Question

Calculate \(\left[\mathrm{CO}_{3}^{2-}\right]\) in a \(0.010-M\) solution of \(\mathrm{CO}_{2}\) in water (usually written as \(\mathrm{H}_{2} \mathrm{CO}_{3}\) ). If all the \(\mathrm{CO}_{3}{ }^{2-}\) in this solution comes from the reaction $$\mathrm{HCO}_{3}^{-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{CO}_{3}^{2-}(a q)$$ what percentage of the \(\mathrm{H}^{+}\) ions in the solution is a result of the dissociation of \(\mathrm{HCO}_{3}^{-} ?\) When acid is added to a solution of sodium hydrogen carbonate \(\left(\mathrm{NaHCO}_{3}\right)\), vigorous bubbling occurs. How is this reaction related to the existence of carbonic acid \(\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)\) molecules in aqueous solution?

Short answer

The concentration of $\mathrm{CO}_{3}^{2-}$ in a $0.010~\text{M}$ solution of $\mathrm{CO}_{2}$ in water is obtained by solving for x in the equilibrium expression: $$K = \frac{x^2}{([H_2CO_3]-x)}$$ After calculating x, the concentration of $\mathrm{CO}_{3}^{2-}$ is equal to x. The percentage of $\mathrm{H}^{+}$ ions from the dissociation of $\mathrm{HCO}_{3}^{-}$ is given by: $$\frac{x}{[H_2CO_3]+x} \times 100$$ When an acid is added to a solution of $\mathrm{NaHCO}_{3}$, the released $\mathrm{H}^{+}$ ions react with $\mathrm{HCO}_{3}^{-}$ ions to form $\mathrm{H}_{2}\mathrm{CO}_{3}$, which decomposes into $\mathrm{CO}_{2}$ gas and water, causing the observed bubbles.

Step-by-step solution

Identify the relevant data and equilibrium constant

We are given that the total concentration of CO2 in water (also referred to as H2CO3) is 0.010 M. We also know the given reaction: $$\mathrm{HCO}_{3}^{-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{CO}_{3}^{2-}(a q)$$ Let's call the equilibrium constant of this reaction K.

Set up the equilibrium expression

According to the law of mass action, the equilibrium constant K is given by: $$K = \frac{[\mathrm{H}^{+}][\mathrm{CO}_{3}^{2-}]}{[\mathrm{HCO}_{3}^{-}]}$$

Define the change in concentrations at equilibrium

Let x be the amount of HCO3^- that dissociates into H+ and CO3^{2-}. The resulting concentrations of HCO3^-, H+, and CO3^{2-} at equilibrium will be: - [HCO3^-] = [H2CO3] - x - [H+] = x - [CO3^{2-}] = x

Substitute the concentrations into the equilibrium expression

Substituting the equilibrium concentrations into the expression for K, we get: $$K = \frac{x \cdot x}{([H_2CO_3] - x)}$$

Solve for x

From this step onwards, you might require to look up the value of K in a textbook or online resources. Knowing K, solve for x in the above equation (a quadratic equation in x). This value of x represents the equilibrium concentration of H+ and CO3^{2-}.

Calculate the concentration of CO3^{2-}

Now that we have the value of x, the concentration of CO3^{2-} is equal to x.

Calculate the percentage of H+ ions from the dissociation of HCO3^-

To find the percentage of H+ ions in the solution resulting from the dissociation of HCO3^-, divide the concentration of H+ ions (which is x) by the total concentration of H+ ions in the solution (H2CO3 + x) and multiply by 100: Percentage = $$\frac{x}{[H_2CO_3] + x} \times 100$$

Connect the reaction to the bubbling of NaHCO3 in acidic solution

When an acid is added to a solution of NaHCO3 (sodium hydrogen carbonate), the acidic H+ ions react with the HCO3^- ions to form H2CO3 (carbonic acid), which then decomposes into CO2 (carbon dioxide) and H2O (water). The release of CO2 gas results in the bubbles observed during the reaction.

Key concepts

Carbonate Ion Concentration

In a chemical solution like one containing carbonic acid (\(\text{H}_2\text{CO}_3\)), it is crucial to identify the concentration of different ions present. The carbonate ion (\(\text{CO}_3^{2-}\)) is a particular ion of interest in aqueous solutions. To determine its concentration, we look at the equilibrium established in the solution.

The dissociation of bicarbonate (\(\text{HCO}_3^{-}\)) into carbonate (\(\text{CO}_3^{2-}\)) provides a basis for this measurement. By setting up equilibrium expressions, one can calculate the precise equilibrium concentration of carbonate ions.

• Let x be the concentration of carbonate ions formed at equilibrium.
• If the equilibrium constant (\(K\)) for the reaction is known, \(x\) can be deduced from solving the equilibrium expression.

Understanding carbonate ion concentrations is essential for predicting the behavior of the solution under various conditions. It helps in determining properties like alkalinity and how a solution might react when perturbed (e.g., by adding acids or bases).

Equilibrium Constant

The equilibrium constant (\(K\)) is a measure of the position of equilibrium in a chemical reaction at a given temperature. For the dissociation of carbonic acid into bicarbonate (\(\text{HCO}_3^{-}\)) and carbonate (\(\text{CO}_3^{2-}\)), the equilibrium constant provides the ratio of these species at equilibrium.

The expression for \(K\) is:\[K = \frac{[\mathrm{H}^{+}][\mathrm{CO}_{3}^{2-}]}{[\mathrm{HCO}_{3}^{-}]}\]

• To find \(K\), you can often look for data in textbooks or scientific literature.
• The value of \(K\) dictates how far the reaction proceeds towards products; the larger \(K\), the more products are formed at equilibrium.

Finding the equilibrium constant is key in many applications, such as calculating solubility, predicting reactivity, and understanding chemical processes in natural systems. Thus, it holds a fundamental role in exploring chemical equilibria.

Carbonic Acid Dissociation

Carbonic acid dissociation is a pivotal reaction in aquatic chemistry, especially in systems exposed to carbon dioxide. It involves several dissociation steps, transforming \(\text{H}_2\text{CO}_3\) into bicarbonate (\(\text{HCO}_3^{-}\)) and then into carbonate (\(\text{CO}_3^{2-}\)). This series of reactions helps maintain the acid-base balance in natural waters.

When examining carbonic acid dissociation:

• The first dissociation of \(\text{H}_2\text{CO}_3\) results in \(\text{HCO}_3^{-}\).
• The second step, where \(\text{HCO}_3^{-}\) dissociates, yields \(\text{CO}_3^{2-}\) ions and is the focal reaction in certain calculations.
• These dissociations are highly dependent on pH and temperature, influencing the concentrations of the resulting ions.

Carbonic acid dissociation is natural and occurs constantly in solution. This is important for buffering capacity and understanding the release of carbon dioxide as seen in reactions involving \(\text{NaHCO}_3\) with added acid, which relates to phenomena like the bubbling in fizzy drinks and effervescent reactions in geology.