Debye-Huckel Limiting Law for Activity Coefficients

The Debye-Huckel limiting law is used to describe the behavior of activity coefficients for ions in very dilute electrolyte solutions. It provides a relationship between the mean ionic activity coefficient ($\gamma_{\pm}$) and the ionic strength ($I$) of the solution. The question focuses on the natural logarithm of the mean ionic activity coefficient, \(\ln \gamma_{\pm}\).

The limiting law states:

\(\ln \gamma_{\pm} = -A' |z_+ z_-| \sqrt{I}\)

Where:

• \(\gamma_{\pm}\) is the mean ionic activity coefficient.
• \(A'\) is a constant specific to the solvent and temperature (equal to \(2.303 \times A\) from the \(\log_{10}\) form).
• \(z_+\) and \(z_-\) are the absolute values of the charges of the cation and anion, respectively.
• \(I\) is the ionic strength of the solution.

For a specific electrolyte in a given solvent at a constant temperature, the term \(A' |z_+ z_-|\) is a constant. Let's call this constant \(B\). Since \(A'\), \(|z_+|\), and \(|z_-|\) are positive, \(B\) is a positive constant.

So, the law can be written as:

\(\ln \gamma_{\pm} = -B \sqrt{I}\)

Ionic Strength Calculation for KCl

The ionic strength \(I\) of a solution is defined by the equation:

\(I = \frac{1}{2} \sum_i c_i z_i^2\)

Where \(c_i\) is the molar concentration of ion \(i\) and \(z_i\) is the charge of ion \(i\).

For a dilute aqueous solution of KCl, which is a strong 1:1 electrolyte, it dissociates almost completely into potassium ions (K⁺) and chloride ions (Cl^-):

\(\text{KCl(aq)} \rightarrow \text{K}^+\text{(aq)} + \text{Cl}^-\text{(aq)}\)

If the molar concentration of the KCl solution is \(C\), then the concentration of K⁺ ions is \(C\) and the concentration of Cl^- ions is \(C\). The charges are \(z_{K^+} = +1\) and \(z_{Cl^-} = -1\).

Substituting these values into the ionic strength formula:

\(I = \frac{1}{2} [ (C \times (+1)^2) + (C \times (-1)^2) ] = \frac{1}{2} [ (C \times 1) + (C \times 1) ] = \frac{1}{2} (C + C) = \frac{1}{2} (2C) = C\)

Thus, for a dilute 1:1 electrolyte like KCl, the ionic strength \(I\) is approximately equal to the molar concentration \(C\).

Effect of Concentration Increase on ln gamma_pm

Substituting \(I=C\) into the simplified Debye-Huckel limiting law equation:

\(\ln \gamma_{\pm} = -B \sqrt{C}\)

Let the initial concentration of the KCl solution be \(C_1\). The initial value of \(\ln \gamma_{\pm}\) is:

\((\ln \gamma_{\pm})_1 = -B \sqrt{C_1}\)

The question states that the concentration is increased 9 folds. So, the new concentration \(C_2\) is \(9\) times the initial concentration:

\(C_2 = 9C_1\)

Now, we calculate the new value of \(\ln \gamma_{\pm}\) using the new concentration \(C_2\):

\((\ln \gamma_{\pm})_2 = -B \sqrt{C_2} = -B \sqrt{9C_1}\)

Using the property of square roots, \(\sqrt{9C_1} = \sqrt{9} \times \sqrt{C_1} = 3 \times \sqrt{C_1}\).

Substitute this back into the expression for \((\ln \gamma_{\pm})_2\):

\((\ln \gamma_{\pm})_2 = -B (3 \sqrt{C_1}) = -3B \sqrt{C_1}\)

Now, we compare \((\ln \gamma_{\pm})_2\) with \((\ln \gamma_{\pm})_1\). We notice that \((\ln \gamma_{\pm})_1 = -B \sqrt{C_1}\). So we can write:

\((\ln \gamma_{\pm})_2 = 3 \times (-B \sqrt{C_1}) = 3 \times (\ln \gamma_{\pm})_1\)

Change in the Value of ln gamma_pm

The relationship \((\ln \gamma_{\pm})_2 = 3 \times (\ln \gamma_{\pm})_1\) shows how the value changes.

According to the Debye-Huckel limiting law, \(\gamma_{\pm}\) is always less than 1 for a dilute electrolyte solution. This means that \(\ln \gamma_{\pm}\) is always a negative value (since the natural logarithm of a number between 0 and 1 is negative).

Let the initial value \((\ln \gamma_{\pm})_1 = V_1\). We know \(V_1\) is negative. The new value is \((\ln \gamma_{\pm})_2 = V_2 = 3 \times V_1\).

Since \(V_1\) is negative, multiplying it by 3 results in a new negative value \(V_2\) whose magnitude is 3 times the magnitude of \(V_1\). For example, if \(V_1 = -0.1\), then \(V_2 = 3 \times (-0.1) = -0.3\). If \(V_1 = -0.05\), then \(V_2 = 3 \times (-0.05) = -0.15\).

In both examples, the value changes from a less negative number to a more negative number (-0.3 is less than -0.1, and -0.15 is less than -0.05). A change from a less negative value to a more negative value represents a decrease in the value.

The magnitude of the original value is \(|V_1|\). The magnitude of the new value is \(|V_2| = |3 V_1| = 3 |V_1|\). So, the magnitude increases by a factor of 3.

Therefore, the value of \(\ln \gamma_{\pm}\) decreases, and its magnitude becomes 3 times the original magnitude. The phrase "Decrease by 3 fold" indicates that the value decreases and the factor associated with the change in magnitude is 3.