20.3 Biological Acids and the Henderson–Hasselbalch Equation

The ###Henderson-Hasselbalch_Equation_0### is a powerful tool for understanding acid-base behavior in biological systems. It helps calculate the ratio of dissociated to undissociated forms of weak acids, crucial for predicting their behavior at different pH levels.

Most carboxylic acids exist as anions at physiological pH due to their low pKa values. This equation allows us to quantify the percentages of protonated and deprotonated species, providing insights into acid-base equilibria and buffer systems in living organisms.

Biological Acids and the Henderson-Hasselbalch Equation

Ratio calculation with Henderson-Hasselbalch equation

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• Henderson-Hasselbalch equation calculates ratio of dissociated (deprotonated) to undissociated (protonated) forms of a weak acid (carboxylic acids)
  • Equation: $pH = pK_a + log([A^-]/[HA])$
    • $pH$: solution pH
    • $pK_a$: negative logarithm of acid dissociation constant $(K_a)$
    • $[A^-]$: concentration of dissociated (deprotonated) acid form
    • $[HA]$: concentration of undissociated (protonated) acid form
• Calculate ratio by solving equation for $log([A^-]/[HA])$, subtracting $pK_a$ from both sides
  • $log([A^-]/[HA]) = pH - pK_a$
• Find ratio $[A^-]/[HA]$ by taking antilog (10 to the power) of both sides
  • $[A^-]/[HA] = 10^{pH - pK_a}$
• Example: Acetic acid $(pK_a = 4.76)$ at pH 5.5
  • $log([A^-]/[HA]) = 5.5 - 4.76 = 0.74$
  • $[A^-]/[HA] = 10^{0.74} ≈ 5.5$, meaning more dissociated than undissociated form
• The Henderson-Hasselbalch equation uses a logarithmic scale to relate pH and pKa

Carboxylic acids as anions at physiological pH

• Physiological pH around 7.4, higher than $pK_a$ of most carboxylic acids (4-5)
• pH higher than $pK_a$ shifts equilibrium towards dissociated (deprotonated) acid form
  • Higher pH indicates lower $H^+$ ion concentration in solution
• At physiological pH, $[A^-]/[HA]$ ratio much greater than 1, majority of carboxylic acid molecules in deprotonated (anionic) form
• Henderson-Hasselbalch equation demonstrates this:
  • If $pH > pK_a$, then $pH - pK_a > 0$, and $10^{pH - pK_a} > 1$
  • Therefore, $[A^-]/[HA] > 1$, deprotonated form dominant
• Example: Lactic acid $(pK_a = 3.86)$ at physiological pH (7.4)
  • $log([A^-]/[HA]) = 7.4 - 3.86 = 3.54$
  • $[A^-]/[HA] = 10^{3.54} ≈ 3467$, vast majority in deprotonated form
• The deprotonated form of the acid acts as a conjugate base in solution

Percentages of acid species at specific pH

• Calculate $[A^-]/[HA]$ ratio using Henderson-Hasselbalch equation
  • $[A^-]/[HA] = 10^{pH - pK_a}$
• Total acid concentration $[A]_T$ equals sum of protonated and deprotonated form concentrations
  • $[A]_T = [HA] + [A^-]$
• Divide ratio equation by $[HA]$:
  • $[A^-]/[HA] + 1 = ([HA] + [A^-])/[HA] = [A]_T/[HA]$
• Calculate protonated form percentage:
  • $\%HA = [HA]/[A]_T * 100\% = 1/([A^-]/[HA] + 1) * 100\%$
• Calculate deprotonated form percentage:
  • $\%A^- = [A^-]/[A]_T * 100\% = ([A^-]/[HA])/([A^-]/[HA] + 1) * 100\%$
• Example: Benzoic acid $(pK_a = 4.20)$ at pH 4.0
  • $[A^-]/[HA] = 10^{4.0 - 4.20} ≈ 0.63$
  • $\%HA = 1/(0.63 + 1) * 100\% ≈ 61.3\%$
  • $\%A^- = 0.63/(0.63 + 1) * 100\% ≈ 38.7\%$

Buffer Solutions and Equilibrium

• Buffer solutions resist changes in pH when small amounts of acid or base are added
• They consist of a weak acid and its conjugate base
• Le Chatelier's principle explains how buffers maintain pH balance
• The equilibrium constant (Ka) determines the strength of the acid-base pair in the buffer
• Titration curves can be used to visualize the buffering capacity of a solution