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# Calculating Ka and Kb

## Discusses methods of calculating ionization constants.

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Calculating Ka and Kb

Who invented the pH meter?

The pH meter was invented because Florida orange growers needed a way to test the acidity of their fruit. The first meter was invented by Arnold Beckman, who went on to form Beckman Instruments. Beckman’s business was very successful and he used much of his fortune to fund science education and research. The Beckman family donated \$40 million to build the Beckman Institute at the University of Illinois, shown above.

### Calculating $K_a$ and $K_b$

The numerical value of  $K_a$ or  $K_b$ can be determined from an experiment. A solution of known concentration is prepared and its pH is measured with an instrument called a pH meter .

A pH meter is a laboratory device that provides quick, accurate measurements of the pH of solutions.

#### Sample Problem: Calculation of an Acid Ionization Constant

A 0.500 M solution of formic acid is prepared and its pH is measured to be 2.04. Determine the  $K_a$ for formic acid.

Step 1: List the known values and plan the problem.

Known

• initial [HCOOH] = 0.500 M
• pH = 2.04

Unknown

• $K_a= \ ?$

First, the pH is used to calculate the [H + ] at equilibrium. An ICE table is set up in order to determine the concentrations of HCOOH and HCOO - at equilibrium. All concentrations are then substituted into the  $K_a$ expression and the  $K_a$ value is calculated.

Step 2: Solve.

$[\text{H}^+]=10^{-\text{pH}}=10^{-2.04}=9.12 \times 10^{-3} \ \text{M}$

Since each formic acid molecule that ionizes yields one H + ion and one formate ion (HCOO - ), the concentrations of H + and HCOO - are equal at equilibrium. We assume that the initial concentrations of each ion are zero, resulting in the following ICE table.

 Concentrations [HCOOH] [H + ] [HCOO − ] Initial 0.500 0 0 Change -9.12 × 10 -3 +9.12 × 10 -3 +9.12 × 10 -3 Equilibrium 0.491 9.12 × 10 -3 9.12 × 10 -3

Now substituting into the  $K_a$ expression gives:

$K_a=\frac{[\text{H}^+][\text{HCOO}^-]}{[\text{HCOOH}]}=\frac{(9.12 \times 10^{-3})(9.12 \times 10^{-3})}{0.491}=1.7 \times 10^{-4}$

The value of  $K_a$ is consistent with that of a weak acid. Two significant figures are appropriate for the answer, since there are two digits after the decimal point in the reported pH.

Similar steps can be taken to determine the  $K_b$ of a base. For example, a 0.750 M solution of the weak base ethylamine (C 2 H 5 NH 2 ) has a pH of 12.31.

$\text{C}_2\text{H}_5\text{NH}_2+ \text{H}_2\text{O} \rightleftarrows \text{C}_2\text{H}_5\text{NH}^+_3 + \text{OH}^-$

Since one of the products of the ionization reaction is the hydroxide ion, we need to first find the [OH ] at equilibrium. The pOH is 14 - 12.31 = 1.69. The [OH ] is then found from 10 -1.69  = 2.04 × 10 -2  M. The ICE table is then set up as shown below.

 Concentrations [C 2 H 5 NH 2 ] [C 2 H 5 NH 3 + ] [OH − ] Initial 0.750 0 0 Change -2.04 × 10 -2 +2.04 × 10 -2 +2.04 × 10 -2 Equilibrium 0.730 2.04 × 10 -2 2.04 × 10 -2

Substituting into the  $K_b$ expression yields the  $K_b$ for ethylamine.

$K_b=\frac{[\text{C}_2\text{H}_5\text{NH}^+_3][\text{OH}^-]}{[\text{C}_2\text{H}_5\text{NH}_2]}=\frac{(2.04 \times 10^{-2})(2.04 \times 10^{-2})}{0.730}=5.7 \times 10^{-4}$

#### Summary

• Calculations of  $K_a$ and  $K_b$ are described.

#### Practice

1. What does an Arrhenius base dissociate to?
2. What does a Brønsted-Lowry base form in water?
3. How is percent ionization determined?

#### Review

1. What approach is used for calculation of ionization constants?
2. What initial assumptions are made?
3. What equilibrium assumptions are made?