Calculating Mean Activity Coefficient for CaF₂

The mean activity coefficient ($\gamma_\pm$) for an electrolyte like $CaF_2$ depends on the individual ionic activity coefficients of its constituent ions ($Ca^{2+}$ and $F^-$) and their stoichiometric coefficients.

Formula for Mean Activity Coefficient

For an electrolyte dissociating into $v_+$ cations and $v_-$ anions ($M_{v_+}^{z_+}X_{v_-}^{z_-}$), the mean activity coefficient is calculated using the formula:

$ \gamma_\pm = (\gamma_+^{v_+}\gamma_-^{v_-})^{\frac{1}{v_+ + v_-}} $

Applying the Formula to CaF₂

In the case of $CaF_2$, it dissociates as follows:

$ CaF_2 \rightarrow Ca^{2+} + 2F^- $

Here:

• The cation is $Ca^{2+}$, with $v_+ = 1$.
• The anion is $F^-$, with $v_- = 2$.
• The given activity coefficient for $Ca^{2+}$ is $\gamma_+ = 0.72$.
• The given activity coefficient for $F^-$ is $\gamma_- = 0.28$.

Step-by-Step Calculation

1. Substitute the values into the formula:

   $ \gamma_\pm = (\gamma_{Ca^{2+}}^{1}\gamma_{F^-}^{2})^{\frac{1}{1 + 2}} $

   $ \gamma_\pm = (0.72^1 \times 0.28^2)^{\frac{1}{3}} $

2. Calculate the term inside the parenthesis:

   $ 0.72 \times (0.28)^2 = 0.72 \times 0.0784 = 0.056448 $

3. Calculate the cube root:

   $ \gamma_\pm = (0.056448)^{\frac{1}{3}} $

   $ \gamma_\pm \approx 0.3836 $

4. Round the result to two decimal places as requested:

   $ \gamma_\pm \approx 0.38 $

Conclusion

The calculated mean activity coefficient for $CaF_2$ is approximately 0.38, which falls within the range of 0.36 to 0.4.