Question

If \(40.00 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}\) is titrated with \(0.100 \mathrm{M}\) \(\mathrm{HCl}\), calculate (a) the \(\mathrm{pH}\) at the start of the titration; (b) the volume of \(\mathrm{HCl}\) required to reach the first equivalence point and the predominant species present at this point; (c) the volume of \(\mathrm{HCl}\) required to reach the second equivalence point and the predominant species present at this point; (d) the pH at the second equivalence point.

Short answer

In summary, the pH at the start of the titration is determined by the basic CO3^2- ion concentration. The volume of HCl needed to reach the first equivalence point is 40.00 mL, and the predominant species at this point are Na+ and HCO3^-. The volume of HCl needed to reach the second equivalence point is 80.00 mL, with the predominant species being Na+ and H2CO3. The pH at the second equivalence point is calculated using the Ka1 value of H2CO3 and the measured H+ ion concentration.

Step-by-step solution

Write the balanced chemical equation for the reaction

The reaction between Na2CO3 (sodium carbonate) and HCl (hydrochloric acid) can be written as the following two equations since sodium carbonate is a diprotic base: 1) Na2CO3 (aq) + HCl (aq) → NaHCO3 (aq) + NaCl (aq) 2) NaHCO3 (aq) + HCl (aq) → H2CO3 (aq) + NaCl (aq)

Calculate the initial pH

We don't need the titration reaction for this part because it's the start of titration. Since Na2CO3 is a strong electrolyte, the main species in the solution are ions Na+ and CO3^2-; in this case, the pH is determined by the basic carbonate ion. The CO3^2- ion reacts with water to form bicarbonate and hydroxide ions: CO3^2- + H2O → HCO3^- + OH^- To find the initial pH, we can use the Kb of the CO3^2- ion (which can be calculated from the Ka1 and Ka2 of H2CO3). In this case, we find the concentration of OH^- ions, and then use the pOH to find the pH.

Volume of HCl needed to reach the first equivalence point

At the first equivalence point, all of the original Na2CO3 has reacted with HCl to form NaHCO3. To find the required volume of HCl to reach this point, we can use stoichiometry. We were given 40.00 mL of 0.100 M Na2CO3 solution, which contains 4.0 mmol of Na2CO3. The stoichiometry from the balanced chemical equation tells us that 1 mole of Na2CO3 reacts with 1 mole of HCl. Therefore, we will need 4.0 mmol of HCl to reach the first equivalence point. Since the HCl solution is 0.100 M, we can calculate the volume needed by using: volume of HCl = (4.0 mmol) / (0.100 M) = 40.00 mL The predominant species present at the first equivalence point will be Na+ and HCO3^- ions.

Volume of HCl needed to reach the second equivalence point

At the second equivalence point, all of the NaHCO3 formed has reacted with HCl to form H2CO3. From the stoichiometry, we know that 1 mole of NaHCO3 reacts with 1 mole of HCl. So, to react with 4.0 mmol of NaHCO3, we would need an additional 4.0 mmol of HCl. This would require another 40.00 mL of the 0.100 M HCl solution, making a total volume of 80.00 mL to reach the second equivalence point. The predominant species present here are Na+ and H2CO3 ions.

Calculate the pH at the second equivalence point

At the second equivalence point, the pH is determined by the weak acid H2CO3. Since H2CO3 is a weak acid, we will need to use its Ka1 value to calculate the concentration of H+ ions in the solution. For this purpose, we will set up an ICE table and solve for the H+ concentration, then use it to find the pH of the solution. From the calculated H+ concentration, we will use the pH formula (pH = -log[H+]) to find the pH at the second equivalence point.

Key concepts

Equivalent point

An equivalence point in a titration is a crucial concept. It is the point at which the amount of titrant added is stoichiometrically equal to the amount of substance present in the solution being titrated. In simpler terms, it is where the acid exactly neutralizes the base, or vice versa.
For a titration involving a diprotic acid like \({\text{H}_2\text{CO}_3}\), there are multiple equivalence points. Each relates to the neutralization of one of the acidic protons.
When titrating sodium carbonate (a diprotic base) with hydrochloric acid:

• The first equivalence point is when the first proton is neutralized to form \({\text{NaHCO}_3}\).
• The second equivalence point occurs when the second proton reacts, resulting in the formation of \({\text{H}_2\text{CO}_3}\).

Understanding equivalence points helps predict the predominant species in solution and is key to performing accurate pH calculations.

Diprotic acid

A diprotic acid, such as carbonic acid, can donate two protons. This characteristic makes their titrations slightly more complex as they involve two equivalence points. When these acids lose one proton, they form an intermediate species that can still donate another proton.
In the context of \({\text{Na}_2\text{CO}_3}\) titrated with \({\text{HCl}}\), sodium carbonate plays the role of a diprotic base. This means it reacts with acids in two stages:

• First, to form sodium bicarbonate \({\text{NaHCO}_3}\)
• Second, to form carbonic acid \({\text{H}_2\text{CO}_3}\)

Tackling diprotic acids requires understanding their two-step dissociation process. Each stage affects pH differently, which makes titration with diprotic acids unique and needs precision.

pH calculation

Calculating pH during an acid-base titration requires a good grasp of several principles. Initially, you consider the strength of the solutions involved.

• At the start, the pH of the base (like \({\text{Na}_2\text{CO}_3}\)) in solution is influenced by its concentration and its tendency to form hydroxide ions in water.
• Between equivalence points, you calculate pH based on the presence of the intermediate species, like bicarbonate \({\text{HCO}_3^-}\).
• At each equivalence point, the pH is determined by the resultant species. For example, weak acids like \({\text{H}_2\text{CO}_3}\) direct pH at the second equivalence point.

To find pH:1. For \({\text{H}_2\text{CO}_3}\), use the acid dissociation constant (Ka) to settle the concentration of hydronium ions.2. Apply the formula \(\text{pH} = -\log[\text{H}^+]\).A detailed understanding will aid in predicting pH changes during real-time titration.

Stoichiometry

Stoichiometry is essential in chemical reactions, especially in titration processes. It allows the calculation of reactants needed and products formed. It is based on the balanced chemical equations and mole-to-mole relationships.
For titration calculations:

• Determine the moles of the initial substance (\(\text{e.g., } \text{Na}_2\text{CO}_3\)) using its volume and molarity.
• Apply balanced equations to find out how many moles of titrant (like \({\text{HCl}}\)) are necessary to reach each equivalence point.
• The stoichiometric coefficients in a balanced equation will tell you the mole ratio between the reactants.

In our example using \({\text{Na}_2\text{CO}_3}\) and \({\text{HCl}}\):1. The first equivalence point requires a 1:1 reaction ratio between \({\text{Na}_2\text{CO}_3}\) and \({\text{HCl}}\) to form \({\text{NaHCO}_3}\).2. The second equivalence point uses another 1:1 ratio with \({\text{NaHCO}_3}\) and \({\text{HCl}}\).By understanding stoichiometry, you can accurately calculate titrant volumes needed for precise titration results.