Chemistry - Ionic Equilibrium Question with Solution | TestHub

In a given mixture, the ionization of two acids can be written as : Let$\mathrm{\alpha },\mathrm{}\mathrm{\beta }$be degree of ionization at same concentration.
$\underset{1-\mathrm{\alpha }}{\mathrm{C}{\mathrm{H}}_3\mathrm{C}\mathrm{O}\mathrm{O}\mathrm{H}}\rightleftharpoons \underset{\mathrm{\alpha }}{\mathrm{C}{\mathrm{H}}_3\mathrm{C}\mathrm{O}{\mathrm{O}}^{-}+}\underset{\mathrm{\alpha }+\mathrm{\beta }}{{\mathrm{H}}^{+}}\mathrm{}$
$\underset{1-\mathrm{\beta }}{{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{C}\mathrm{O}\mathrm{O}\mathrm{H}}\rightleftharpoons \underset{\mathrm{\beta }}{{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{C}\mathrm{O}{\mathrm{O}}^{-}+}\underset{\mathrm{\alpha }+\mathrm{\beta }}{{\mathrm{H}}^{+}}\mathrm{}$
$\therefore {\text{K}}_{\text{Acetic}\text{acid}}\text{=}\frac{\left[\text{α}\right]\left[\text{α}\text{+}\text{β}\right]\text{·c}}{\left[\text{1}-\text{α}\right]}$
${\text{K}}_{\text{Propionic}\text{acid}}\text{=}\frac{\left[\text{β}\right]\left[\text{α}\text{+}\text{β}\right]\cdot \text{c}}{\left[\text{1}-\text{β}\right]}$
$\therefore \frac{{\text{K}}_{\text{Acetic}\text{acid}}}{{\text{K}}_{\text{Propionic}\text{acid}}}\text{=}\frac{\text{α}}{\text{1}-\text{α}}\text{×}\frac{\left[\text{1}-\text{β}\right]}{\text{β}}$
or$\frac{\mathrm{\alpha }}{1-\mathrm{\alpha }}=\frac{1·75}{1·3}\times \left[\frac{\mathrm{\beta }}{1-\mathrm{\beta }}\right]\mathrm{}$