$ is the square root of the ionic strength, which can also be written as $I^\frac{1}{2}$.

Comparing this theoretical formula with the given options:

• Option 1: $log \gamma_{\pm} = -0.509|z_{+}Z_{-}|I^2$ (Incorrect exponent for $I$)
• Option 2: $log \gamma_{\pm} = -0.509|z_{+}Z_{-}|I$ (Incorrect exponent for $I$)
• Option 3: $log \gamma_{\pm} = 0.509|z_{+}Z_{-}|I^\frac{1}{2}$ (Incorrect sign)
• Option 4: $log \gamma_{\pm} = -0.509|z_{+}Z_{-}|I^\frac{1}{2}$ (Matches the correct form of the Debye-Hückel limiting law)

Therefore, the correct representation of the Debye-Hückel limiting law is $log \gamma_{\pm} = -0.509|z_{+}Z_{-}|I^\frac{1}{2}$. The negative sign indicates that the actual activity is less than predicted by ideal behavior due to attractive forces between ions and oppositely charged ions in the ionic atmosphere.