From the equilibrium concentrations given, calculate \(K_{\mathrm{a}}\) for each of the weak acids and \(K_{\mathrm{b}}\) for each of the weak bases. (a) \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}:\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.34 \times 10^{-3} \mathrm{M}$$\left[\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}\right]=1.34 \times 10^{-3} \mathrm{M}$$\left[\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right]=9.866 \times 10^{-2} \mathrm{M}\). (b) \(\mathrm{ClO}^{-}:\left[\mathrm{OH}^{-}\right]=4.0 \times 10^{-4} \mathrm{M}\) \([\mathrm{HClO}]=2.38 \times 10^{-4} \mathrm{M}$$\left[\mathrm{ClO}^{-}\right]=0.273 \mathrm{M}\). (c) \(\mathrm{HCO}_{2} \mathrm{H}:\left[\mathrm{HCO}_{2} \mathrm{H}\right]=0.524 \mathrm{M}\) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=9.8 \times 10^{-3} \mathrm{M}\) \(\left[\mathrm{HCO}_{2}^{-}\right]=9.8 \times 10^{-3} \mathrm{M}\). (d) \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{3}^{+}: \quad\left[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{3}^{+}\right]=0.233 \mathrm{M}\) \(\left[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{2}\right]=2.3 \times 10^{-3} \mathrm{M}\) \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=2.3 \times 10^{-3} \mathrm{M}\).

Step by step solution

01

Calculate the acid dissociation constant (Ka) for CH3CO2H

To calculate the Ka value for CH3CO2H, use the formula Ka = [H3O+][CH3CO2-]/[CH3CO2H]. Substitute the given concentrations into the formula: Ka = (1.34 x 10^-3 M)(1.34 x 10^-3 M) / 9.866 x 10^-2 M.

02

Calculate the base dissociation constant (Kb) for ClO-

For the weak base ClO-, use the Kb formula: Kb = [OH-][HClO]/[ClO-]. Insert the values given: Kb = (4.0 x 10^-4 M)(2.38 x 10^-4 M) / 0.273 M.

03

Calculate the acid dissociation constant (Ka) for HCO2H

For HCO2H, use the formula Ka = [H3O+][HCO2-]/[HCO2H]. Plug in the concentrations: Ka = (9.8 x 10^-3 M)(9.8 x 10^-3 M) / 0.524 M.

04

Calculate the base dissociation constant (Kb) for C6H5NH3+

Use the concentration provided to get the Kb for the conjugate base of C6H5NH3+: Kb = [C6H5NH2][H3O+]/[C6H5NH3+]. Input the values: Kb = (2.3 x 10^-3 M)(2.3 x 10^-3 M) / 0.233 M.

05

Perform the calculations for each substance

Compute the numerical values for Ka and Kb using the multiplication and division of the concentrations given for CH3CO2H, ClO-, HCO2H and C6H5NH3+

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weak Acid Equilibrium

In chemistry, weak acids do not fully dissociate in solution, which sets up an equilibrium between the undissociated acid (HA) and its dissociated ions (H⁺ or H₃O⁺ and A^-). The equilibrium for a weak acid can be represented by the chemical equation:
HA <==> H⁺ + A^-.
For a given weak acid equilibrium, the acid dissociation constant (Ka) quantifies the strength of the acid. It is the product of the concentrations of the ions divided by the concentration of the initial acid, at equilibrium. The expression is as follows:
Ka = [H⁺][A^-]/[HA].
The higher the Ka value, the stronger the acid (but still weak when compared to strong acids). Understanding this equilibrium is crucial for predicting the behavior of weak acids in various chemical scenarios.

Weak Base Equilibrium

Analogous to weak acids, weak bases only partially dissociate in water to yield hydroxide ions (OH^-) and their corresponding conjugate acids (BH⁺). This partial dissociation creates an equilibrium that can be written as:
B + H₂O <==> BH⁺ + OH^-.
The base dissociation constant (Kb) reflects the base's strength by describing this equilibrium. Kb is defined by the equation:
Kb = [BH⁺][OH^-]/[B].
Like Ka, the larger the Kb, the stronger the base. However, it is important to remember that 'stronger' still refers within the context of weak bases, not in comparison to strong bases.

Equilibrium Concentration Calculation

The equilibrium concentrations of products and reactants in a weak acid or base reaction can be determined through an ICE table (Initial, Change, Equilibrium) or directly calculated when the equilibrium concentrations are directly provided, as seen in the exercise. By plugging these values into the Ka or Kb expression, the respective constant can be evaluated. Calculating these concentrations requires careful stoichiometric analysis as well as an understanding of the assumptions involved in the calculations of weak acid or weak base equilibria, such as the neglect of water's ion concentration compared to that of the acid or base in solution.

Ka and Kb Expressions

The acid dissociation constant (Ka) and the base dissociation constant (Kb) are expressions that are used to describe the strength of weak acids and bases, respectively. These constants are derived from the equilibrium concentrations of the ions and the undissociated weak acid or base. They are valuable because they allow chemists to predict how a compound will behave in a solution, compare acidities and basicities of different compounds, and calculate pH. As an example, the Ka for acetic acid (CH₃CO₂H) is the ratio of the concentration of acetate ions (CH₃CO₂^-) and hydronium ions (H₃O⁺) to the concentration of undissociated acetic acid, while the Kb for ammonia (NH₃) would relate the concentrations of ammonium ions (NH₄⁺) and hydroxide ions (OH^-) to the concentration of undissociated ammonia.