Problem 21 - Chemistry The Central Science

Question

(a) Calculate the pH of a buffer that is \(0.12 \mathrm{M}\) in lactic acid and \(0.11 M\) in sodium lactate. (b) Calculate the pH of a buffer formed by mixing \(85 \mathrm{~mL}\) of \(0.13 \mathrm{M}\) lactic acid with \(95 \mathrm{~mL}\) of \(0.15 \mathrm{M}\) sodium lactate.

Step-by-Step Solution

Answer

The pH of the first buffer is calculated using the Henderson-Hasselbalch equation to be \(pH = pK_a + \log{\frac{0.11}{0.12}}\), where \(pK_a = -\log_{10}(1.4 \times 10^{-4})\). For the second buffer, the concentrations of lactic acid and sodium lactate after mixing are \(\mathrm{[HA]} = \frac{0.01105}{0.18}\) and \(\mathrm{[A^-]} = \frac{0.01425}{0.18}\), respectively. The pH of this buffer is calculated using the same equation: \(pH = pK_a + \log{\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}}\), where the concentrations are the calculated values.

Step 1: Finding the pKa of lactic acid

Lactic acid has a chemical formula: \(\mathrm{CH_3CH(OH)COOH}\). The dissociation of lactic acid can be presented by the following equation: \(\mathrm{CH_3CH(OH)COOH} \rightleftharpoons \mathrm{H^+ + CH_3CH(OH)COO^-} \) The acid dissociation constant (K_a) for lactic acid is approximately \(1.4 \times 10^{-4}\). We will use this value to find the pKa of lactic acid: \(pK_a = -\log_{10}{K_a} = -\log_{10}(1.4 \times 10^{-4})\) We can plug this value into a calculator to find out the pKa.

Step 2: Calculate the pH of the first buffer

Now that we have the pKa of lactic acid, we can use the Henderson-Hasselbalch equation to find the pH of the first buffer: \(pH = pK_a + \log{\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}}\) The concentration of lactic acid (HA) is 0.12 M, and the concentration of sodium lactate (A-) is 0.11 M. Plugging these values and the pKa of lactic acid into the equation, we find the pH.

Step 3: Calculate the pH of the second buffer

For the second part of the problem, we need to find the pH of the buffer system formed by mixing volumes of lactic acid and sodium lactate. We will use the given volumes and concentrations to find the overall concentration of lactic acid and sodium lactate in the final buffer and then apply the Henderson-Hasselbalch equation. Amount of lactic acid (in moles) = volume of lactic acid \(\times\) concentration of lactic acid = \((85 \times 10^{-3})L \times 0.13 M = 0.01105\) moles. Amount of sodium lactate (in moles) = volume of sodium lactate \(\times\) concentration of sodium lactate = \((95\times 10^{-3})L \times 0.15 M = 0.01425\) moles. Total volume of the buffer = volume of lactic acid + volume of sodium lactate = \((85 + 95) mL = 180 mL = 0.18 L\) Now we can find the final concentrations in the mixed buffer: Concentration of lactic acid, \(\mathrm{[HA]} = \frac{0.01105 \, \text{moles}}{0.18L}\) Concentration of sodium lactate, \(\mathrm{[A^-]} = \frac{0.01425\, \text{moles}}{0.18L}\) We will use these concentrations to calculate the pH using the Henderson-Hasselbalch equation: \(pH = pK_a + \log{\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}}\) Plug in the pKa of lactic acid and the calculated concentrations for lactic acid, and sodium lactate into the equation to find the pH.

Key Concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation plays a critical role in buffer chemistry, as it relates the pH of a solution to the pKa and the concentrations of the acid and its conjugate base. In mathematical terms, the equation is expressed as follows:\[pH = pK_a + \log{\frac{[A^-]}{[HA]}} \]In this equation:

• \([HA]\) represents the concentration of the acid (in this case, lactic acid).
• \([A^-]\) represents the concentration of the conjugate base (sodium lactate).

Using this equation, you can calculate the pH of a buffer system if you know the pKa of the acid and the concentrations of the acid and base in the solution. It shows that the pH is closest to the pKa value when the concentrations of the acid and its base are similar. This balance makes buffers effective at resisting changes in pH when acids or bases are added.

pH calculation

Calculating pH using the Henderson-Hasselbalch equation involves several steps. First, you need the pKa value of the acid.
For lactic acid, this comes from its dissociation constant, which is a measure of the acid's strength. Knowing the pKa is crucial, as it is directly used in the calculation.
Once you have the pKa, you calculate log ratio of the concentrations of the conjugate base to the acid. Evaluate as follows:\[pH = pK_a + \log{\frac{[A^-]}{[HA]}} \]Ensure you substitute the correct values into the equation:

• Use the accurate concentration values for the calculations. For example, if the solution contains 0.12 M lactic acid and 0.11 M sodium lactate, replace \([HA]\) with 0.12 and \([A^-]\) with 0.11.
• Carry out the calculation to receive the resulting pH.

Understanding this process is essential, especially when working with buffer problems in chemistry, because it allows for the maintenance of pH in a narrow range, crucial in many biological and chemical processes.

lactic acid dissociation

Lactic acid is a common organic acid with the formula \(\mathrm{CH_3CH(OH)COOH}\). When dissolved in water, it can lose a proton to form its conjugate base, lactate, according to the equation:\[\mathrm{CH_3CH(OH)COOH} \rightleftharpoons \mathrm{H^+} + \mathrm{CH_3CH(OH)COO^-}\]This process is an example of acid dissociation, where the acid releases a proton (\(\mathrm{H^+}\)) into the solution, increasing the hydrogen ion concentration.
The extent of dissociation of an acid is quantified by its dissociation constant (\(K_a\)), which for lactic acid is about \(1.4 \times 10^{-4}\). This value indicates that lactic acid is a weak acid since it does not completely dissociate in water.
To find the pKa, which is often more useful in various calculations, take the negative logarithm of \(K_a\):\[pK_a = -\log_{10}(K_a)\]This results in a pKa value around 3.85, representing a crucial part of determining the pH of buffer solutions containing lactic acid.

acid-base chemistry

Acid-base chemistry is fundamental for understanding buffers, which help maintain pH stability in solutions. The basic concept involves acids (proton donors) and bases (proton acceptors).
When you combine these in a buffer, you form a solution that resists pH changes when small amounts of acids or bases are added. This is due to the equilibrium formed between the acid and its conjugate base.

For example, in the buffering system involving lactic acid and sodium lactate:

• Lactic acid donates protons to form lactate.
• The sodium lactate acts as a base, accepting protons.

This dynamic creates a buffer solution, which is very useful in conditions where maintaining a constant pH is essential, such as in physiological processes or certain industrial applications. Understanding how the equilibrium shifts in response to added acids or bases is key to mastering acid-base chemistry and buffer solution calculations.