Acids and Bases Equilibria - Buffers

Buffer Solutions

• Resist changes in pH when an acid or base is added.
• Act by neutralizing the added acid or base.
• Contain either:
  • Significant amounts of a weak acid and its conjugate base.
  • Significant amounts of a weak base and its conjugate acid.
  • Example: Blood contains a mixture of H2CO3 and HCO_3^-.

Acidic Buffer Solutions

• Must contain significant amounts of both a weak acid and its conjugate base.
• If a strong base is added:
  • It is neutralized by the weak acid (e.g., HC2H3O_2) in the buffer.
  • NaOH(aq) + HC2H3O2(aq) \rightarrow H2O(l) + NaC2H3O_2(aq)
  • If the amount of NaOH added is less than the amount of acetic acid present, the pH change is small.
• If a strong acid is added:
  • It is neutralized by the conjugate base (e.g., NaC2H3O_2) in the buffer.
  • HCl(aq) + NaC2H3O2(aq) \rightarrow HC2H3O2(aq) + NaCl(aq)
  • If the amount of HCl is less than the amount of NaC2H3O_2 present, the pH change is small.

Common Ion Effect

• Example demonstrating pH changes:
  • 0. 100 M HC2H3O_2: pH = 2.9
  • 0. 100 M NaC2H3O_2: pH = 8.9
  • 0. 100 M HC2H3O2 + 0.100 M NaC2H3O2: pH = 4.7

Henderson-Hasselbalch Equation

• An equation derived from the K_a expression that allows the calculation of the pH of a buffer solution.
• Calculates the pH of a buffer from the pK_a and initial concentrations of the weak acid and the salt of its conjugate base, provided the "x is small" approximation is valid.

Using the Henderson-Hasselbalch Equation

• The equation is generally suitable when the "x is small" approximation is applicable.
• The "x is small" approximation is generally valid when:
  • The initial concentrations of the acid and salt are not very dilute.
  • The K_a is fairly small.
• For most problems, initial acid and salt concentrations should be 100 to 1000 times larger than the value of K_a.

Calculating Buffer pH - Equilibrium Approach

• Steps:
  1. Write the balanced equation for the ionization of the acid and prepare an ICE (Initial, Change, Equilibrium) table.
  2. Represent the change in the concentration of H_3O^+ with the variable x. Express changes in other concentrations in terms of x.
  3. Sum each column to determine the equilibrium concentrations in terms of initial concentrations and x.
  4. Substitute the expressions for the equilibrium concentrations into the expression for the acid ionization constant.
     • In most cases, the approximation that x is small can be made.
  5. Substitute the value of the acid ionization constant into the K_a expression and solve for x.
     • Confirm that x is small by calculating the ratio of x to the number it was subtracted from in the approximation.
     • The ratio should be less than 0.05 (or 5%) for the approximation to be valid.
  6. Determine the H_3O^+ concentration from the calculated value of x and substitute into the pH equation to find pH.

Example Calculation

• Calculate the pH of a buffer solution that is 0.050 M in benzoic acid (HC7H5O2) and 0.150 M in sodium benzoate (NaC7H5O2).
• For benzoic acid, K_a = 6.5 \times 10^{-5}.

Henderson-Hasselbalch Approach

• Determine which component is the acid and which is the base and substitute their concentrations into the Henderson–Hasselbalch equation to calculate pH.

pH Change After Adding Acid or Base

• Calculating the new pH after adding acid or base requires breaking the problem into two parts:
  1. A stoichiometry calculation for the reaction of the added chemical with one of the ingredients of the buffer to reduce its initial concentration and increase the concentration of the other.
     • Added acid reacts with A^- to make more HA.
     • Added base reacts with the HA to make more A^-.
  2. An equilibrium calculation of [H_3O^+] using the new initial values of [HA] and [A^-].

