The molar conductivity of acetic acid solution at infinite dilution is 390 S cm$^2$ mol$^{-1}$. What is the molar conductivity of 0.01 M acetic acid solution (in S cm$^2$ mol$^{-1}$)? (Given: $K_a(\text{CH}_3\text{COOH})=1.8\times10^{-5}$, assume $1-\alpha \approx 1$)
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For weak electrolytes, the degree of dissociation $\alpha$ can be found using Ostwald's dilution law: $K_a = \frac{C\alpha^2}{1-\alpha}$. For very weak electrolytes, this simplifies to $\alpha = \sqrt{K_a/C}$. The molar conductivity is then found using Kohlrausch's law in the form $\Lambda_m^C = \alpha \Lambda_m^\circ$.