Question

The electrolyte AB2 ionises in water as
AB2 \( \rightleftharpoons \) \( A^{2+} + 2B^{-} \)
The mean ionic activity coefficient (\( \gamma_{\pm} \)) is

• A. \( \gamma_{A^{2+}} \gamma_{B^{-}} \)
• B. \( \gamma_{A^{2+}}^{2} \gamma_{B^{-}} \)
• C. \( \gamma_{A^{2+}}^{3} \gamma_{B^{-}}^{2} \)
• D. \( \left( \gamma_{A^{2+}}^{2} + 2 \gamma_{B^{-}} \right)^{-1/2} \)

Hint:

The mean ionic activity coefficient for a dissociating electrolyte is the geometric mean of the activity coefficients of the ions involved, with each ion's coefficient weighted by its stoichiometric coefficient.

Answer 1

The Correct Option is D

Step 1: Understanding the ionic activity coefficient. 
The mean ionic activity coefficient \( \gamma_{\pm} \) is a measure of the effective concentration of ions in solution. It is defined as the geometric mean of the activity coefficients of the cations and anions involved. For a compound like \( \text{AB}_2 \), which dissociates into one \( A^{2+} \) ion and two \( B^{-} \) ions, the mean ionic activity coefficient is given by the formula: 
\[ \gamma_{\pm} = \left( \gamma_{A^{2+}}^{2} + 2 \gamma_{B^{-}} \right)^{-1/2} \] 
 

Step 2: Analyzing the options. 
(A) \( \gamma_{A^{2+}} \gamma_{B^{-}} \): Incorrect — This does not account for the correct number of ions and their corresponding coefficients. 
(B) \( \gamma_{A^{2+}}^{2} \gamma_{B^{-}} \): Incorrect — This does not represent the correct formula for the mean ionic activity coefficient. 
(C) \( \gamma_{A^{2+}}^{3} \gamma_{B^{-}}^{2} \): Incorrect — This is not the correct formula for the mean ionic activity coefficient. 
(D) \( \left( \gamma_{A^{2+}}^{2} + 2 \gamma_{B^{-}} \right)^{-1/2} \): Correct — This is the correct formula for the mean ionic activity coefficient in this case. 
 

Step 3: Conclusion. 
The correct answer is (D) as it accurately represents the mean ionic activity coefficient for the dissociation of \( AB_2 \).