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, \( \dfrac{x}{m} = k C^{1/n} \).
If the plot of \( \log_{10}\!\left(\dfrac{x}{m}\right) \) 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.
