Question

In the titration of \(10.00 \mathrm{mL}\) of \(0.04050 \mathrm{M} \mathrm{HCl}\) with \(0.01120 \mathrm{M} \mathrm{Ba}(\mathrm{OH})_{2}\) in the presence of the indicator 2,4-dinitrophenol, the solution changes from colorless to yellow when 17.90 mL of the base has been added. What is the approximate value of \(\mathrm{p} K_{\mathrm{HIn}}\) for 2,4 -dinitrophenol? Is this a good indicator for the titration?

Short answer

The approximate value of \(pK_{\mathrm{HIn}}\) for 2,4 -dinitrophenol is slightly less than 7. 2,4 -dinitrophenol is a suitable indicator for this titration, although it is not the ideal one since its color change does not exactly coincide with the equivalence point.

Step-by-step solution

Identify the stoichiometry of the reaction

We need to first understand the reaction between \(HCl\) and \(Ba(OH)_2\). This reaction can be represented by the equation \(2HCl + Ba(OH)_2 \rightarrow BaCl_2 + 2H_2O\). Hence, the ratio of \(HCl\) to \(Ba(OH)_2\) in a balanced reaction is 2:1.

Calculate the moles of \(HCl\) and find equivalence point

The volume of \(HCl\) is given as 10.00 mL and its molarity as 0.04050 M. We can calculate the moles of \(HCl\) using the formula Molarity = moles/volume. Therefore, moles of \(HCl\) = volume * molarity = 10.00 mL * 0.04050 M = 0.000405 mol. At the equivalence point, all the \(HCl\) would have reacted with \(Ba(OH)_2\) based on the 2:1 ratio, meaning the moles of \(Ba(OH)_2\) needed is half the moles of \(HCl\), which is 0.0002025 mol.

Calculate volume of \(Ba(OH)_2\) required

Next, we calculate the volume of \(Ba(OH)_2\) required to reach the equivalence point by using the formula Volume = moles/Molarity. Therefore, Volume = 0.0002025 mol/0.01120 M = 18.08 mL.

Calculate the pH at equivalence point

The pH at the equivalence point of a titration of a strong acid with a strong base is 7.

Find \(pK_{\mathrm{HIn}}\)

Based on the color change of the indicator, the pH at which 2,4-dinitrophenol changes color is the same as the \(pK_{\mathrm{HIn}}\). Since the color change occurred before the equivalence point (17.90 mL compared to 18.08 mL), the pH is slightly less than 7. Since pH decreases with increasing H+ concentration, \(pK_{\mathrm{HIn}}\) would be slightly less than 7.

Evaluate the suitability of the indicator

A good indicator for a titration would change color at the same pH as the equivalence point. In this case, the \(pK_{\mathrm{HIn}}\) is slightly less than the pH at equivalence point, and so 2,4-dinitrophenol is not a perfect indicator for the titration but still a fairly good one since the difference is small.

Key concepts

Stoichiometry

Stoichiometry plays a crucial role in acid-base titration as it helps us determine the precise relationships between reactants in a chemical reaction. In the given exercise, we're dealing with a titration between hydrochloric acid (HCl) and barium hydroxide (Ba(OH)_2). The balanced chemical equation for this reaction is:\[ 2 \text{HCl} + \text{Ba(OH)}_2 \rightarrow \text{BaCl}_2 + 2 \text{H}_2\text{O} \]

• This equation tells us that two moles of HCl react with one mole of Ba(OH)_2.
• Understanding this stoichiometric relationship is vital for accurately calculating the moles of each reactant required for the reaction to reach its endpoint.

To determine how much Ba(OH)_2 is needed to neutralize a given amount of HCl, we first calculate the moles of HCl using its volume and molarity. Knowing the stoichiometry, we can then deduce the moles of Ba(OH)_2 needed to neutralize these moles of HCl, using a simple ratio of 2:1. This foundational concept ensures that we know just the right amount of titrant (Ba(OH)_2) to add to reach the equivalence point effectively.

pH Calculation

pH calculation is a central concept in acid-base titrations, as it helps us understand the acidity or basicity of a solution at various points during the titration. To comprehend this, we must first understand that pH is a measure of hydrogen ion concentration in a solution. During the titration process, when the acid HCl is neutralized by the base Ba(OH) _2 , the pH of the solution changes from acidic (low pH) towards neutral (pH 7) and eventually to basic (high pH) when excess base is added. The equivalence point in this strong acid-strong base titration occurs where the pH is equal to 7 because both the acid and base completely neutralize each other.

• Initially, before adding any base, the pH is determined by the concentration of HCl.
• As Ba(OH) _2 is added, the pH begins to increase more rapidly as we get closer to the equivalence point.
• At this point, theoretically, the pH is 7 in a perfect scenario.

Calculating the exact pH before and after the equivalence point helps us understand how close we are to completing the reaction and when an appropriate indicator will signal this event with a color change.

Titration Indicators

Titration indicators are substances that help observe the completion of a titration by changing color at certain pH levels. In this exercise, 2,4-dinitrophenol is used as an indicator. Here’s what you need to know about indicators:

• Indicators work by changing color when the pH of a solution reaches a specific value, known as the pK_{ ext{HIn}} .
• The pK_{ ext{HIn}} of 2,4-dinitrophenol is slightly less than 7.
• This means it will change color when the pH is slightly less than 7, which happens just before reaching the equivalence point in this titration.

For a strong acid-strong base titration, an ideal indicator would change color precisely at the equivalence point pH of 7. While 2,4-dinitrophenol doesn’t meet this requirement perfectly, it still provides a fairly good estimate of when the endpoint is approached. Choosing the right indicator depends on how closely the pK_{ ext{HIn}} matches the expected pH at the equivalence point. Thus, knowing when an indicator will change its color is key to accurately detecting the endpoint of a titration.