Question

According to Debye–Hückel limiting law, the mean molal activity coefficient for 0.87 g K\(_2\)SO\(_4\) (molar mass = 174 g mol\(^{-1}\)) in 1 kg of water at 25 °C is _________ (rounded off to two decimal places). 
 

Hint:

For \(\mathrm{A_xB_y}\) electrolytes, use \(m_i=\nu_i m\) in \(I=\tfrac12\sum m_i z_i^2\). At 25 °C, \(A=0.509\) in the Debye–Hückel limiting law.

Answer 1

Step 1: Molality.
Moles of salt \(= \frac{0.87}{174} = 0.005\ \text{mol}\). In 1 kg water, \(m = 0.005\ \text{mol kg}^{-1}\).
Step 2: Ionic strength \(I\).
\(\mathrm{K_2SO_4 \to 2K^+ + SO_4^{2-}}\). Using \(I = \tfrac12 \sum m_i z_i^2\):
\[ I=\tfrac12[(2m).(1)^2 + (m).(2)^2] = \tfrac12(0.01 + 0.02)=0.015. \] Step 3: Debye–Hückel limiting law (25 °C).
\[ \log_{10}\gamma_{\pm} = -A|z_+z_-|\sqrt{I} = -0.509\times 2 \times \sqrt{0.015}. \] \[ \sqrt{0.015}=0.1225⇒ \log_{10}\gamma_{\pm}\approx -0.1247. \] Step 4: Evaluate \(\gamma_{\pm}\).
\[ \gamma_{\pm}=10^{-0.1247}\approx 0.75 \ (\text{to two decimals}). \]