Question

One mole of a weak acid HA was dissolved in 2.0 \(\mathrm{L}\) of solution. After the system had come to equilibrium, the concentration of HA was found to be 0.45 \(\mathrm{M} .\) Calculate \(K_{\mathrm{a}}\) for \(\mathrm{HA}\) .

Short answer

The dissociation constant Ka of the weak acid HA is approximately \(5.56 \times 10^{-3}\).

Step-by-step solution

Write the equilibrium reaction for the dissociation of the weak acid (HA)

The general reaction for the dissociation of a weak acid in water is given below: \[HA + H_2O \rightleftharpoons H_3O^+ + A^-\] In this reaction, HA is the weak acid, H2O is water, H3O+ is the hydronium ion, and A- is the conjugate base of the weak acid.

Determine the initial concentrations

The initial concentration of HA can be calculated using the relationship: \[Initial \thinspace concentration \thinspace of \thinspace HA = \frac{number \thinspace of \thinspace moles \thinspace of \thinspace HA}{volume \thinspace of \thinspace solution}\] We are given that 1 mole of HA is dissolved in 2.0 L of solution, so: \[Initial \thinspace concentration \thinspace of \thinspace HA = \frac{1 \thinspace mol}{2.0 \thinspace L} = 0.5 \thinspace M\] The initial concentration of H3O+ and A- is 0 because no dissociation has taken place yet.

Create an ICE table

Before writing the ICE table, it is important to know the changes that occur for each species in the reaction. As the equilibrium reaction progresses, the concentration of HA decreases, while the concentration of H3O+ and A- increases. Here’s the ICE table for the weak acid (HA) dissociation reaction: | | Initial (M) | Change (M) | Equilibrium (M) | |-------------|-------------|-------------|-----------------| | HA | 0.50 | -x | 0.50 - x | | H3O+ | 0 | +x | x | | A- | 0 | +x | x |

Calculate the change in concentration (x)

We are given that the equilibrium concentration of HA is 0.45 M. We can use this information to calculate the change in concentration (x) of HA: \[0.50 - x = 0.45\] Solving for x: \[x = 0.50 - 0.45 = 0.05 \thinspace M\] Now that we have x, we can also find the equilibrium concentration of H3O+ and A-: - Final concentration of H3O+: x = 0.05 M - Final concentration of A-: x = 0.05 M

Calculate Ka

Now that we have the equilibrium concentrations, we can calculate the dissociation constant (Ka) for the weak acid. The equilibrium expression for this reaction is: \[K_a = \frac{[H_3O^+][A^-]}{[HA]}\] By substituting the equilibrium concentrations that we found in Step 4: \[K_a = \frac{(0.05)(0.05)}{(0.45)} = \frac{0.0025}{0.45} = 5.56 \times 10^{-3}\] So the dissociation constant Ka of HA is approximately \(5.56 \times 10^{-3}\).

Key concepts

Equilibrium

Equilibrium in chemistry refers to the state where the rate of the forward reaction is equal to the rate of the reverse reaction. This balance means that the concentrations of reactants and products remain constant over time, even though the reactions continue to occur on a molecular level. In the context of a weak acid dissociation, equilibrium is reached when the weak acid, represented by HA, dissociates into its ions H extsubscript{3}O extsuperscript{+} and A extsuperscript{-}, and the system stabilizes where both reactants and products coexist in solution.

• The concept of equilibrium doesn't mean that the amounts of reactants and products are equal, but that their concentrations are stable.
• This state can be represented by an equilibrium reaction, often denoted by a double arrow, such as: \[ \mathrm{HA} + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{H}_3\mathrm{O}^+ + \mathrm{A}^- \]
• Changes in conditions like temperature or pressure can shift the equilibrium either to the right (favoring products) or to the left (favoring reactants), according to Le Chatelier's Principle.

Understanding equilibrium is vital as it allows us to predict how alterations in conditions can affect the ratio of products to reactants in a chemical reaction.

Dissociation Constant

The dissociation constant, specifically for acids, is known as the acid dissociation constant (K_a). It quantifies the strength of a weak acid by measuring its degree of dissociation in water.

For a weak acid HA that dissociates in water, the reaction is written as:\[ \mathrm{HA} \rightleftharpoons \mathrm{H}_3\mathrm{O}^+ + \mathrm{A}^- \]The equilibrium expression for K_a reflects the ratio of the concentrations of the resulting ions over the concentration of the undissociated acid:\[ K_a = \frac{[\mathrm{H}_3\mathrm{O}^+][\mathrm{A}^-]}{[\mathrm{HA}]} \]This formula shows how much of the acid has dissociated at equilibrium. A larger K_a value indicates a stronger weak acid, which means it dissociates more in solution, releasing more H extsubscript{3}O extsuperscript{+} ions.

• Weak acids typically have small K_a values, reflecting partial dissociation.
• K_a values are central in calculating pH and in predicting the extent to which an acid-base reaction will occur.
• Concentration units should always be expressed in molarity (M).

Grasping K_a is crucial, as it provides a comprehensive understanding of an acid's behavior in solution.

ICE Table

An ICE table is a crucial tool for organizing data when dealing with equilibrium problems. The acronym stands for Initial, Change, and Equilibrium, which are the stages of concentration changes during a chemical reaction proceeding to equilibrium.

• Initial: These are the concentrations of reactants and products at the beginning, before any reaction occurs. Typically, for a weak acid, the starting concentration of the acid is known, while the products start at zero.
• Change: This column describes how the concentrations change as the system moves towards equilibrium. The change is generally denoted by a variable, such as \( x \).
• Equilibrium: This column summarizes the concentrations when the reaction has reached equilibrium. It embodies the initial concentration plus or minus the change.

For the weak acid dissociation (HA ), the table would look something like this:
- HA: Initial = 0.5 M, Change = -x, Equilibrium = 0.5 - x
- H extsubscript{3}O extsuperscript{+}: Initial = 0, Change = +x, Equilibrium = x
- A extsuperscript{-}: Initial = 0, Change = +x, Equilibrium = xUsing an ICE table simplifies determining how much of each species is present at equilibrium and is fundamental for calculating equilibrium constants like K_a.

Equilibrium Constant

The equilibrium constant (K_a for acids) is a quantitative representation of the equilibrium state of a reaction. It depends on the concentrations of the products and reactants in a reaction at equilibrium.
For a weak acid dissociation into H extsubscript{3}O extsuperscript{+} and A extsuperscript{-}, the equilibrium equation is:\[ \mathrm{HA} + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{H}_3\mathrm{O}^+ + \mathrm{A}^- \]The equilibrium constant K_a is expressed as:\[ K_a = \frac{[\mathrm{H}_3\mathrm{O}^+][\mathrm{A}^-]}{[\mathrm{HA}]} \]Here:

• The brackets [ ] represent concentration in molarity (M).
• A higher K_a value suggests a greater extent of dissociation, indicating a stronger weak acid.

The value of K_a is specific to each weak acid at a given temperature, reflecting how far the reaction proceeds at equilibrium. Understanding how to calculate and interpret K_a is essential, as it informs decisions in chemical production, environmental processes, and even within biological systems like the human body.