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?

Step by step solution

01

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.

02

Set up the equilibrium expression

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

03

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

04

Substitute the concentrations into the equilibrium expression

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

05

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-}.

06

Calculate the concentration of CO3^{2-}

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

07

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 = $$\frac{x}{[H_2CO_3] + x} \times 100$$

08

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.

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