Question

A buffer consists of 0.20 -M propanoic acid \(\left(K_{\mathrm{a}}=1.4 \times 10^{-5}\right)\) and \(0.30-\mathrm{M}\) sodium propanoate. (a) Calculate the pH of this buffer. (b) Calculate the pH after the addition of \(1.0 \mathrm{~mL}\) of \(0.10-\mathrm{M}\) \(\mathrm{HCl}\) to \(0.010 \mathrm{~L}\) of the buffer. (c) Calculate the \(\mathrm{pH}\) after the addition of \(3.0 \mathrm{~mL}\) of \(1.0-\mathrm{M}\) \(\mathrm{HCl}\) to \(0.010 \mathrm{~L}\) of the buffer.

Short answer

(a) pH = 5.03; (b) pH = 4.99; (c) Highly acidic (buffer capacity exceeded).

Step-by-step solution

Identify the System and Formula

The buffer contains propanoic acid and sodium propanoate. Use the Henderson-Hasselbalch equation to calculate the pH of a buffer: \[ \text{pH} = \text{pKa} + \log\left(\frac{[A^-]}{[HA]}\right) \] where \( [A^-] = 0.30 \ M \) (conjugate base), \( [HA] = 0.20 \ M \) (acid), and \( K_a = 1.4 \times 10^{-5} \).

Calculate pKa

First, calculate the pKa from the given Ka: \[ \text{pKa} = -\log(K_a) = -\log(1.4 \times 10^{-5}) \approx 4.85 \]

Calculate Initial pH using Henderson-Hasselbalch Equation

Substitute the values into the Henderson-Hasselbalch equation: \[ \text{pH} = 4.85 + \log\left(\frac{0.30}{0.20}\right) = 4.85 + \log(1.5) \approx 4.85 + 0.176 = 5.03 \]

Calculate pH after Adding 0.1 mL of 0.10 M HCl

Calculate the moles of HCl added: \[ \text{moles of HCl} = 0.001 \ L \times 0.10 \ M = 0.0001 \ mol \] Calculate the moles of propanoic acid and sodium propanoate in 10 mL of the buffer: \[ \text{moles of propanoic acid (HA)} = 0.20 \ M \times 0.010 \ L = 0.002 \ mol \] \[ \text{moles of sodium propanoate (A^-)} = 0.30 \ M \times 0.010 \ L = 0.003 \ mol \] Adding HCl causes some of the base to convert to acid: \[ [HA] = 0.002 + 0.0001 = 0.0021 \ mol \] \[ [A^-] = 0.003 - 0.0001 = 0.0029 \ mol \] Calculate the new pH using the Henderson-Hasselbalch equation: \[ \text{pH} = 4.85 + \log\left(\frac{0.0029}{0.0021}\right) \approx 4.85 + 0.136 = 4.99 \]

Calculate pH after Adding 3.0 mL of 1.0 M HCl

Calculate the moles of HCl: \[ \text{moles of HCl} = 0.003 \ L \times 1.0 \ M = 0.003 \ mol \] Initially, the buffer contains: \[ \text{moles of HA} = 0.002 \ mol, \quad \text{moles of A^-} = 0.003 \ mol \] Adding the HCl completely converts the base to acid and adds extra HCl to convert a portion of the remaining base: \[ [A^-] = 0.003 - 0.003 = 0 \] \[ [HA] = 0.002 + 0.003 = 0.005 \ mol \] All propanoate is converted to propanoic acid, and the buffer capacity is exceeded. The system no longer behaves as a buffer, and pH is determined by the excess HCl, resulting in a significantly lower pH. In this case, roughly pH matches that of a strong acid solution: for simplicity, assume near completeness of acidicity (pH highly acidic).

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental tool in chemistry for calculating the pH of a buffer solution. It is an equation derived from the relationship between the concentrations of an acid and its conjugate base. The equation is expressed as: \[ \text{pH} = \text{pKa} + \log\left(\frac{[A^-]}{[HA]}\right) \]where

• \( \text{pH} \) is the measure of acidity or basicity.
• \( \text{pKa} \) is the negative logarithm of the acid dissociation constant \( K_a \).
• \([A^-]\) is the concentration of the conjugate base.
• \([HA]\) is the concentration of the acid.

By adjusting the ratio of the acid and its conjugate base in a solution, you can predict and control the pH within a specified range. This is particularly useful in buffer solutions used to resist changes in pH upon the addition of small amounts of acids or bases. The Henderson-Hasselbalch equation simplifies the process of finding the pH for systems involving weak acids and their salts.

Acid-Base Chemistry

Acid-base chemistry is an area of chemistry that explores the reactions between acids and bases and their properties. Acids are substances that can donate a proton \((H^+)\), while bases are substances that can accept a proton. This proton exchange is a fundamental aspect of many chemical reactions. Understanding this concept allows you to analyze various chemical reactions, including buffer systems.In the case of a buffer system consisting of a weak acid like propanoic acid and its salt (such as sodium propanoate), the buffer works by neutralizing small quantities of added acids (HCl) or bases. When an acid is added to the buffer, the base component (sodium propanoate) will react with the added HCl to form more of the weak acid (propanoic acid), preventing a large shift in pH. This ability to stabilize pH is crucial for many biological and chemical systems.

Buffer Capacity

Buffer capacity refers to the ability of a buffer solution to maintain its pH level when small amounts of acid or base are added. It is determined by the concentrations of the acid and its conjugate base within the buffer solution. The closer the concentrations, the higher the buffer capacity. In practical terms, a buffer's capacity is maximum when the pH of the solution is equal to the pKa of the acid involved. This is due to the fact that at this point, there are equal amounts of acid and conjugate base available to neutralize any added acids or bases. However, once an excess of strong acid or base is added, exceeding the buffer capacity, the pH will change significantly. For instance, when 3.0 mL of 1.0 M HCl is added to the propanoic acid-sodium propanoate buffer, the buffer capacity is overwhelmed, leading to a depletion of the conjugate base, and a sharp drop in pH.

pH Calculation

The calculation of pH in a buffer solution involves understanding both the acid and base components within the buffer. Initially, the Henderson-Hasselbalch equation is used to determine the baseline pH of the buffer. As in the example provided:

• The pKa value is calculated from the acidic dissociation constant \(K_a\).
• Substitute the concentrations of the acid \([HA]\) and conjugate base \([A^-]\) into the equation.

Through this method, the pH before adding any additional acids or bases can be established. For added convenience, pH adjustments due to the addition of an acid such as HCl can also be computed with the same equation, by updating the concentrations of \([HA]\) and \([A^-]\) accordingly.After the addition of acids or bases that exceed the buffer capacity, the pH will significantly deviate from the calculated pH. As buffer capacity is exceeded, calculations become more complex, often requiring more in-depth methods such as titration curves to predict accurately.

Propanoic Acid

Propanoic acid, also known as propionic acid, is a weak acid represented by the formula CH₃CH₂COOH. It is a common component in biochemical applications, especially in the formation of buffer systems due to its quantifiable acidity. Its usefulness as a part of a buffer system comes from its accessible dissociation constant \(K_a\), helping in calculations like those demonstrated in buffer pH determination.In buffers, propanoic acid donates protons \((H^+)\) when a base is added and accepts protons when an acid is introduced, thanks to its ability to exist in equilibrium with its conjugate base (propanoate ion). This dual role is essential in maintaining the pH of the buffer solution. For instance, in our exercise, propanoic acid is paired with sodium propanoate, creating a solution capable of resisting pH changes and demonstrating fundamental principles of acid-base chemistry.