Phenol red is an indicator with a \(\mathrm{pK}_{\mathrm{a}}\) of \(7.4\). It is yellow in acid solution and red in alkaline. (a) What is its \(K_{\mathrm{a}}\) ? (b) What is its \(\mathrm{pH}\) range? (c) What would its color be at \(\mathrm{pH}\) 7.4?

The relationship between Ka and pKa is given by the formula: \(pK_a = -\log_{10}(K_a)\). First, we need to use the given pKa value to calculate the Ka value as follows: - Rearrange the equation for Ka: \(K_a = 10^{-pK_a}\). - Plug in the given pKa value, 7.4, and calculate Ka: \(K_a = 10^{-7.4}\).

Now we can calculate the Ka value using the formula: \(K_a = 10^{-7.4}\). Evaluating the expression gives us: \(K_a \approx 3.98 \times 10^{-8}\).

The pH range of an indicator can be estimated by taking the pKa value ±1. This is because the pH range over which an indicator changes color depends on the concentration of the acidic and basic forms of the indicator in the solution. So, for Phenol Red, the pH range will be: \(pH = pK_a \pm 1\) \(pH = 7.4 \pm 1\) The pH range of Phenol Red is approximately 6.4 to 8.4.

At a pH value equal to the pKa of the indicator (in this case, 7.4), the concentration of the acidic form and the basic form of the indicator are equal. Phenol Red is yellow in an acidic solution and red in an alkaline solution. Since the pH value, 7.4, is equal to the pKa value, the solution is at the transition point between the acidic and basic forms of Phenol Red. At this point, the indicator will exhibit a color that is an intermediate between the acidic yellow and the basic red colors. The resulting color at pH 7.4 is orange. In summary: - (a) The Ka value for Phenol Red is approximately \(3.98 \times 10^{-8}\). - (b) The pH range for Phenol Red is approximately 6.4 to 8.4. - (c) The color of Phenol Red at pH 7.4 is orange.