Acid-Base Equilibrium p 2 and pH Calculations

Strong Acids and Bases

• In aqueous solutions of strong acids, there are two sources of hydronium ions ($H_3O^+$

  • The acid itself.

  • Water.

• Similarly, in aqueous solutions of strong bases, there are two sources of hydroxide ions ($OH^−$):

  • The base itself.

  • Water.

• For strong acids and bases, the contribution of water to the total concentration of hydronium or hydroxide ions is typically negligible.

  • The concentration of hydronium ions from the acid shifts the water autoionization equilibrium (Kw) such that the hydronium ion concentration from water is too small to be significant.

  • This is generally true except in very dilute solutions (typically less than $1 Imes 10^{−4}$ M).

Finding pH of Strong Acid or Strong Base Solutions

• For monoprotic strong acids, the concentration of hydronium ions ($[H_3O^+]$) is equal to the concentration of the acid ($[HAcid]$).

  • However, for polyprotic acids, other ionizations can generally be ignored, with the notable exception of sulfuric acid ($H2SO4$ ), where the second ionization cannot be ignored.

  • Example: A 0.10 M solution of HCl has $[H_3O^+] = 0.10$ M, and therefore, the pH is 1.00.

• For strong ionic bases, the concentration of hydroxide ions ($[OH^−]$) is equal to the number of hydroxide ions multiplied by the concentration of the base.

  • For molecular bases with multiple lone pairs, only one lone pair typically accepts a proton, and the other reactions can generally be ignored.

  • Example: A 0.10 M solution of $Ca(OH)_2$ has $[OH^−] = 0.20$ M, and the pH is 13.30.

Weak Acids

• In aqueous solutions of weak acids, there are two sources of hydronium ions ($H_3O^+$

  • The weak acid.

  • Water.

• However, determining the concentration of $H_3O^+$ is complicated because the acid only undergoes partial ionization.

• Calculating the concentration of $H_3O^+$ requires solving an equilibrium problem for the reaction that defines the acidity of the acid:

  $HAcid + H2O riptleRightleftharpoons Acid^− + H3O^+$

Example: Finding the pH of a Weak Acid

• Problem: Find the pH of a 0.200 M $HNO_2$(aq) solution at 25°C.

• Step 1: Write the reaction for the acid with water.

  $HNO2 + H2O riptleRightleftharpoons NO2^− + H3O^+$

• Step 2: Construct an ICE (Initial, Change, Equilibrium) table for the reaction.

• Step 3: Enter the initial concentrations, assuming that the concentration of $H_3O^+$ from water is approximately 0.

  • Since no products are initially present, $Q_c = 0$, and the reaction is proceeding forward.

• Step 4: Represent the Change in concentration in terms of x.

• Step 5: Sum the columns to find the equilibrium concentrations in terms of x.

• Step 6: Substitute into the equilibrium constant expression.

  $Ka = frac{[NO2^-][H3O^+]}{[HNO2]}$

• Step 7: Determine the value of $K_a$ from a table.

  • For $HNO<em>2$, $K</em>a = 4.6 Imes 10^{−4}$

• Step 8: Because $K<em>a$ is very small, approximate the $[HNO</em>2]<em>{eq}$ with $[HNO</em>2]_{initial}$ and solve for x.

  $4.6 Imes 10^{−4} = frac{x^2}{0.200}$

• Step 9: Check if the approximation is valid by seeing if x < 5% of $[HNO<em>2]</em>{init}$.

  • If true, the approximation is valid.

• Step 10: Substitute x into the equilibrium concentration definitions and solve.

• Step 11: Substitute $[H_3O^+]$ into the formula for pH and solve.

  $pH = -log[H_3O^+]$

• Step 12: Check: Substitute the equilibrium concentrations back into the equilibrium constant expression and compare the calculated $K<em>a$ to the given $K</em>a$.

Practice Problem: pH of Nicotinic Acid

• Problem: What is the pH of a 0.012 M solution of nicotinic acid, $HC<em>6H</em>4NO<em>2$? ($K</em>a = 1.4 Imes 10^{−5}$ at 25°C).

• Follow the same steps as in the previous example:

  • Write the reaction: $HC<em>6H</em>4NO<em>2 + H</em>2O riptleRightleftharpoons C<em>6H</em>4NO<em>2^− + H</em>3O^+$

  • Create an ICE table.

  • Enter initial concentrations, assuming $[H_3O^+]$ from water is negligible.

  • Represent changes in terms of x and calculate equilibrium concentrations.

  • Use the small $K_a$ approximation if valid.

  • Solve for x, calculate $[H_3O^+]$ and then pH.

  • Finally, check by plugging the values back into the $K_a$ expression.

Example: Another Weak Acid pH Calculation

• Problem: Find the pH of a 0.100 M $HClO_2$(aq) solution at 25°C.

• Write the reaction: $HClO<em>2 + H</em>2O riptleRightleftharpoons ClO<em>2^− + H</em>3O^+$

• Create an ICE table with initial concentrations.

• Determine the value of $K<em>a$ for $HClO</em>2$ from a table ($K_a = 1.1 Imes 10^{−2}$).

• Here, the approximation is found to be invalid, with x NOT < 5% of the initial concentration. In such cases, you must solve for x using the quadratic formula.

• Solve for x, then calculate the hydronium ion concentration and, finally, the pH.

• Check the answer by substituting the equilibrium concentrations back into the $K_a$ expression to verify that they match.

Example: Determining $K_a$ from pH

• Problem: What is the $K_a$ of a weak acid if a 0.100 M solution has a pH of 4.25?

• Use the given pH to find the equilibrium $[H<em>3O^+]$ . $[H</em>3O^+] = 10^{-pH}$. $[H_3O^+] = 10^{-4.25} = 5.6 Imes 10^{-5}$

• Write the reaction for the acid with water: $HA + H<em>2O riptleRightleftharpoons A^− + H</em>3O^+$

• Construct an ICE table.

• Enter the initial concentrations and the equilibrium $[H_3O^+]$.

• Fill in the rest of the table using the $[H_3O^+]$ value as a guide.

• If the difference is insignificant, assume $[HA]<em>{equil} = [HA]</em>{initial}$.

• Substitute the values into the $K<em>a$ expression and compute $K</em>a$.

• Final Answer: $K_a = 3.1 x 10^{-8}$

Practice Problem: Determining $K_a$ for Nicotinic Acid

• Problem: What is the $K<em>a$ of nicotinic acid, $HC</em>6H<em>4NO</em>2$, if a 0.012 M solution of it has a pH of 3.40?

• Follow similar steps as above:

  • Use the pH to find the equilibrium $[H_3O^+]$.

    • $[H<em>3O^+] = 10^{-pH}$. $[H</em>3O^+] = 10^{-3.40} = 4.0 Imes 10^{-4}$

  • Write the reaction. $HA + H<em>2O riptleRightleftharpoons A^− + H</em>3O^+$

  • Construct an ICE table.

  • Enter the initial concentrations and $[H<em>3O^+]</em>{equil}$.

  • Complete the table using the hydronium ion concentration.

  • Calculate $K_a$.