Question

Preparation of a Phosphate Buffer Phosphoric acid \(\left(\mathrm{H}_{3} \mathrm{PO}_{4}\right)\), a triprotic acid, has three \(\mathrm{p} K_{\mathrm{a}}\) values: \(2.14,6.86\), and 12.4. What molar ratio of \(\mathrm{HPO}_{4}^{2-}\) to \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\)in solution would produce a \(\mathrm{pH}\) of \(7.0 ?\) Hint: Only one of the \(\mathrm{p} K_{\mathrm{a}}\) values is relevant here.

Short answer

The molar ratio is approximately 1.38:1 of \(\mathrm{HPO}_4^{2-}\) to \(\mathrm{H}_2\mathrm{PO}_4^-\).

Step-by-step solution

Identify the Relevant pKa

Phosphoric acid, being a triprotic acid, can lose protons stepwise, and each step has a different pKa value: 2.14, 6.86, and 12.4. When we want to create a buffer with a pH of 7.0, we should use the pKa value that is closest to this pH. Therefore, the relevant pKa is 6.86, which corresponds to the equilibrium between \( \mathrm{H}_2\mathrm{PO}_4^- \) and \( \mathrm{HPO}_4^{2-} \).

Use the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is given by: \[ \mathrm{pH} = \mathrm{p}K_{\mathrm{a}} + \log \left( \frac{[\mathrm{A}^-]}{[\mathrm{HA}]} \right) \]Where \([\mathrm{A}^-]\) is the concentration of the conjugate base and \([\mathrm{HA}]\) is the concentration of the acid. Here, \([\mathrm{A}^-] = [\mathrm{HPO}_4^{2-}]\) and \([\mathrm{HA}] = [\mathrm{H}_2\mathrm{PO}_4^-]\). Plug in the desired pH and the relevant pKa value:

Solve for the Molar Ratio

Set the pH to 7.0 and \(\mathrm{p}K_{\mathrm{a3}}\) to 6.86 in the equation:\[ 7.0 = 6.86 + \log \left( \frac{[\mathrm{HPO}_4^{2-}]}{[\mathrm{H}_2\mathrm{PO}_4^-]} \right) \]Subtract 6.86 from both sides to get:\[0.14 = \log \left( \frac{[\mathrm{HPO}_4^{2-}]}{[\mathrm{H}_2\mathrm{PO}_4^-]} \right) \]

Evaluate the Logarithm

To solve for the ratio \( \frac{[\mathrm{HPO}_4^{2-}]}{[\mathrm{H}_2\mathrm{PO}_4^-]} \), exponentiate both sides with base 10:\[10^{0.14} = \frac{[\mathrm{HPO}_4^{2-}]}{[\mathrm{H}_2\mathrm{PO}_4^-]} \]Using a calculator, find \(10^{0.14} \approx 1.38\).

Conclusion

The molar ratio of \(\mathrm{HPO}_4^{2-}\) to \(\mathrm{H}_2\mathrm{PO}_4^-\) required to achieve a pH of 7.0 is approximately 1.38 to 1. This means that the concentration of \(\mathrm{HPO}_4^{2-}\) should be 1.38 times the concentration of \(\mathrm{H}_2\mathrm{PO}_4^-\) in the buffer solution.

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful tool in chemistry for understanding the relationship between pH, pKa, and the concentration of acid and its conjugate base. It essentially states that the pH of a solution is equal to the pKa of the acid plus the logarithm of the ratio of the concentration of the conjugate base \[ [A^-] \] to the concentration of the acid \[ [HA] \].

This equation is especially helpful when preparing buffer solutions. Buffers are solutions that resist changes in pH upon the addition of small amounts of acids or bases. By allowing chemists to calculate the exact ratio of acid to conjugate base needed to achieve a desired pH, the Henderson-Hasselbalch equation is vital in buffer preparation.

Using this equation in the context of a phosphate buffer, we set \( ext{pH} = ext{p}K_a + ext{log} rac{[A^-]}{[HA]} \). This equation, when rearranged, allows us to solve for the concentration ratio that gives us a particular pH in the buffer system.

pKa values

pKa values are crucial in understanding how an acid dissociates in a solution. Each type of dissociation, where a proton is donated to the solution, has a corresponding pKa value. They are unique to each acid and indicate the strength of an acid.

A lower pKa value means a stronger acid, as it signifies that the acid more readily donates a proton. Conversely, a higher pKa value implies a weaker acid.

In the case of phosphoric acid, a triprotic acid, there are three pKa values because it can lose three protons, each with increasing difficulty. For the preparation of a phosphate buffer, identifying the correct pKa value closest to the desired pH is essential. For a pH of 7.0, the pKa value of 6.86 is most relevant.

The knowledge of pKa values allows chemists to predict at what pH levels specific ionic species will dominate in a solution.

buffer solutions

Buffer solutions play an essential role in maintaining the pH of a system. These solutions are a combination of a weak acid and its conjugate base (or a weak base and its conjugate acid). By absorbing excess H⁺ or OH⁻ ions, buffers stabilize the pH of a solution.

In biological systems, many processes rely on precise pH conditions. Buffers are crucial in scenarios like enzyme activity, where any shift in pH can lead to significant changes in activity rates.

When preparing a phosphate buffer, one must consider the ratio of \([HPO_4^{2-}]\) to \([H_2PO_4^-]\) and use the Henderson-Hasselbalch equation to maintain the solution at a desired pH. The balance of these compounds ensures that the overall pH of the buffer solution can resist changes when exposed to external influences.

triprotic acids

Triprotic acids, like phosphoric acid \( (H_3PO_4) \), are capable of donating three protons, making them complex yet versatile in biological and chemical systems. For each proton lost, there is a corresponding \( ext{p}K_a \) value.

In a triprotic acid like phosphoric acid, the first dissociation is the easiest, with a pKa of 2.14, followed by a pKa of 6.86 for the second proton, and finally 12.4 for the third. To prepare a buffer at a specific pH, it is necessary to use the pKa value closest to the pH desired. This ensures the components in the solution are balanced to maintain that pH.

Understanding the sequential release of protons in triprotic acids helps in calculating the precise molar ratios needed in solution to achieve a desired pH level. This characteristic makes triprotic acids a powerful tool in preparing buffered solutions in both laboratory and industrial settings.