Calculating Mean Ionic Activity Coefficient using Debye-Hückel Limiting Law
This solution outlines the calculation of the mean ionic activity coefficient for $ZnSO_4$ using the Debye-Hückel limiting law (DHLL).
Identify Given Parameters
- Electrolyte: $ZnSO_4$ (aq)
- Molality ($m$): 0.001 molal
- Temperature: 298 K
- Debye-Hückel constant ($A$): 0.509 $molal^{-½}$
- Dissociation: $ZnSO_4 \rightarrow Zn^{2+} + SO_4^{2-}$
- Ionic charges: $z_{+} = +2$, $z_{-} = -2$
- Absolute value of product of charges: $|z_{+} z_{-}| = |(+2)(-2)| = 4$
Calculate Ionic Strength
The ionic strength ($I$) quantifies the concentration of ions in a solution. It is calculated using the molalities ($m_i$) and charges ($z_i$) of all ions present.
Formula for Ionic Strength:
\( I = \frac{1}{2} \sum_{i} m_i z_i^2 \)
For $ZnSO_4$, the molality of $Zn^{2+}$ and $SO_4^{2-}$ ions are both 0.001 molal.
Calculation:
\( I = \frac{1}{2} [m_{Zn^{2+}} z_{Zn^{2+}}^2 + m_{SO_4^{2-}} z_{SO_4^{2-}}^2] \)
\( I = \frac{1}{2} [0.001 \times (2)^2 + 0.001 \times (-2)^2] \)
\( I = \frac{1}{2} [0.001 \times 4 + 0.001 \times 4] \)
\( I = \frac{1}{2} [0.008] \)
\( I = 0.004 \) molal
Apply Debye-Hückel Limiting Law
The Debye-Hückel limiting law provides an empirical relationship between the mean ionic activity coefficient ($\gamma_{\pm}$) and the ionic strength ($I$) for dilute solutions.
Formula for DHLL (using base-10 logarithm):
\( \log_{10} \gamma_{\pm} = - A |z_{+} z_{-}| \sqrt{I} \)
First, calculate the square root of the ionic strength:
\( \sqrt{I} = \sqrt{0.004} \approx 0.063246 \)
Substitute the known values into the DHLL formula:
\( \log_{10} \gamma_{\pm} = - (0.509) \times (4) \times (0.063246) \)
\( \log_{10} \gamma_{\pm} = - (2.036) \times (0.063246) \)
\( \log_{10} \gamma_{\pm} \approx -0.128777 \)
Final Result
To determine the mean ionic activity coefficient ($\gamma_{\pm}$), we calculate the antilogarithm of the result obtained above:
\( \gamma_{\pm} = 10^{-0.128777} \)
\( \gamma_{\pm} \approx 0.74351 \)
The calculated mean ionic activity coefficient is approximately 0.7435, which falls within the expected range of 0.73 to 0.75.
Important Questions from Debye Huckel Theory
The electrolyte solution that has the smallest Debye-length at 298 K is
The amount of Ba(NO3)2 (molecular weight 261.32 amu) required to be added to 500 g of a 0.11 mol kg-1 solution of KNO3 in order to raise its ionic strength to 1.00 is approximately:
- Debye - Huckel limiting law in :
- The ionic activity coefficients of $Ca^{2+}$ and $F^-$ are 0.72 and 0.28, respectively. The mean activity coefficient of $CaF_2$ is ________ (up to two decimal places)
- For a 1 molal aqueous NaCl solution, the mean ionic activity coefficient ($\gamma_\pm$) and the Debye- Hückel Limiting Law constant (A) are related as
