Question

Methyl red has a \(\mathrm{pK}_{\mathrm{a}}\) of 5.0 and is red in its acid form and yellow in its basic form. If several drops of this indicator are placed in a 25.0-mL sample of \(0.100 \mathrm{M} \mathrm{HCl}\), what color will the solution appear? If \(0.100 \mathrm{M} \mathrm{NaOH}\) is slowly added to the \(\mathrm{HCl}\) sample, in what \(\mathrm{pH}\) range will the indicator change color?

Short answer

The initial color will be red since the solution pH is 1, which is lower than the \(\mathrm{pK}_{\mathrm{a}}\) of methyl red. The color change will occur approximately from pH 4.0 to pH 6.0 as NaOH is added.

Step-by-step solution

Determine the Initial pH

First, determine the pH of the initial solution of 0.100 M HCl, which is a strong acid and dissociates completely in water. The pH is calculated using the formula \(\mathrm{pH} = -\log([\mathrm{H}^+])\). Since the concentration of HCl is 0.100 M, the concentration of hydronium ions \( [\mathrm{H}^+] \) is also 0.100 M. Therefore, the pH is \(\mathrm{pH} = -\log(0.100)\).

Calculate pH of HCl Solution

Calculating the pH gives \(\mathrm{pH} = -\log(0.100) = 1\). Since the pH is significantly lower than the \(\mathrm{pK}_{\mathrm{a}}\) of methyl red, the solution will be acidic enough for the indicator to be in its acid form, which is red.

Identify the Color Change pH Range

The color change of methyl red occurs around its \(\mathrm{pK}_{\mathrm{a}}\), which is 5.0. Typically, the color change range for a pH indicator is about plus or minus 1 pH unit from its \(\mathrm{pK}_{\mathrm{a}}\). Therefore, the color change for methyl red will occur approximately from pH 4.0 to pH 6.0.

Key concepts

pH Calculation

Understanding how to calculate pH is fundamental in chemistry, especially when dealing with solutions of acids and bases. The pH scale is used to quantify the acidity or basicity of an aqueous solution. The scale generally runs from 0 to 14, with 7 being neutral. A pH less than 7 indicates acidity, and a pH greater than 7 indicates basicity.

The pH of a solution can be calculated using the formula: \[\text{pH} = -\log([\mathrm{H^+}])\] Here, \([\mathrm{H^+}]\) represents the concentration of hydrogen ions in moles per liter (Molarity). In the context of our exercise, the initial solution of 0.100 M HCl (a strong acid) will have a pH calculated by simply taking the negative logarithm of its molarity, which yields a pH of 1. This calculation shows that the solution is highly acidic.

pKa Value

The pKa value is a crucial concept in acid-base chemistry and indicates the strength of an acid. It is defined as the negative base-10 logarithm of the acid dissociation constant (Ka) of a solution, telling us how readily an acid donates its proton (H+). The lower the pKa value of an acid, the stronger it is, meaning it dissociates more completely in solution.

Mathematically, it's expressed as: \[\text{pKa} = -\log(\text{Ka})\] In our exercise, the pKa value given for methyl red is 5.0. This pKa value is of particular importance because it is close to the middle of the pH scale, making methyl red useful for detecting changes in pH around the neutral zone of the pH scale.

Color Change of Indicators

Acid-base indicators are substances that exhibit different colors at different pH levels due to changes in their molecular structure. The color change typically occurs over a range of pH values, which is usually about plus or minus 1 pH unit from the indicator's pKa value. This range is where the solution changes from being mostly one form of the indicator to mostly the other.

For instance, methyl red transitions from red to yellow over a pH range centered around its pKa of 5. Knowing this range is crucial for accurate titration experiments, as it allows chemists to determine the endpoint of a reaction. In the exercise provided, methyl red would start to noticeably change from red to yellow as the pH moves from around 4.0 to 6.0.

Methyl Red

Methyl red is a pH indicator commonly used in titration because of its clearly defined color change. In an acidic environment, it appears red due to the presence of excess hydrogen ions that stabilize the structure causing the red form. As the environment becomes less acidic and moves towards basic, the indicator changes to its yellow form.

In our scenario, methyl red is added to an HCl solution with a pH of 1, so it initially shows its acidic color which is red. As NaOH is added and the solution becomes less acidic, there will be a gradual shift toward yellow. This shift can be used to visually estimate the pH of the solution as it crosses the pKa value of methyl red.

Strong Acid Dissociation

Strong acids, such as hydrochloric acid (HCl), dissociate completely in aqueous solution to produce hydronium ions (H3O+) and their corresponding anions. The term 'strong' refers to the acid's ability to completely ionize in water. This property of strong acids results in a low pH value for their solutions.

The dissociation of HCl is described by the equation: \[\text{HCl} (aq) \longrightarrow \text{H}^+ (aq) + \text{Cl}^- (aq)\] Because the dissociation is complete, the concentration of hydronium ions in the solution equals the initial concentration of HCl, facilitating straightforward pH calculations as demonstrated in the exercise.