Question

The \(\mathrm{pH}\) of a sodium acetate-acetic acid buffer is \(4.50 .\) Calculate the ratio \(\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right] /\left[\mathrm{CH}_{3} \mathrm{COOH}\right] .\)

Short answer

The ratio \(\frac{[\mathrm{CH}_3\mathrm{COO}^{-}]}{[\mathrm{CH}_3\mathrm{COOH}]}\) is approximately 0.55.

Step-by-step solution

Write the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is useful for calculating the pH of a buffer solution. It is given by: \[ \text{pH} = \text{pKa} + \log\left(\frac{[\mathrm{A}^{-}]}{[\mathrm{HA}]}\right) \] where \([\mathrm{A}^{-}]\) is the concentration of the conjugate base and \([\mathrm{HA}]\) is the concentration of the acid.

Identify the pKa of Acetic Acid

Acetic acid (\(\mathrm{CH}_3\mathrm{COOH}\)) has a pKa value of approximately 4.76. This is a known constant for acetic acid and will be used in the Henderson-Hasselbalch equation.

Substitute Known Values into the Equation

Substitute the given pH value and the pKa value of acetic acid into the equation: \[ 4.50 = 4.76 + \log\left(\frac{[\mathrm{CH}_3\mathrm{COO}^{-}]}{[\mathrm{CH}_3\mathrm{COOH}]}\right) \]

Solve for the Logarithmic Ratio

Rearrange the equation to solve for the logarithm: \[ \log\left(\frac{[\mathrm{CH}_3\mathrm{COO}^{-}]}{[\mathrm{CH}_3\mathrm{COOH}]}\right) = 4.50 - 4.76 = -0.26 \]

Calculate the Ratio

Convert the logarithmic expression into a ratio by taking the antilog: \[ \frac{[\mathrm{CH}_3\mathrm{COO}^{-}]}{[\mathrm{CH}_3\mathrm{COOH}]} = 10^{-0.26} \] Using a calculator, this equals approximately 0.55.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a powerful tool in chemistry that helps us understand the pH of a buffer solution.
It's particularly useful when you deal with solutions containing a weak acid and its conjugate base.
This equation is expressed as follows: \[ \text{pH} = \text{pKa} + \log\left(\frac{[\mathrm{A}^{-}]}{[\mathrm{HA}]}\right) \]- **\(\text{pH}\)**: It measures the acidity or basicity of the solution.- **\(\text{pKa}\)**: This is a constant that represents the acid strength. - **\([\mathrm{A}^{-}]\)**: The concentration of the conjugate base.- **\([\mathrm{HA}]\)**: The concentration of the weak acid. Using this equation, you can quickly calculate the pH if you know the concentrations of the acid and its conjugate base, along with the pKa of the acid. It's a practical approach for designing buffer solutions with desired pH levels.
This method provides clarity, especially when adjusting the mixture's components to stabilize the pH.

pH

pH is a scale that indicates how acidic or basic a solution is, ranging from 0 to 14.
In chemistry, understanding pH is crucial because reactions often depend on the acidity or basicity of the environment.
- A pH of 7 is neutral. Water is a common example. - A pH less than 7 indicates an acidic solution. - A pH greater than 7 shows a basic or alkaline solution. In the context of buffer solutions, a buffer's main role is to maintain a stable pH when small amounts of acid or base are added. Buffers are particularly important in biological and chemical processes where maintaining a certain pH is critical.
In the exercise example, we are dealing with a buffer solution with a pH of 4.50, which suggests it is slightly acidic.

pKa of Acetic Acid

The pKa value of an acid like acetic acid is a measure of its strength, giving insight into how easily the acid donates hydrogen ions in solution.
Acetic acid, a common acid found in vinegar, has a pKa of approximately 4.76. This is known as its dissociation constant and helps in evaluating the acid's behavior in a solution.
- **Why is pKa important?** - It indicates how strong or weak an acid is. - Acids with lower pKa values are stronger because they dissociate more in solution. - pKa values are essential when using the Henderson-Hasselbalch equation to calculate the pH of buffer solutions. When using acetic acid in a buffer, knowing its pKa helps us anticipate how it will interact with its conjugate base, allowing for precise pH adjustments.

Conjugate Base and Acid Ratio

The ratio of conjugate base to acid, \(\left(\left[\mathrm{CH}_3\mathrm{COO}^{-}\right]/\left[\mathrm{CH}_3\mathrm{COOH}\right]\right)\), is central to understanding buffer solutions.
This ratio shows how much of the conjugate base is present compared to the acid, directly influencing the buffer's pH.
- **Impact on pH:** - Changing this ratio alters the pH of the buffer. - More conjugate base increases pH, making it more basic. - More acid decreases pH, making it more acidic.For effective buffers, it's crucial that the concentrations of both the acid and its conjugate base are significant enough to resist changes in pH.
In the exercise, the calculated ratio (approximately 0.55) indicates a balance favoring slightly more of the acid rather than the base, which aligns with the acidic pH of 4.50.
Understanding this ratio is key to preparing buffers that maintain specific pH levels.