Question

At \(900^{\circ} \mathrm{C}, K_{\mathrm{p}}=1.04\) for the reaction $$\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$$.At a low temperature, dry ice (solid \(\mathrm{CO}_{2}\) ), calcium oxide, and calcium carbonate are introduced into a 50.0 -L reaction chamber. The temperature is raised to \(900^{\circ} \mathrm{C},\) resulting in the dry ice converting to gaseous \(\mathrm{CO}_{2}\). For the following mixtures, will the initial amount of calcium oxide increase, decrease, or remain the same as the system moves toward equilibrium at \(900^{\circ} \mathrm{C} ?\) a. \(655 \mathrm{g} \mathrm{CaCO}_{3}, 95.0 \mathrm{g} \mathrm{CaO}, P_{\mathrm{CO}_{2}}=2.55 \mathrm{atm}\) b. \(780 \mathrm{g} \mathrm{CaCO}_{3}, 1.00 \mathrm{g} \mathrm{CaO}, P_{\mathrm{CO}_{2}}=1.04 \mathrm{atm}\) c. \(0.14 \mathrm{g} \mathrm{CaCO}_{3}, 5000 \mathrm{g} \mathrm{CaO}, P_{\mathrm{CO}_{2}}=1.04 \mathrm{atm}\) d. \(715 \mathrm{g} \mathrm{CaCO}_{3}, 813 \mathrm{g} \mathrm{CaO}, P_{\mathrm{CO}_{2}}=0.211 \mathrm{atm}\)

Short answer

a. The amount of calcium oxide will decrease. b. The amount of calcium oxide will remain the same. c. The amount of calcium oxide will remain the same. d. The amount of calcium oxide will increase.

Step-by-step solution

Write down the Kp expression for the given reaction

For the given reaction: \(\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)\), the Kp expression is the pressure of CO2 (since solid concentrations are not accounted for): \[K_p = P_{CO_2}\]

Calculate Qp for each mixture

Calculate Qp for each mixture using the given amounts and the reaction chamber size (50.0 L): a. \( P_{CO_2}=2.55 \) atm \(Q_{p(a)} = 2.55\) b. \( P_{CO_2}=1.04 \) atm \(Q_{p(b)} = 1.04\) c. \( P_{CO_2}=1.04 \) atm \(Q_{p(c)} = 1.04\) d. \( P_{CO_2}=0.211 \) atm \(Q_{p(d)} = 0.211\)

Compare Qp to Kp for each mixture and determine the direction of the reaction

Compare the calculated Qp values against the given Kp value (1.04) to determine if the reaction will move towards equilibrium: a. \(Q_{p(a)} = 2.55\) > \(K_p = 1.04\) → The reaction needs to move left towards higher calcium carbonate concentration and lower CO2 and calcium oxide concentrations. Therefore, the amount of calcium oxide will decrease. b. \(Q_{p(b)} = 1.04\) = \(K_p = 1.04\) → This mixture is already at equilibrium, so the amount of calcium oxide will remain the same. c. \(Q_{p(c)} = 1.04\) = \(K_p = 1.04\) → This mixture is also already at equilibrium, so the amount of calcium oxide will remain the same. d. \(Q_{p(d)} = 0.211\) < \(K_p = 1.04\) → The reaction needs to move right towards higher CO2 and calcium oxide concentrations and lower calcium carbonate concentration. Therefore, the amount of calcium oxide will increase.

Key concepts

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept in chemical equilibrium that explains how a system at equilibrium responds to external changes. It states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change and restore a new balance. Let's understand how this applies to gaseous equilibria:

• When pressure, temperature, or concentration is altered, a system will adjust to minimize the effect of the change.
• For example, in the given reaction, if the pressure of carbon dioxide increases (let's say above the equilibrium constant pressure at 1.04 atm), the reaction will shift towards the production of reactants (CaCO₃).
• Conversely, if pressure decreases, the equilibrium shifts towards forming more products (CaO and CO₂).

Understanding Le Chatelier's Principle helps predict the direction in which the equilibrium will shift when a change occurs, ensuring that students can gauge the system’s response effectively.

Reaction Quotient

The Reaction Quotient (Q) is a crucial tool in determining the direction a chemical reaction will take to reach equilibrium. Unlike the equilibrium constant, which is a fixed value at a given temperature, Q is calculated at any point in time to establish where the system is relative to its equilibrium state. Here’s how it works:

• If Q < K, the reaction will proceed in the forward direction until equilibrium is reached.
• If Q = K, the system is at equilibrium, and no shift occurs.
• If Q > K, the reaction will proceed in reverse to consume excess products and form reactants.

For the provided chemical system, calculating Q_p from the partial pressure of CO₂ allows for quick assessment of whether the system has too many products or reactants. This is essential for predicting how quantities of substances like calcium oxide will change over time.

Equilibrium Constant

The Equilibrium Constant (K) is a numerical representation of the ratio of product concentrations to reactant concentrations at equilibrium, with each raised to the power of their respective coefficients in the balanced equation. In gaseous reactions, K_p applies, where the constant is expressed in terms of partial pressures.

• For the reaction that we are considering, K_p is defined solely by the pressure of CO₂ because solids do not affect the equilibrium constant. Thus, K_p = P_CO2.
• This constant is a key indicator of the position of equilibrium and will help determine which direction a reaction needs to move to achieve balance.
• The value of K_p can also help to predict whether reactions are product-favored or reactant-favored at the temperature in question.

Understanding the equilibrium constant in terms of its physical and mathematical significance is pivotal for solving equilibrium problems and gauging reaction progress.

Gaseous Equilibria

Gaseous equilibria involve reactions where gases are either reactants or products, making equilibrium considerations slightly different from those involving only solids and liquids. Due to the nature of gases, their concentrations are often expressed in terms of partial pressures. Here's what you need to know about gaseous equilibria:

• Partial pressures are utilized in the equilibrium expression (such as K_p) to represent the concentrations of gaseous substances.
• Changes in pressure and volume can significantly affect the position of equilibrium in gaseous systems.
• In reactions involving gases, such as the decomposition of calcium carbonate to calcium oxide and carbon dioxide, these changes are often intuitive thanks to Le Chatelier’s Principle.

By understanding how gases behave under changing conditions, students can predict shifts in equilibria more confidently, allowing them to tackle problems involving gaseous reactions with greater ease.