Problem 54 - Chemistry: The Central Science in SI Units, Global Edition

Question

A \(0.100 \mathrm{M}\) solution of bromoacetic acid \(\left(\mathrm{BrCH}_{2} \mathrm{COOH}\right)\) is 13.2\% ionized. Calculate \(\left[\mathrm{H}^{+}\right],\left[\mathrm{BrCH}_{2} \mathrm{COO}^{-}\right],\left[\mathrm{BrCH}_{2} \mathrm{COOH}\right]\) and \(K_{a}\) for bromoacetic acid.

Step-by-Step Solution

Answer

The concentrations of the ions and the undissociated bromoacetic acid are: \(\left[\mathrm{H}^{+}\right] = 0.0132\,\mathrm{M}\) \(\left[\mathrm{BrCH}_{2}\mathrm{COO}^{-}\right] = 0.0132\,\mathrm{M}\) \(\left[\mathrm{BrCH}_{2}\mathrm{COOH}\right] = 0.0868\,\mathrm{M}\) \(\text{and the }K_{a}\text{ for bromoacetic acid is:}\) \(K_{a} = 2.0 \times 10^{-3}\)

Step 1: Determine the initial concentration of bromoacetic acid

The initial concentration is given as \(0.100\,\mathrm{M}\).

Step 2: Calculate the concentration of dissociated bromoacetic acid

We are given that the solution is 13.2% ionized. We can calculate the concentration of dissociated bromoacetic acid by multiplying the initial concentration by the percent ionization: \(\left[\mathrm{BrCH}_{2}\mathrm{COO}^{-}\right] = 0.100\,\mathrm{M} \times 0.132 = 0.0132\,\mathrm{M}\)

Step 3: Calculate the concentration of undissociated bromoacetic acid

Since the solution is 13.2% ionized, we can say that it still contains 86.8% of the initial bromoacetic acid. We can calculate the concentration of the undissociated bromoacetic acid by multiplying the initial concentration by the remaining percent: \(\left[\mathrm{BrCH}_{2}\mathrm{COOH}\right] = 0.100\,\mathrm{M} \times 0.868 = 0.0868\,\mathrm{M}\)

Step 4: Calculate the concentration of hydrogen ions

Since the concentration of hydrogen ions is equal to the concentration of bromoacetate ions, the concentration of hydrogen ions is: \(\left[\mathrm{H}^{+}\right] = 0.0132\,\mathrm{M}\)

Step 5: Calculate the \(K_{a}\) of bromoacetic acid

Using the ionization equation of a weak acid, we can calculate the \(K_{a}\) as follows: \(K_{a} = \frac{\left[\mathrm{H}^{+}\right]\left[\mathrm{BrCH}_{2}\mathrm{COO}^{-}\right]}{\left[\mathrm{BrCH}_{2}\mathrm{COOH}\right]}\) Substitute the values obtained in Steps 2, 3 and 4 into the above equation: \(K_{a} = \frac{0.0132\,\mathrm{M}\times 0.0132\,\mathrm{M}}{0.0868\,\mathrm{M}}= 2.0 \times 10^{-3}\) Thus, the concentrations of the ions and the undissociated bromoacetic acid are: \(\left[\mathrm{H}^{+}\right] = 0.0132\,\mathrm{M}\) \(\left[\mathrm{BrCH}_{2}\mathrm{COO}^{-}\right] = 0.0132\,\mathrm{M}\) \(\left[\mathrm{BrCH}_{2}\mathrm{COOH}\right] = 0.0868\,\mathrm{M}\) and the \(K_{a}\) for bromoacetic acid is: \(K_{a} = 2.0 \times 10^{-3}\)

Key Concepts

Weak Acid Ionization

Weak acids only partially dissociate in water. This means they don’t completely break apart into ions. Understanding weak acid ionization is key to predicting how acid solutions behave. When bromoacetic acid is added to water, it only ionizes a little.
This is because weak acids like bromoacetic acid establish an equilibrium between ionized and non-ionized forms. In this case, the equilibrium consists of hydrogen ions \((\mathrm{H}^{+})\), bromoacetate ions \((\mathrm{BrCH}_{2}\mathrm{COO}^{-})\), and undissociated bromoacetic acid \((\mathrm{BrCH}_{2}\mathrm{COOH})\).
Since the degree of ionization impacts the acidity of the solution, knowing how weak acids ionize is essential to doing calculations involving acids.

Percent Ionization

The percent ionization of an acid solution tells us what fraction of the original acid molecules ionized. In our example, the bromoacetic acid solution is 13.2% ionized.
This means if we started with a specific amount of bromoacetic acid, only 13.2% of it would dissociate into ions. This percentage is calculated by the formula:

• Percent Ionization = \( \frac{\text{Ionized Concentration}}{\text{Initial Concentration}} \times 100\% \)

Knowing the percent ionization helps us determine how much acid is active in solution. It also provides insights into acid strength. With only a small percentage ionizing, bromoacetic acid is a weak acid.

Equilibrium Concentrations

Once a weak acid like bromoacetic acid is added to water, it reaches equilibrium between the ionized and non-ionized forms. The concentrations at equilibrium are critical for various calculations.
In our scenario, we calculated the concentrations at equilibrium for each species:

• The concentration of hydrogen ions \(\left[\mathrm{H}^{+}\right]\) is 0.0132 M.
• The concentration of bromoacetate ions \(\left[\mathrm{BrCH}_{2}\mathrm{COO}^{-}\right]\) is also 0.0132 M.
• The concentration of undissociated bromoacetic acid \(\left[\mathrm{BrCH}_{2}\mathrm{COOH}\right]\) is 0.0868 M.

These figures allow us to quantify how much of the acid remains undissociated and how much turns into ions. Equilibrium concentrations provide a complete picture of the chemical scenario once it has reached equilibrium.

Chemical Equilibrium Calculations

Chemists use equilibrium concentrations to calculate other important values, like the acid dissociation constant, \( K_{a} \). This constant quantifies the strength of an acid in a solution. The equilibrium expression for calculating \( K_{a} \) of bromoacetic acid is:

• \( K_{a} = \frac{\left[\mathrm{H}^{+}\right]\left[\mathrm{BrCH}_{2}\mathrm{COO}^{-}\right]}{\left[\mathrm{BrCH}_{2}\mathrm{COOH}\right]} \)

Using the equilibrium concentrations we found, the \( K_{a} \) of bromoacetic acid comes out to be \( 2.0 \times 10^{-3} \).
This tiny value reaffirms that bromoacetic acid is a weak acid, as weak acids have lower \( K_{a} \) values. Mastering these calculations helps in assessing acid strength and predicting reaction outcomes.