Question

At a given temperature, \(0.45\,\text{g}\) of acetic acid in \(50\,\text{mL}\) of water is shaken with \(1.0\,\text{g}\) of charcoal and the pH of the resulting solution is \(3.0\). Assume, the adsorption of acetic acid from the aqueous solution by charcoal follows Freundlich isotherm,

\[ \frac{x}{m} = kC^{1/n} \]
If the plot of \(\log_{10}(x/m)\) against \(\log_{10} C\) gives a straight line with slope \(1\), the value of \(k\) in \(\text{L mol}^{-1}\) is _______.

Given: The molar mass of acetic acid is \(60\,\text{g mol}^{-1}\).
The acid dissociation constant of acetic acid is \(1.0 \times 10^{-5}\) at the given temperature.
\(x\) is the mass (in grams) of acetic acid adsorbed.
\(m\) is the mass (in grams) of charcoal.
\(C\) is the equilibrium concentration of acetic acid in the solution after the adsorption is complete.
\(k\) and \(n\) are constants for acetic acid–charcoal system at the given temperature.

Hint:

For adsorption problems, carefully distinguish between initial and equilibrium concentrations. Use pH to find $[H^+]$ and then $K_a$ to find the equilibrium concentration of the undissociated acid. Account for the amount adsorbed ($x$) by subtracting the equilibrium amount in solution from the initial amount.

Answer 1