Example: Stoichiometry and Equilibrium

• A 1.0 L buffer solution contains 0.100 mol HC2H3O2 and 0.100 mol NaC2H3O2.
• The value of Ka for HC2H3O2 is 1.8 \times 10^{-5}.
• Because the initial amounts of acid and conjugate base are equal, the pH of the buffer is equal to pK_a = -\log(1.8 \times 10^{-5}) = 4.74.
• Calculate the new pH after adding 0.010 mol of solid NaOH to the buffer.
• For comparison, calculate the pH after adding 0.010 mol of solid NaOH to 1.0 L of pure water. (Ignore any small changes in volume upon addition of the base.)

Equilibrium Calculation

• Write the balanced equation for the ionization of the acid and use it as a guide to prepare an ICE table.
• Use the amounts of acid and conjugate base from part I as the initial amounts of acid and conjugate base in the ICE table.
• Substitute the expressions for the equilibrium concentrations of acid and conjugate base into the expression for the acid ionization constant.
• Make the x is small approximation and solve for x.
• Calculate the pH from the value of x, which is equal to [H_3O^+].
• As long as the x is small approximation is valid, you can substitute the quantities of acid and conjugate base after the addition (from part I) into the Henderson–Hasselbalch equation and calculate the new pH.

pH Change Comparison

• The buffer solution changed from pH = 4.74 to pH = 4.83 upon the addition of the base (a small fraction of a single pH unit).
• In contrast, the pure water changed from pH = 7.00 to pH = 12.00, five whole pH units (a factor of 10^5).
• Adding base should make the solution more basic (higher pH); adding acid should make the solution more acidic (lower pH).

Basic Buffers

• B:(aq) + H_2O(l) \rightleftharpoons H:B^+(aq) + OH^-(aq)
• Buffers can also be made by mixing a weak base, (B:), with a soluble salt of its conjugate acid, H:B^+Cl^-
• H2O(l) + NH3 (aq) \rightleftharpoons NH_4^+ (aq) + OH^- (aq)

Henderson-Hasselbalch Equation for Basic Buffers

• The Henderson–Hasselbalch equation is generally written for a chemical reaction with a weak acid reactant and its conjugate base as a product.
• The chemical equation of a basic buffer is written with a weak base as a reactant and its conjugate acid as a product.
• B: + H_2O \rightleftharpoons H:B^+ + OH^-

Relationship between pKa and pKb

• Just as there is a relationship between the Ka of a weak acid and Kb of its conjugate base, there is also a relationship between the pKa of a weak acid and the pKb of its conjugate base.
• Ka \cdot Kb = K_w = 1.0 \times 10^{-14}
• \log(Ka \cdot Kb) = -\log(K_w) = 14
• \log(Ka) + -\log(Kb) = 14
• pKa + pKb = 14

Buffering Effectiveness

• A good buffer should be able to neutralize moderate amounts of added acid or base.
• The buffering capacity is the amount of acid or base a buffer can neutralize.
• The buffering range is the pH range the buffer can be effective.
• The effectiveness of a buffer depends on two factors:
  • The relative amounts of acid and base, and
  • The absolute concentrations of acid and base.

Buffering Capacity

• Buffering capacity is the amount of acid or base that can be added to a buffer without causing a large change in pH.
• The buffering capacity increases with increasing absolute concentration of the buffer components.

Relative Buffer Component Concentrations

• As the [base]:[acid] ratio approaches 1, the ability of the buffer to neutralize both added acid and base improves.
• Buffers that need to work mainly with added acid generally have [base] > [acid].
• Buffers that need to work mainly with added base generally have [acid] > [base].

Effectiveness of Buffers

• A buffer will be most effective when the [base]:[acid] = 1 (equal concentrations of acid and base).
• A buffer will be effective when 0.1 < [base]:[acid] < 10.
• A buffer will be most effective when the [acid] and the [base] are large.

Buffering Range

• A buffer will be effective when 0.1 < [base]:[acid] < 10.
• Substituting into the Henderson–Hasselbalch equation we can calculate the maximum and minimum pH at which the buffer will be effective.
• Lowest pH: pH = pKa + \log(0.1) = pKa - 1
• Highest pH: pH = pKa + \log(10) = pKa + 1
• Therefore, the effective pH range of a buffer is pK_a \pm 1.
• When choosing an acid to make a buffer, choose one whose pK_a is closest to the pH of the buffer.