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 \(\mathrm{pH}\) of the solution (assume that the added \(\mathrm{NaOH}\) does not change the volume of the solution).

Short answer

To double the pH of the 1.00 L of 1.00 M acetic acid solution, approximately \(2.14 \times 10^{-8} \mathrm{g}\) of sodium hydroxide (NaOH) must be added.

Step-by-step solution

Calculate the initial pH of the acetic acid solution

First, let's determine the initial pH of the 1.00 M HC2H3O2 solution. Since this is a weak acid, we will use the acid dissociation constant formula: \(K_a = \frac{[H^+][A^-]}{[HA]}\) For acetic acid, the value of Ka is 1.8 × 10⁻⁵. We can set up an equation to solve for \([H^+]\). \(1.8 \times 10^{-5} = \frac{[H^+][1 - H^+]}{1 - H^+}\) Solving for \([H^+]\), we obtain: \([H^+] = 4.24 \times 10^{-3}\) Now, we can find the initial pH using the equation: pH = -log([H^+]) Plugging in the obtained value for \([H^+]\), we get: Initial pH = -log(4.24 × 10⁻³) = 2.37

Determine the new pH

Next, we need to double the initial pH to find the new pH of the solution. Since the initial pH was 2.37, the new pH will be: New pH = 2 × 2.37 = 4.74

Calculate the moles of NaOH needed

To achieve the new pH of the solution, we need to neutralize some of the acetic acid molecules by using the following reaction: \(HC_2H_3O_2 + OH^-(NaOH) -> NaC_2H_3O_2 + H_2O\) First, let's calculate the \([OH^-]\) concentration needed to achieve the new pH of 4.74 using the relationship: \(pH + pOH = 14\) We can find the pOH concentration as: pOH = 14 - 4.74 = 9.26 Now, we can calculate the \([OH^-]\) concentration by using: \([OH^-] = 10^{-pOH}\) \([OH^-] = 10^{-9.26} = 5.49 \times 10^{-10} \mathrm{M}\) Since the volume of the solution remains constant at 1.00 L, we can find the moles of OH⁻ required: moles of OH⁻ required = \([OH^-]\) × volume moles of OH⁻ required = (5.49 × 10⁻¹⁰ M) × (1.00 L) = 5.49 × 10⁻¹⁰ moles

Calculate the mass of NaOH required

Finally, we can calculate the mass of NaOH required, given its molar mass (22.99 g/mol for Na and 15.999 g/mol for OH): mass of NaOH required = moles of NaOH × molar mass of NaOH mass of NaOH required = (5.49 × 10⁻¹⁰ moles) × (22.99 + 15.999) g/mol mass of NaOH required = 5.49 × 10⁻¹⁰ × 38.989 g = 2.14 × 10⁻⁸ g Thus, the mass of sodium hydroxide that must be added to the 1.00 L of 1.00 M acetic acid solution to double the pH of the solution is approximately 2.14 × 10⁻⁸ g.

Key concepts

Sodium Hydroxide

Sodium hydroxide, commonly known as NaOH and sometimes referred to as "lye", is an essential compound in chemistry. It is widely used in various industries, including the production of paper, textiles, and soaps. Most importantly, for our exercise, it functions as a strong base. In chemical reactions, sodium hydroxide dissociates completely in water to release hydroxide ions (\(OH^-\)), which are critical in neutralizing acids.
When added to an acetic acid solution, sodium hydroxide reacts to form water and sodium acetate, which helps in shifting the pH level of the solution.
In our exercise, the addition of sodium hydroxide is pivotal to increase the solution's pH level, effectively making the environment less acidic. When working on such calculations, knowing the precise amount of NaOH to add ensures the desired change in pH.

Acid Dissociation Constant

The acid dissociation constant, often denoted as \(K_a\), is a numerical representation of the strength of a weak acid in solution. It gives insight into how well an acid can donate protons in an aqueous solution. The formula used for calculating the \(K_a\) is:

• \(K_a = \frac{[H^+][A^-]}{[HA]}\)

For acetic acid, the \(K_a\) value is a small number, precisely \(1.8 \times 10^{-5}\). This indicates that acetic acid doesn't fully dissociate in solution, classifying it as a weak acid. In our problem, we use this constant to calculate the initial concentration of hydrogen ions \([H^+]\) in the acetic acid solution.
These calculations are crucial as they set the foundation to determine the initial pH and to understand how the subsequent addition of a strong base will affect the acidity of the solution.

Acetic Acid

Acetic acid is a common weak acid found in many household products, such as vinegar, and is represented by the formula \(HC_2H_3O_2\) or simply \(CH_3COOH\). As a weak acid, it partially dissociates into acetate ions and hydrogen ions in water.
In the exercise, we started with a 1.00 M acetic acid solution. The challenge is to alter the solution's pH by adding sodium hydroxide without altering the volume. Understanding how acetic acid behaves as a weak acid helps in predicting how it will react when introduced to a strong base.
The reversible reaction between acetic acid and sodium hydroxide produces sodium acetate and transforms acetic acid molecules into water molecules, thus influencing the solution's pH.

pOH

pOH is a measure of the hydroxide ion concentration in a solution, and it complements the concept of pH. Both these values are interrelated through the formula:

• \(pH + pOH = 14\)

In our exercise, understanding and calculating the pOH was crucial for determining how much sodium hydroxide was needed to adjust the pH to the desired level. By calculating the new pOH, derived from the desired pH of the solution, we can comprehend the level of basicity required.
The formula used to find the concentration of hydroxide ions \([OH^-]\) is:

• \([OH^-] = 10^{-pOH}\)

For our target new pH of 4.74, a corresponding pOH is calculated as 9.26. Through this understanding, it's clear how precise calculations of pOH contribute to accurate pH supplements in solutions.