Question

Calculate the concentration of all species in a \(0.155 \mathrm{M}\) solution of \(\mathrm{H}_{2} \mathrm{CO}_{3}\)

Short answer

[H2CO3] ≈ 0.155 M, [H+] ≈ [HCO3-] ≈ square root of (Ka1 × [H2CO3]) = square root of (4.3 × 10^-7 × 0.155), [CO3^2-] ≈ 0

Step-by-step solution

Recognize the compound and its dissociation

Identify that carbonic acid \( \mathrm{H}_{2} \mathrm{CO}_{3} \) is a weak diprotic acid that can dissociate in two steps:\ First dissociation: \( \mathrm{H}_{2} \mathrm{CO}_{3} \rightleftharpoons \mathrm{H}^{+} + \mathrm{HCO}_{3}^{-} \)\ Second dissociation: \( \mathrm{HCO}_{3}^{-} \rightleftharpoons \mathrm{H}^{+} + \mathrm{CO}_{3}^{2-} \)\ Since it is a weak acid, we'll focus on the first dissociation primarily because the second dissociation is much weaker.

Write the equilibrium expressions

For the first dissociation equilibrium, the acid dissociation constant \( K_{a1} \) expression is:\ \( K_{a1} = \frac{[\mathrm{H}^{+}][\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]} \)\ For the second dissociation, the acid dissociation constant \( K_{a2} \) expression is:\ \( K_{a2} = \frac{[\mathrm{H}^{+}][\mathrm{CO}_{3}^{2-}]}{[\mathrm{HCO}_{3}^{-}]} \)\ Note that the hydrogen ion concentration is involved in both equilibrium expressions and that \( [\mathrm{H}_{2} \mathrm{CO}_{3}] \) changes very little due to the weak nature of the acid.

Determine the acid dissociation constants

Consult a chemistry reference to find the acid dissociation constants for carbonic acid. Typically, \( K_{a1} \) for \( \mathrm{H}_{2} \mathrm{CO}_{3} \) is approximately \( 4.3 \times 10^{-7} \) and \( K_{a2} \) is much smaller, about \( 5.6 \times 10^{-11} \). Given the small value of \( K_{a2} \) compared to \( K_{a1} \) and the initial concentration, we can assume that \( [\mathrm{CO}_{3}^{2-}] \) is negligible.

Set up the ICE table for the first dissociation

For the first dissociation \( \mathrm{H}_{2} \mathrm{CO}_{3} \rightleftharpoons \mathrm{H}^{+} + \mathrm{HCO}_{3}^{-} \), set up an ICE (Initial, Change, Equilibrium) table:\ Initial concentrations: \( [\mathrm{H}_{2} \mathrm{CO}_{3}] = 0.155 \) M, \( [\mathrm{H}^{+}] = 0 \) M, \( [\mathrm{HCO}_{3}^{-}] = 0 \) M.\ Changes in concentration: \( [\mathrm{H}^{+}] \) and \( [\mathrm{HCO}_{3}^{-}] \) will both increase by \( x \) M, and \( [\mathrm{H}_{2} \mathrm{CO}_{3}] \) will decrease by \( x \) M.\ Equilibrium concentrations: \( [\mathrm{H}_{2} \mathrm{CO}_{3}] = 0.155 - x \) M, \( [\mathrm{H}^{+}] = x \) M, \( [\mathrm{HCO}_{3}^{-}] = x \) M.

Solve for concentrations using the equilibrium constant

Substitute the equilibrium concentrations into the \( K_{a1} \) expression and solve for \( x \) (ignoring \( x \) against 0.155 M due to the small \( K_{a1} \)): \( K_{a1} = \frac{[\mathrm{H}^{+}][\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2} \mathrm{CO}_{3}]} = \frac{x \cdot x}{0.155} \)\ Solve this quadratic equation to find \( x \) and hence \( [\mathrm{H}^{+}] \) and \( [\mathrm{HCO}_{3}^{-}] \). The concentration of \( [\mathrm{CO}_{3}^{2-}] \) remains approximately zero.

Key concepts

Acid Dissociation Constant

The acid dissociation constant, denoted as Ka, is a quantitative measure of the strength of an acid in solution. It specifically refers to the equilibrium constant for the dissociation of an acid into its conjugate base and a hydrogen ion (proton). In the context of carbonic acid (H2CO3), this constant is crucial for understanding how much of the acid dissociates in water.
For any weak acid, the dissociation process can be represented by: HA ⇌ H^+ + A^- and the acid dissociation constant expression is given by: Ka = [H^+][A^-] / [HA] where [HA] is the concentration of the weak acid, [H^+] is the concentration of hydrogen ions, and [A^-] is the concentration of the conjugate base. A large Ka value indicates a strong acid that dissociates easily, while a small Ka value characterizes a weak acid with limited dissociation. Most diprotic acids, such as carbonic acid, have two dissociation constants, Ka1 and Ka2, each representing one of the two proton losses. The first dissociation constant (Ka1) tends to be larger than the second (Ka2), illustrating that the first proton is easier to lose than the second.

ICE Table

An ICE table, which stands for Initial, Change, Equilibrium, is a systematic way to keep track of the changes in concentrations of reactants and products during a chemical reaction that has reached equilibrium.

• Initial concentrations are those present before any reaction occurs.
• Change refers to the increase or decrease in concentrations as the system moves towards equilibrium.
• Equilibrium concentrations are those when the system is at equilibrium, where the forward and reverse reactions occur at the same rate.

In the problem-solving process involving a weak acid like (H2CO3), an ICE table helps you visualize and quantify the concentration shifts during the dissociation process. By systematically applying the acid dissociation constant, we can determine the concentrations of all species at equilibrium. This table is especially useful when dealing with weak acids or bases, as they do not dissociate completely and require equilibrium considerations.

Weak Diprotic Acid

A weak diprotic acid, such as carbonic acid (H2CO3), is an acid that can donate two protons (hydrogen ions) and does so in two distinct steps, each with its own equilibrium constant.

First Dissociation Step

In the first step, one proton is donated to form the conjugate base: H2CO3 ⇌ H^+ + HCO3^- This is represented by the acid dissociation constant Ka1, which is generally higher than Ka2, indicating that the first proton dissociates more readily than the second.

Second Dissociation Step

The second dissociation step involves the newly formed conjugate base releasing another proton: HCO3^- ⇌ H^+ + CO3^2- and is represented by the constant Ka2. Due to the weakness of the second dissociation, in most cases, it can be considered negligible in calculations, simplifying the analysis of the system. Understanding these steps is critical to solving problems involving weak diprotic acids and in predicting the pH and buffer capacity of solutions.