Question

Calculate the \(\mathrm{pH}\) of a solution that has an acetic acid concentration of \(0.050 \mathrm{M}\) and a sodium acetate concentration of \(0.075 \mathrm{M}.\)

Short answer

The pH of the solution is approximately 4.92.

Step-by-step solution

Write down the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful formula to calculate the pH of a buffer solution. It is given by: \[ \mathrm{pH} = \mathrm{pK_a} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \] where - \([\text{A}^-]\) is the concentration of the conjugate base (sodium acetate in this case), - \([\text{HA}]\) is the concentration of the acid (acetic acid), and - \(\mathrm{pK_a}\) is the negative logarithm of the acid dissociation constant.

Find the pKa of acetic acid

The \(\mathrm{K_a}\) for acetic acid is approximately \(1.8 \times 10^{-5}\). To find \(\mathrm{pK_a}\), we take the negative logarithm of \(\mathrm{K_a}\): \[ \mathrm{pK_a} = -\log(1.8 \times 10^{-5}) \approx 4.74 \]

Substitute known values into the equation

Now substitute the concentrations and \(\mathrm{pK_a}\) value into the Henderson-Hasselbalch equation: \[ \mathrm{pH} = 4.74 + \log \left( \frac{0.075}{0.050} \right) \]

Calculate the ratio and its logarithm

First, calculate the ratio of the concentrations: \[ \frac{0.075}{0.050} = 1.5 \]Then find the logarithm of the ratio: \[ \log(1.5) \approx 0.176 \]

Compute the final pH

Add the log value to the \(\mathrm{pK_a}\) to find the \(\mathrm{pH}\): \[ \mathrm{pH} = 4.74 + 0.176 = 4.916 \]

Provide the final answer

So, the \(\mathrm{pH}\) of the solution is approximately 4.92.

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a pivotal tool in chemistry used to determine the pH of buffer solutions. This equation links pH, pKa, and the ratio of concentrations of conjugate base to weak acid in the solution. It is expressed as follows: \[ \mathrm{pH} = \mathrm{pK_a} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \]In this equation,

• \([\text{A}^-]\): Concentration of the conjugate base.
• \([\text{HA}]\): Concentration of the acid.
• 'pK_a': Negative log of the acid dissociation constant.

This equation is especially useful for buffering systems, which is often the case for solutions containing a weak acid and its conjugate base. Understanding the contributions of each component to the solution's pH helps predict how it will react to the addition of other acids or bases. As seen in the exercise with acetic acid and sodium acetate, these concentrations provide the necessary data inputs for our equation.

pKa of acetic acid

The pKa value is a crucial parameter when examining acid-base equilibria as it indicates the strength of an acid. For acetic acid, a common weak acid often found in vinegar, the Ka (acid dissociation constant) is approximately \(1.8 \times 10^{-5}\). The pKa is derived from the Ka by taking the negative logarithm:\[ \mathrm{pK_a} = -\log(1.8 \times 10^{-5}) \approx 4.74 \]This value indicates that acetic acid is weakly dissociated in water, hence its higher pKa compared to stronger acids. The pKa helps in predicting how acetic acid will behave in various chemical environments, especially when engaged in buffer solutions where it stabilizes pH against minor perturbations. By understanding pKa, we gain insights into the equilibrium position of the dissociation reaction of acetic acid in aqueous solutions.

Acid-base equilibrium

Acid-base equilibrium plays a vital role in chemistry by describing the stability and reactivity of acidic and basic species in a solution. It involves the dissociation of acids into protons ( H⁺) and their conjugate bases in aqueous solutions, forming an equilibrium state. This equilibrium can be influenced by factors such as concentration changes and temperature. For buffer solutions, like in the pH calculation example with acetic acid and sodium acetate, the equilibrium maintains the solution's pH around a certain value, despite dilution or the addition of small amounts of acids or bases.

• Weak Acid: Partially dissociates in solution and establishes an equilibrium between its ionized and non-ionized forms.
• Conjugate Base: Results from the dissociation and can recombine with protons to reform the acid.

Understanding these equilibria allows chemists to predict and control the effects of adding acids, bases, or changing conditions within buffer solutions, making it an essential concept in designing experiments and industrial applications.