An aqueous solution contains 0.02 mol kg-1 NaCl and 0.03 mol kg-1 Ca(NO3)2. The logarithm of the mean ionic activity coefficient (logγ±) of this solution at 25°C is
Ionic Strength
The ionic strength of a solution quantifies the total concentration of ions, taking into account the charge of each ion. For a mixed solution containing multiple electrolytes, the total ionic strength is the sum of the contributions from all ions present.
The formula for calculating ionic strength \(I\) is given by:
\[ I = \frac{1}{2} \sum_i m_i Z_i^2 \]where \(m_i\) is the molality of ion \(i\) (in mol kg\(^{-1}\)) and \(Z_i\) is the charge of ion \(i\).
Ionic Strength from NaCl
The solution contains 0.02 mol kg\(^{-1}\) of NaCl. When NaCl dissolves in water, it dissociates completely into Na\(^+\) and Cl\(^-\) ions.
- For Na\(^+\) ions: molality \(m_{\text{Na}^+}\) = 0.02 mol kg\(^{-1}\), charge \(Z_{\text{Na}^+}\) = +1.
- For Cl\(^-\) ions: molality \(m_{\text{Cl}^-}\) = 0.02 mol kg\(^{-1}\), charge \(Z_{\text{Cl}^-}\) = -1.
The contribution to the ionic strength from NaCl is:
\[ I_{\text{NaCl}} = \frac{1}{2} (m_{\text{Na}^+} Z_{\text{Na}^+}^2 + m_{\text{Cl}^-} Z_{\text{Cl}^-}^2) \] \[ I_{\text{NaCl}} = \frac{1}{2} ((0.02)(+1)^2 + (0.02)(-1)^2) \] \[ I_{\text{NaCl}} = \frac{1}{2} (0.02 + 0.02) = \frac{1}{2} (0.04) = 0.02 \, \text{mol kg}^{-1} \]Ionic Strength from Ca(NO\(_3\))\(_2\)
The solution also contains 0.03 mol kg\(^{-1}\) of Ca(NO\(_3\))\(_2\). This salt dissociates into Ca\(^{2+}\) and two NO\(_{3}^-\) ions.
- For Ca\(^{2+}\) ions: molality \(m_{\text{Ca}^{2+}}\) = 0.03 mol kg\(^{-1}\), charge \(Z_{\text{Ca}^{2+}}\) = +2.
- For NO\(_{3}^-\) ions: molality \(m_{\text{NO}_3^-}\) = 2 \(\times\) 0.03 = 0.06 mol kg\(^{-1}\), charge \(Z_{\text{NO}_3^-}\) = -1.
The contribution to the ionic strength from Ca(NO\(_3\))\(_2\) is:
\[ I_{\text{Ca(NO}_3)_2} = \frac{1}{2} (m_{\text{Ca}^{2+}} Z_{\text{Ca}^{2+}}^2 + m_{\text{NO}_3^-} Z_{\text{NO}_3^-}^2) \] \[ I_{\text{Ca(NO}_3)_2} = \frac{1}{2} ((0.03)(+2)^2 + (0.06)(-1)^2) \] \[ I_{\text{Ca(NO}_3)_2} = \frac{1}{2} ((0.03)(4) + (0.06)(1)) \] \[ I_{\text{Ca(NO}_3)_2} = \frac{1}{2} (0.12 + 0.06) = \frac{1}{2} (0.18) = 0.09 \, \text{mol kg}^{-1} \]Total Ionic Strength
The total ionic strength \(I\) of the mixed aqueous solution is the sum of the ionic strengths contributed by each salt:
\[ I = I_{\text{NaCl}} + I_{\text{Ca(NO}_3)_2} \] \[ I = 0.02 + 0.09 = 0.11 \, \text{mol kg}^{-1} \]Mean Ionic Activity Coefficient
The mean ionic activity coefficient (\(\gamma_{\pm}\)) accounts for the non-ideal behavior of ions in solution due to interionic interactions. The logarithm of the mean ionic activity coefficient (\(\log \gamma_{\pm}\)) is related to the ionic strength \(I\) of the solution. For dilute solutions, the Debye-Hückel limiting law provides this relationship, although the options suggest a simplified form is expected.
Based on the format of the given options, the logarithm of the mean ionic activity coefficient corresponds to the negative square root of the total ionic strength:
\[ \log \gamma_{\pm} = -\sqrt{I} \]Using the calculated total ionic strength \(I = 0.11 \, \text{mol kg}^{-1}\):
\[ \log \gamma_{\pm} = -\sqrt{0.11} \]This value matches one of the provided options.
Important Questions from Electrochemistry
- At $298 \, K$, given the standard electrode potentials: $E^\circ_{Cu^{2+}/Cu} = 0.34 \, V$, $E^\circ_{Zn^{2+}/Zn} = -0.76 \, V$, $E^\circ_{Fe^{2+}/Fe} = -0.44 \, V$, and $E^\circ_{Ag^{+}/Ag} = 0.80 \, V$.
Based on these values, which of the following reactions is NOT expected to occur spontaneously under standard conditions? - Which of the following processes is required for extracting metal from cinnabar ore?
You are given three metals 'X', 'Y' and 'Z'. Metal 'X' is found to react with an aqueous solution of both YSO 4and ZSO 4whereas metal 'Z' is found to react only with aqueous solution of YSO 4. Based on these observations, select the correct statement from the following.
The mobility of a divalent cation in water is 8 × 10-8 m2 V-1 s-1. The effective radius of the ion is (viscosity of water = 1 cP : c = 1.6 × 10-19 C)
The electrical double layer model among the following that consists of both fixed and diffuse layers is
