Understanding the Debye-Hückel Limiting Law
The Debye-Hückel limiting law provides a relationship between the activity of ions in a solution and their concentration. It is especially useful for dilute solutions and helps in calculating the activity coefficients of electrolyte solutions.
Activity Coefficient Calculation
The activity coefficient (γ) for an electrolyte can be calculated using the formula:
\[ \log(γ) = - \frac{A z^2 \sqrt{I}}{1 + B a \sqrt{I}} \]
Where:
- A = Debye-Hückel constant (0.51 for water at 25°C)
- z = charge of the ion
- I = ionic strength of the solution
- B = constant related to the size of the ions (not specified here)
- a = effective diameter of the ions in nm (not specified here)
Calculating Ionic Strength (I)
For a 0.1 M CaCl₂ solution:
- Ca²⁺ has a charge of +2
- Cl⁻ has a charge of -1
The ionic strength (I) can be calculated as:
\[ I = \frac{1}{2} \sum c_i z_i^2 \]
Where \( c_i \) is the concentration and \( z_i \) is the charge.
For CaCl₂:
- \( I = \frac{1}{2}[(0.1)(2^2) + (0.2)(1^2)] \)
- \( I = \frac{1}{2}(0.4 + 0.2) = 0.3 \, M \)
Calculating Activity Coefficient (γ)
Assuming the size constants (B and a) are negligible for this calculation, we can simplify:
\[ \log(γ) = - A z^2 \sqrt{I} \]
\[ \log(γ) = - 0.51 \times (2^2) \times \sqrt{0.3} \]
\[ \log(γ) = - 0.51 \times 4 \times 0.5477 \]
\[ \log(γ) = -1.116 \]
Thus,
\[ γ = 10^{-1.116} \approx 0.077 \]
Calculating Activity (a)
The activity (a) of Ca²⁺ is given by:
\[ a = c \cdot γ \]
\[ a = 0.1 \cdot 0.077 \approx 0.0077 \, M \]
Conclusion
The activity of a 0.1 M CaCl₂ solution is approximately 0.0077 M, indicating the effective concentration of ions in the solution considering interactions among them.