A few drops of each of the indicators shown in the accompanying table were placed in separate portions of a \(1.0 \mathrm{M}\) solution of a weak acid, \(\mathrm{HX}\). The results are shown in the last column of the table. What is the approximate \(\mathrm{pH}\) of the solution containing HX? Calculate the approximate value of \(K_{\alpha}\) for \(\mathrm{HX}\). \begin{tabular}{|lclcc|} \hline Indicator & Color of Hin & Color of \(\mathrm{In}^{-}\) & \(\mathrm{p} \boldsymbol{K}_{\mathrm{a}}\) of Hin & Color of \(1.0 \mathrm{M} \mathrm{HX}\) \\\ \hline Bromphenol blue & Yellow & Blue & \(4.0\) & Blue \\ Bromcresol purple & Yellow & Purple & \(6.0\) & Yellow \\ Bromcresol green & Yellow & Blue & \(4.8\) & Green \\ Alizarin & Yellow & Red & \(6.5\) & Yellow \\ \hline \end{tabular}

We have four indicators with four different color changes at different \(\text{pH}\) values. To determine the approximate \(\text{pH}\) range of the \(\text{HX}\) solution, we need to look at the color change observed for each indicator in the presence of the \(\text{HX}\) solution. Bromphenol blue: \(\text{pK}_\text{a} = 4.0\,\text{and}\,\text{Color} = \text{Blue}\) \\ Bromcresol purple: \(\text{pK}_\text{a} = 6.0\,\text{and}\,\text{Color} = \text{Yellow}\) \\ Bromcresol green: \(\text{pK}_\text{a} = 4.8\,\text{and}\,\text{Color} = \text{Green}\) \\ Alizarin: \(\text{pK}_\text{a} = 6.5\,\text{and}\,\text{Color} = \text{Yellow}\) From this information, we can infer that the \(\text{pH}\) range of the \(\text{HX}\) solution is between \(4-6\). Thus, an approximate pH of the solution could be: \[ \text{pH} \approx 5 \]

Since the weak acid \(\text{HX}\) has an approximate \(\text{pH}\) of \(5\), we can use the weak acid equilibrium expression to calculate the value of its dissociation constant \(K_{\alpha}\): \[ \text{HX} \rightleftharpoons \text{H}^{+} + \text{X}^{-} \] \[ K_{\alpha} = \frac{[\text{H}^{+}][\text{X}^{-}]}{[\text{HX}]} \] Since the \(\text{pH}\) is approximately \(5\), we know that the concentration of \(\text{H}^{+}\) is \(10^{-5}\,\text{M}\). Using the approximation that \(\text{HX}\) dissociates very slightly, we can assume that the concentration of \(\text{HX}\) remaining is approximately equal to the initial concentration, which is \(1.0\,\text{M}\). Also, the concentration of \(\text{X}^{-}\) is equal to the concentration of \(\text{H}^{+}\), which is \(10^{-5}\,\text{M}\). Plugging these values into the equilibrium expression for \(K_{\alpha}\), we have: \[ K_{\alpha} \approx \frac{(10^{-5})(10^{-5})}{1.0} \] \[ K_{\alpha} \approx 10^{-10} \] So, the approximate value of the dissociation constant \(K_{\alpha}\) for \(\text{HX}\) is \(10^{-10}\).