Question

Write the equilibrium constant expression for each of these heterogeneous systems. (a) \(\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g})+\mathrm{O}_{3}(\mathrm{~g}) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{~s})+\mathrm{O}_{2}(\mathrm{~g})\) (b) \(\mathrm{C}(\mathrm{s})+2 \mathrm{~N}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{~N}_{2}(\mathrm{~g})\) (c) \(\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\)

Short answer

(a) \( K_c = \frac{[ ext{O}_2]}{[ ext{N}_2 ext{O}_4][ ext{O}_3]} \); (b) \( K_c = \frac{[ ext{CO}_2][ ext{N}_2]^2}{[ ext{N}_2 ext{O}]^2} \); (c) \( K_c = [ ext{H}_2 ext{O}] \).

Step-by-step solution

Understanding Heterogeneous Equilibria

In a heterogeneous equilibrium, reactants and products are in different phases. For such systems, only the concentrations of gases and solutes in aqueous solution appear in the equilibrium constant expression. Pure solids and liquids have constant concentrations and do not appear in the expression.

System (a): Ignoring Solids

For the reaction \( ext{N}_2 ext{O}_4( ext{g}) + ext{O}_3( ext{g}) \rightleftharpoons ext{N}_2 ext{O}_5( ext{s}) + ext{O}_2( ext{g})\), N₂O₅ is a solid and will not appear in the expression. The equilibrium constant expression is: \[ K_c = \frac{[ ext{O}_2]}{[ ext{N}_2 ext{O}_4][ ext{O}_3]} \]

System (b): Ignoring Solids

For the reaction \( ext{C}( ext{s}) + 2 ext{N}_2 ext{O}( ext{g}) \rightleftharpoons ext{CO}_2( ext{g}) + 2 ext{N}_2( ext{g})\), carbon (C) is a solid and is ignored in the expression. The equilibrium constant expression is: \[ K_c = \frac{[ ext{CO}_2][ ext{N}_2]^2}{[ ext{N}_2 ext{O}]^2} \]

System (c): Ignoring Liquids

In the reaction \( ext{H}_2 ext{O}( ext{l}) \rightleftharpoons ext{H}_2 ext{O}( ext{g})\), liquid water is not included in the expression. The equilibrium constant is expressed as: \[ K_c = [ ext{H}_2 ext{O}] \]

Key concepts

Heterogeneous Equilibria

In the fascinating world of chemical reactions, a heterogeneous equilibrium occurs when the reactants and products are in different phases. Phases can be solids, liquids, or gases. In calculations involving heterogeneous equilibria, pure solids and liquids are usually not included in the equilibrium constant expression.

This is because their concentrations remain constant during the reaction. They are essentially fixed since adding more of a solid or a pure liquid does not change its concentration, given that concentration depends on the amount and volume, which for solids and pure liquids does not change with usual reaction conditions.

So, when you write the expression for the equilibrium constant (usually denoted as \( K_c \) or \( K_p \), depending on whether concentrations or partial pressures are used), you should focus only on the gases and aqueous solutions involved. This simplifies the calculations significantly, allowing chemists to better understand and predict the behavior of reactions.

Concentration of Gases

When dealing with gaseous reactions, the equilibrium constant expression involves the concentrations of gases. Concentration in this case is usually expressed in terms of molarity (moles per liter) or sometimes in terms of partial pressures. Partial pressures may lead to expressions denoted as \( K_p \). Most of the time, the letters \( P \) (for pressure) or \( c \) (for concentration) are used to distinguish between these two.

For equilibrium calculations, it’s crucial to have a clear understanding of partial pressures, especially if gases are involved in the reaction. Partial pressure is the pressure that a gas would exert if it alone occupied the whole container. This is related to the molar concentration through the ideal gas law, which is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Using the ideal gas equation, you can easily convert between concentration (mol/L) and partial pressure (atm), which helps when expressing or solving for the equilibrium constant in reactions where gases are present.

Equilibrium Expressions

The equilibrium expression is a concise way to express the equilibrium state of a chemical reaction. It gives the ratio of the concentrations of products to reactants, each raised to the power of their coefficients in the balanced equation. This mathematical relation is key to predicting how changes in conditions affect the reaction.

To write the equilibrium expression, follow these steps:

• Identify the reactants and products involved in the gas phase, as those will appear in the equilibrium constant.
• Exclude solids and pure liquids from the expression as their concentration does not change during the reaction.
• For products, place their concentrations in the numerator and for reactants in the denominator of the expression.
• Raise each concentration to the power of its respective stoichiometric coefficient from the balanced chemical equation.

The resulting expression represents the equilibrium constant \( K \), reflecting how far a reaction proceeds towards products. If the value of \( K \) is large, it indicates that the reaction heavily favors the products. Conversely, a small value of \( K \) suggests that reactants are predominant, and the reaction does not proceed much forward. Understanding these principles is essential for mastering equilibrium calculations and manipulating chemical reactions to achieve desired outcomes.