Question

Calculate the mass of sodium hydroxide that must be added to 1.00 \(\mathrm{L}\) of \(1.00-M \mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\) to double the pH of the solution (assume that the added NaOH does not change the volume of the solution).

Short answer

The mass of sodium hydroxide (NaOH) required to double the pH of the 1.00 L solution of 1.00 M acetic acid (HC₂H₃O₂) can be calculated by following these steps: 1. Determine the initial pH using the Ka of acetic acid (1.8 x 10⁻⁵) and the given concentration. 2. Calculate the new pH by doubling the initial pH. 3. Find the moles of NaOH needed for neutralization based on [H⁺] ion concentration change. 4. Convert the moles of NaOH to mass using the molar mass of NaOH (40 g/mol). Following these steps, we can find the mass of NaOH needed to achieve the desired pH change.

Step-by-step solution

Find the initial pH of the acetic acid solution.

To calculate the initial pH of the acetic acid solution, we first need to determine the Ka (acid dissociation constant) for acetic acid. The Ka of acetic acid is 1.8 x 10⁻⁵. Then, we can write the equilibrium expression for acetic acid's ionization: \[HC_{2}H_{3}O_{2} \rightleftharpoons H^{+} + C_{2}H_{3}O_{2}^{-}\] Since the initial concentration of acetic acid is given as 1.00 M, we can set up the Ka expression: \[Ka = \frac{[H^{+}][C_{2}H_{3}O_{2}^{-}]}{[HC_{2}H_{3}O_{2}]}\] Plugging in the given values, we get: \[1.8 \times 10^{-5} = \frac{(x)(x)}{1 - x}\] Assuming that x is very small compared to 1, we can ignore the x in the denominator, so we get: \[1.8 \times 10^{-5} = x^2\] Now, we will solve for x (which represents the [H⁺] ion concentration): \[x = \sqrt{1.8\times10^{-5}}\]

Calculate the initial pH and double it.

Now that we have found the [H⁺] ion concentration, we can use the pH formula to calculate the pH of the acetic acid solution: \[ pH = -\log{[H^{+}]} \] Plug the calculated [H⁺] into the pH formula to find the initial pH: \[ pH = -\log{\sqrt{1.8\times10^{-5}}} \] Once we have the initial pH, we can double it to find the new pH: \[ new\ pH = 2 \times initial\ pH \]

Determine the moles of NaOH and acetic acid after neutralization.

Now that we have the new pH, we can use it to determine the concentration of [H⁺] ions: \[ [H^{+}] = 10^{-new\ pH} \] As NaOH neutralizes acetic acid on a 1:1 basis, the moles of NaOH required to neutralize the moles of [H⁺] ions can be determined directly from the concentration change. \[moles\ of\ NaOH = moles\ of\ H^{+}_{final} - moles\ of\ H^{+}_{initial}\]

Calculate the mass of NaOH

Now that we have the moles of NaOH needed, we can convert it to mass by using the molar mass of NaOH. The molar mass of NaOH is 22.99 g/mol (Na) + 15.99 g/mol (O) + 1.01 g/mol (H) = 40 g/mol: \[mass\ of\ NaOH = moles\ of\ NaOH \times molar\ mass\ of\ NaOH\] This will give us the mass of NaOH that must be added to the 1.00 L solution of 1.00 M acetic acid to double the pH of the solution.

Key concepts

Acid-Base Neutralization

Acid-base neutralization is a chemical process where an acid and a base react to form water and a salt, essentially cancelling each other's acidity and basicity. In this context, when sodium hydroxide (NaOH) is added to a solution of acetic acid (CH₃COOH), they undergo neutralization. The reaction proceeds as follows:

• The hydroxide ions (OH⁻) from NaOH react with the hydrogen ions (H⁺) from acetic acid.
• This forms water (H₂O), and acetic acid loses its acidic property.

This neutralization helps to increase the pH of the solution. Knowing how much base to add is crucial to achieve a specific pH change, such as doubling the initial pH in this exercise. Understanding these reactions helps in various applications, from industry processes to everyday tasks like determining the acidity level in food.

Acetic Acid

Acetic acid, with the chemical formula CH₃COOH, is a common weak acid known for giving vinegar its distinctive sour taste and odor. In solutions, it partially dissociates into acetate ions (CH₃COO⁻) and hydrogen ions (H⁺):

• This dissociation is not complete because acetic acid's ionization is limited by its Ka value.
• The equilibrium constant expression \[ Ka = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]} \] measures its tendency to dissociate.

In this exercise, understanding the degree of dissociation helps determine the initial pH of the acetic acid solution, which is used to calculate how much sodium hydroxide is needed to alter the pH. Since it's a weak acid, changes in its concentration are straightforward to track, which allows for accurate calculations when neutralizing it with strong bases like NaOH.

Sodium Hydroxide

Sodium hydroxide, or NaOH, is a strong base used in many chemical reactions, known for its ability to fully dissociate in water. This complete ionization makes it highly effective for neutralizing acids. When NaOH is added to an acetic acid solution, it dissociates to produce sodium ions (Na⁺) and hydroxide ions (OH⁻):

• The OH⁻ ions readily neutralize H⁺ ions from the acid.
• This reaction forms water, reducing the acidity of the solution and increasing its pH.

In our problem, we calculate the precise amount of NaOH needed to achieve the desired change in pH. This involves converting the calculated number of moles of NaOH into mass using its molar mass (40 g/mol). Sodium hydroxide's capability to completely neutralize acids makes it an important tool in both laboratory and industrial settings.

Ka (Acid Dissociation Constant)

The acid dissociation constant, Ka, is a measure of an acid's strength — i.e., its ability to donate protons (H⁺ ions). For acetic acid, the Ka value is 1.8 × 10⁻⁵, indicating that it is a weak acid. The Ka value helps in calculating the concentration of hydrogen ions in the solution, and thus, the pH:

• From the equilibrium expression \[ Ka = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]} \], we can deduce that only a fraction of acetic acid molecules ionize in solution.
• Understanding Ka is crucial for predicting the initial pH of the solution before and after neutralization.

When performing calculations, the Ka helps determine how the balance of acetic acid and its ions shifts with changes in pH. This is vital for calculating how much sodium hydroxide is needed to achieve the target pH change in this scenario, illustrating the importance of equilibrium constants in acid-base chemistry.