Question

Two drops of indicator $\mathrm{HIn}(K_{\mathrm{a}}=1.0\times {10}^{-9})$, where HIn is yellow and ${\mathrm{In}}^{-}$ is blue, are placed in $100.0\mathrm{mL}$ of $0.10\mathrm{M}\mathrm{HCl}$. a. What color is the solution initially? b. The solution is titrated with $0.10M\mathrm{NaOH}$. At what $\mathrm{pH}$ will the color change (yellow to greenish yellow) occur? c. What color will the solution be after $200.0\mathrm{mL}\mathrm{NaOH}$ has been added?

Short answer

a. The initial color of the solution is yellow. b. The color change from yellow to greenish-yellow occurs at pH = 9. c. The final color of the solution after 200.0 mL of NaOH is added will be blue.

Step-by-step solution

a. Initial color of the solution

The HCl solution is acidic, which means it has a high concentration of H+ ions. When the indicator, HIn, is added to the acidic solution, it will primarily be present in its acidic form, which is yellow. Therefore, the initial color of the solution will be yellow.

b. Estimating the pH at which the color change occurs

To determine the pH at which the color change from yellow to greenish-yellow occurs, we need to find the pH at which the concentration of HIn and In- are equal. We are given the dissociation constant, Ka, for the indicator: $K_a=1.0\times {10}^{-9}$ The equation for Ka is: $$K_a=\frac{[H^{+}][I{n}^{-}]}{[HIn]}$$ When the concentrations of HIn and In- are equal, the equation becomes: $$K_a=[H^{+}]$$ Next, we need to find the pH at which this equilibrium occurs. The definition of pH is: $$pH=-\log [H^{+}]$$ Combining the two equations, we get: $$pH=-\log (K_a)=-\log (1.0\times {10}^{-9})=9$$ Therefore, the color change will occur at a pH of 9.

c. Final color of the solution after 200 mL of NaOH is added

To determine the final color, we need to consider the amounts of HCl and NaOH. The initial moles of HCl are given by: Moles of HCl = Initial concentration × Volume = 0.10 M × 0.100 L = 0.010 mol After adding 200.0 mL (0.200 L) of 0.10 M NaOH, we have: Moles of NaOH = Concentration × Volume = 0.10 M × 0.200 L = 0.020 mol Since the moles of NaOH exceed the moles of HCl, the remaining solution will be basic. The difference in moles of NaOH and HCl is: Moles of OH- ions = Moles of NaOH - Moles of HCl = 0.020 mol - 0.010 mol = 0.010 mol In the basic solution, the concentration of OH- ions is given by: $$[O{H}^{-}]=\frac{0.010\text{ mol}}{0.100\text{ L}+0.200\text{ L}}=\frac{0.010\text{ mol}}{0.300\text{ L}}=0.033\text{ M}$$ We can now calculate the pOH using: $$pOH=-\log [O{H}^{-}]=-\log (0.033)=1.48$$ Since pH and pOH are related by: $$pH+pOH=14$$ We find the final pH of the solution: $$pH=14-pOH=14-1.48=12.52$$ As the final pH is above 9, the indicator will be present as In-, which has a blue color. Thus, the final color of the solution will be blue.

Key concepts

pH Calculation

The concept of pH is crucial in understanding the acidity or basicity of a solution. It measures the concentration of hydrogen ions $H^{+}$ in a solution, which determines how acidic or basic it is. The pH scale ranges from 0 to 14:

• A pH less than 7 indicates an acidic solution.
• A pH of 7 indicates a neutral solution.
• A pH greater than 7 indicates a basic solution.

To calculate pH, use the formula: $$pH=-\log [H^{+}]$$ For example, if the concentration of $H^{+}$ ions in a solution is $1.0\times {10}^{-9}$, the pH is $$pH=-\log (1.0\times {10}^{-9})=9$$ This tells us the solution is around neutral to slightly basic. Understanding pH helps predict the behavior of a solution in reactions such as acid-base titrations.

Chemical Indicators

Chemical indicators are substances added to a solution to visually signal the acidic or basic nature of the environment. They change color at specific pH levels, making them useful in titrations to determine endpoint.
Here's how indicators work:

• Different indicators change color at different pH ranges, hence choosing an appropriate indicator is crucial.
• During a titration, the indicator will change color once the solution reaches a certain pH, known as the equivalence point.
• For example, phenolphthalein turns from colorless to pink as a solution changes from acidic to basic.

In the exercise we considered, a chemical indicator $\mathrm{HIn}$ was used, turning from yellow to greenish-yellow at a pH of 9. This shift indicates a chemical reaction has progressed sufficiently, hinting at a balance between the amounts of acid and base present.

Dissociation Constant (Ka)

The dissociation constant $K_a$ helps measure the strength of an acid in solution. It represents the equilibrium constant for the dissociation of an acid into its ions. A larger $K_a$ value indicates a stronger acid capable of dissociating more in solution. Conversely, a smaller $K_a$ suggests a weaker acid.
The formula for $K_a$ is: $$K_a=\frac{[H^{+}][A^{-}]}{[HA]}$$ where $HA$ is the acid, $H^{+}$ is the hydrogen ion, and $A^{-}$ is the conjugate base.
In our specific example, $K_a$ for the indicator $\mathrm{HIn}$ was $1.0\times {10}^{-9}$, highlighting it as a weak acid. This value influences the pH at which the indicator changes color. Knowing $K_a$ can be crucial in calculating pH for buffer solutions and during titration analysis, assisting in determining the completion of the reaction.