Question

An important environmental and physiological reaction is the formation of carbonic acid, \(\mathrm{H}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) from carbon dioxide and water. Write the equilibrium constant expression for this reaction in terms of activities. Convert that expression into an equilibrium constant expression containing concentrations and pressures.

Short answer

The equilibrium constant expression in terms of activities is \[K = \frac{a(H_{2}CO_{3})}{a(CO_{2})a(H_{2}O)}\] , and in terms of concentrations and pressures it is \[K = \frac{[H_{2}CO_{3}]}{P(CO_{2})[H_{2}O]} \]

Step-by-step solution

Understand the Reaction and Write it Down

The reaction is between carbon dioxide (CO2) and water (H2O) to form carbonic acid (H2CO3). The reaction can be written as: \[CO_{2(g)} + H_{2O(l)} \longleftrightarrow H_{2}CO_{3(aq)} \]

Write the Equilibrium Constant Expression in Terms of Activities

The equilibrium expression in terms of activities can be written based on the law of mass action. For the reaction mentioned above, the equilibrium constant expression can be written as: \[K = \frac{a(H_{2}CO_{3})}{a(CO_{2})a(H_{2}O)}\] where \(K\) is the equilibrium constant and \(a(Y)\) represents the activity of species \(Y\) in the reaction.

Convert the Equilibrium Constant Expression into Concentration and Pressure Terms

The activity of a species in solution is approximately equal to its molar concentration, and the activity of a gas is approximately equal to its partial pressure. Therefore, the equilibrium constant expression can be converted into concentration and pressure terms as follows: \[K = \frac{[H_{2}CO_{3}]}{P(CO_{2})[H_{2}O]}\] where \([Y]\) represents the concentration of species \(Y\), and \(P(Y)\) represents the partial pressure of species \(Y\) in the reaction.

Key concepts

Carbonic Acid Formation

The formation of carbonic acid (\(\mathrm{H}_{2}\mathrm{CO}_{3}\)) is a crucial reaction both environmentally and physiologically. This reaction occurs when carbon dioxide (\(\mathrm{CO}_{2}\)) dissolves in water (\(\mathrm{H}_{2}\mathrm{O}\)). The resulting product is carbonic acid, which is a weak acid. It plays an important role in maintaining pH levels in bodily fluids and also contributes to the carbon cycle in nature.
The process can be summarized by the chemical reaction: \[\mathrm{CO}_{2(g)} + \mathrm{H}_{2}\mathrm{O}_{(l)} \longleftrightarrow \mathrm{H}_{2}\mathrm{CO}_{3(aq)}\] This reversible reaction indicates that carbonic acid can decompose back into carbon dioxide and water. The reaction reaches an equilibrium point where the formation and decomposition rates of carbonic acid are equal.

Activities in Equilibrium

When discussing chemical equilibria, 'activities' of the species involved provide a more accurate expression of concentrations. Activities account for the non-ideal behaviors of real chemical solutions, where concentration alone might not suffice.
For ideal conditions, like dilute solutions, activities approximate closely to concentrations, but as solutions become more concentrated, these diverge. An activity coefficient can be applied to correct concentrations to activities. For gases, activity is particularly linked to the partial pressure.

• The activity of a solvent, like water often in excess, is typically indexed to 1.
• Incorporating activities into the equilibrium expression ensures the equation aligns with real-world chemical behavior.

Using the concept of activities, the equilibrium expression for carbonic acid formation is:\[K = \frac{a(\mathrm{H}_{2}\mathrm{CO}_{3})}{a(\mathrm{CO}_{2})a(\mathrm{H}_{2}\mathrm{O})}\]This expression takes into account activities rather than mere concentrations.

Law of Mass Action

The law of mass action forms the basis for writing equilibrium constant expressions. It states that the rate of a chemical reaction is proportional to the product of the activities (or concentrations) of the reactants each raised to a power equal to their coefficients in the balanced chemical equation.
This law underpins how we develop expressions to predict how altering conditions will affect the reaction equilibrium. For instance, when the equation for carbonic acid formation is applied, this law dictates:

• The equilibrium constant (\(K\)) provides insight into the balance between reactants and products at equilibrium.
• A fully balanced equation is crucial, as each species’ coefficient becomes its exponent in the expression.

The law of mass action hence allows chemists to understand the dynamics of various reactions under equilibrium conditions and predict potential shifts.

Concentration and Pressure Terms

When converting from activities to concentrations and partial pressures, we simplify the equilibrium expression while still considering the physical state of the reactants and products.
For the formation of carbonic acid:

• Aqueous concentrations are denoted by brackets, like \([\mathrm{H}_{2}\mathrm{CO}_{3}]\).
• Gas pressures, such as for carbon dioxide, are expressed in terms of partial pressure (\(P(\mathrm{CO}_{2})\)).

The equilibrium constant, now in terms of concentrations and pressures, helps to better relate to measurable quantities:\[K = \frac{[\mathrm{H}_{2}\mathrm{CO}_{3}]}{P(\mathrm{CO}_{2})[\mathrm{H}_{2}\mathrm{O}]}\] This expression is practical for laboratory settings where concentrations and pressures are more directly measured compared to activities. The shift from using activities to concentrations and pressures is common where conditions approximate ideality or where concentration and pressures more accurately define the involved species.