Question

A few drops of each of the indicators shown in the accompanying table were placed in separate portions of a \(1.0-M\) solution of a weak acid, 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 \(K_{\mathrm{a}}\) for HX. $$ \begin{array}{|lcccc|} \hline \text { Indicator } & \begin{array}{c} \text { Color } \\ \text { of Hin } \end{array} & \begin{array}{c} \text { Color } \\ \text { of In }^{-} \end{array} & \begin{array}{c} \mathrm{pK}_{\mathrm{a}} \\ \text { of Hin } \end{array} & \begin{array}{c} \text { Color of } \\ 1.0 \mathrm{MHX} \end{array} \\ \text { Bromphenol blue } & \text { Yellow } & \text { Blue } & 4.0 & \text { Blue } \\ \text { Bromcresol purple } & \text { Yellow } & \text { Purple } & 6.0 & \text { Yellow } \\ \text { Bromcresol green } & \text { Yellow } & \text { Blue } & 4.8 & \text { Green } \\ \text { Alizarin } & \text { Yellow } & \text { Red } & 6.5 & \text { Yellow } \\\ \hline \end{array} $$

Short answer

The approximate pH of the weak acid HX is 5.0, and its approximate Ka value is \(10^{-10}\).

Step-by-step solution

Check the color of each indicator in the presence of HX

Observe the last column of the table, which shows the color of each indicator when exposed to the 1.0 M solution of HX. Note that Bromphenol blue turns blue, Bromcresol purple turns yellow, Bromcresol green turns green, and Alizarin turns yellow.

Determine the pH range of HX

Using the colors observed from each indicator and referring to their respective pKa values, we can establish a pH range for the solution of HX: - Since Bromphenol blue turns blue, the pH is higher than its pKa of 4.0. - Bromcresol green turns green, which means its pH is around its pKa of 4.8. - Bromcresol purple and Alizarin both turn yellow, which means the pH is lower than their pKa values of 6.0 and 6.5, respectively. Thus, we can conclude that the pH of the HX solution is somewhere between 4.8 and 6.0, and we can approximate it as 5.0 as the middle value.

Calculate the concentration of HX

Since we know that the concentration of the HX solution is 1.0 M, we can use the formula for Ka to determine the concentration of hydrogen ions (H+) and the weak base ions (X-). The formula is: \[ K_a = \frac{[H^+] [X^-]}{[HX]}\] However, we know that \([H^+]=[X^-]\), so we can simplify the equation to: \[ K_a = \frac{[H^+]^2}{[HX]}\]

Calculate Ka using the pH value

Now that we determined an approximate pH value of 5.0, we can calculate the concentration of hydrogen ions: \[[H^+] = 10^{-\text{pH}} = 10^{-5}\] Then, substitute the values for H+ and HX into the simplified Ka equation: \[K_a = \frac{(10^{-5})^2}{1}\] Solve for Ka: \[K_a = 10^{-10}\] So, the approximate value of Ka for HX is \(10^{-10}\).

Key concepts

Acid Dissociation Constant

Understanding the acid dissociation constant (\( K_a \)) is fundamental to grasping acid-base chemistry. The acid dissociation constant is a quantitative measure of the strength of an acid in solution and is defined as the equilibrium constant for the dissociation of a weak acid into its conjugate base and a proton (H+). In the case of a weak acid, HX, the dissociation reaction is represented as:

\[ HX \rightleftharpoons H^+ + X^- \]

The equilibrium expression for this reaction is:

\[ K_a = \frac{[H^+][X^-]}{[HX]} \]

The larger the value of the acid dissociation constant, the more the acid dissociates to form H+ and X-, and the stronger the acid. Conversely, a smaller value of \( K_a \) indicates a weaker acid. In the problem scenario, we determine the concentration of H+ ions and then use that to calculate the \( K_a \) for the weak acid HX. This constant is important for predicting the degree of dissociation and understanding the behavior of the acid in various chemical contexts.

pH Calculation

The pH is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It is calculated as the negative logarithm of the hydrogen ion concentration:\[ \text{pH} = -\log[H^+] \]

In this scenario, we use chemical indicators to estimate the pH range of a weak acid solution. Chemical indicators change color at specific pH values, and hence are utilizable to estimate the pH of a solution. Once the pH range is identified using the indicators, the middle value within this range gives us an approximate pH of the solution. We then convert this pH value into hydrogen ion concentration and proceed with further calculations related to the acid dissociation constant. Calculation of pH is not only essential for solving chemistry problems but is also vital in many practical applications like water quality testing and regulation of biochemical processes.

Chemical Indicators

Chemical indicators are substances that change color at certain pH levels due to a change in their structure and are invaluable for estimating the pH of a solution. Each indicator has a specific pH range over which it changes color, known as the transition range, which correlates to the pKa of the indicator.

For the problem at hand, we look at an array of indicators with different color changes at various pH levels. By observing the color of the indicators in the presence of the weak acid HX and correlating them with their respective pKa values, we are able to deduce a pH range for the acid solution. Using this approach doesn't give an exact pH value but allows us to approximate the value within a range. It's crucial to select an appropriate indicator that undergoes a color change within the pH range of interest for a given acid or base in various laboratory and industrial applications.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of reactants and products in a reversible chemical reaction when the rates of the forward and reverse reactions are equal, leading to no net change in the amounts of substances. In the context of weak acid dissociation, once the system has reached equilibrium, the concentrations of the weak acid (HX), and its dissociated ions (H+ and X-) remain constant.

To find the equilibrium concentration of these ions, we start with the initial concentration of the weak acid and the ±assumption that the concentration of H+ and X- produced upon dissociation is equal, as they come from the same molecule of acid. Then, by using the pH value or chemical indicators as a guide, we can calculate the equilibrium concentrations of the ions. This information allows us to understand how the acid behaves in solution, predict reaction outcomes, and calculate the pH of solutions in a variety of chemical processes.