Question

At \(900^{\circ} \mathrm{C}, K_{\mathrm{C}}=0.0108\) for the reaction $$\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$$ A mixture of \(\mathrm{CaCO}_{3}, \mathrm{CaO},\) and \(\mathrm{CO}_{2}\) is placed in a 10.0 - \(\mathrm{L}\) vessel at \(900^{\circ} \mathrm{C}\) . For the following mixtures, will the amount of \(\mathrm{CaCO}_{3}\) increase, decrease, or remain the same as the system approaches equilibrium? \begin{equation} \begin{array}{l}{\text { (a) } 15.0 \mathrm{g} \mathrm{CaCO}_{3}, 15.0 \mathrm{g} \mathrm{CaO}, \text { and } 4.25 \mathrm{gCO}_{2}} \\ {\text { (b) } 2.50 \mathrm{g} \mathrm{CaCO}_{3}, 25.0 \mathrm{g} \mathrm{CaO}, \text { and } 5.66 \mathrm{g} \mathrm{CO}_{2}} \\ {\text { (a) } 30.5 \mathrm{g} \mathrm{CaCO}_{3}, 25.5 \mathrm{g} \mathrm{CaO}, \text { and } 6.48 \mathrm{g} \mathrm{CO}_{2}}\end{array} \end{equation}

Short answer

For the given mixtures, the amount of CaCO3 will change as follows: (a) decrease (b) increase (c) increase

Step-by-step solution

(a) Calculate moles of CO2 in the first mixture

Given, 4.25 g of CO2. To determine the number of moles, we will use the formula: moles = mass / molar mass moles = 4.25 g / 44.01 g/mol = 0.0966 mol

(b) Calculate moles of CO2 in the second mixture

Given, 5.66 g of CO2. To determine the number of moles, we will use the formula: moles = mass / molar mass moles = 5.66 g / 44.01 g/mol = 0.1286 mol

(c) Calculate moles of CO2 in the third mixture

Given, 6.48 g of CO2. To determine the number of moles, we will use the formula: moles = mass / molar mass moles = 6.48 g / 44.01 g/mol = 0.1472 mol ##Step 2: Compute the reaction quotient Qc for each mixture## Next, we will calculate the reaction quotient Qc for each mixture using the formula: Qc = [CO2]

(a) Calculate Qc for the first mixture

Qc = [0.0966 mol CO2 / 10 L] = 0.00966

(b) Calculate Qc for the second mixture

Qc = [0.1286 mol CO2 / 10 L] = 0.01286

(c) Calculate Qc for the third mixture

Qc = [0.1472 mol CO2 / 10 L] = 0.01472 ##Step 3: Compare Qc to Kc to determine the change in CaCO3## Finally, we will compare the Qc value of each mixture to the Kc value (0.0108) to determine if the CaCO3 will increase, decrease, or remain the same as the system approaches equilibrium:

(a) Determine the change in CaCO3 for the first mixture

Qc < Kc, therefore the reaction will proceed to the right to reach equilibrium. This means that the amount of CaCO3 will decrease.

(b) Determine the change in CaCO3 for the second mixture

Qc > Kc, therefore the reaction will proceed to the left to reach equilibrium. This means that the amount of CaCO3 will increase.

(c) Determine the change in CaCO3 for the third mixture

Qc > Kc, therefore the reaction will proceed to the left to reach equilibrium. This means that the amount of CaCO3 will increase.

Key concepts

Reaction Quotient

The reaction quotient, often represented as \( Q_c \) for reactions in terms of concentrations, is a numerical value that provides insight into the current state of a chemical reaction. It is calculated in a similar fashion to the equilibrium constant \( K_c \), but for a system that hasn't reached equilibrium.
By comparing \( Q_c \) to \( K_c \), which is the equilibrium constant, we can predict the direction the reaction will need to shift to reach equilibrium.

For our given example, the reaction describes the decomposition of calcium carbonate (\(\text{CaCO}_3\)) into calcium oxide (\(\text{CaO}\)) and carbon dioxide (\(\text{CO}_2\)). Here, since \( \text{CaCO}_3 \) and \( \text{CaO} \) are solids, they don't appear in the \( Q_c \) expression. Only gaseous or aqueous species are included.

• Calculation: For this reaction, the reaction quotient is \( Q_c = [\text{CO}_2] \).
• If \( Q_c < K_c \), the reaction will proceed in the forward direction to increase \([\text{CO}_2]\), decreasing \(\text{CaCO}_3\).
• If \( Q_c > K_c \), the reaction will shift in the reverse direction, increasing \(\text{CaCO}_3\).

Equilibrium Constant

The equilibrium constant, denoted as \( K_c \) for concentration, is a fundamental concept in chemical equilibrium. It quantifies the ratio of concentrations of products to reactants at equilibrium for a given reversible reaction at a specific temperature.
The value of \( K_c \) is unique for every reaction at a particular temperature and does not change regardless of the initial concentrations of reactants or products.
In our example reaction:\[ \text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g) \] \( K_c \) is given as 0.0108. This signifies the concentration of \(\text{CO}_2\) at equilibrium when \(\text{CaCO}_3\) decomposes.

• The equilibrium position tells us where the balance lies between reactants and products; whether reactants or products are favored in this balance.
• A \(K_c\) much less than 1 suggests that at equilibrium, the reactants are favored over products, and vice versa for a \(K_c\) larger than 1.

Thus, knowing \(K_c\) helps us understand the stability and progression of a reaction under equilibrium conditions.

Le Chatelier's Principle

Le Chatelier's Principle is a foundational principle in chemistry that predicts how a system at equilibrium will respond to external changes. When an equilibrium system experiences a disturbance like changes in concentration, pressure, or temperature, the system will adjust itself to partially counteract the effect of the disturbance.
This principle can be applied to our reaction with the equilibrium constant \( K_c = 0.0108 \) at \( 900^{\circ} \text{C} \).

• If we add more \(\text{CO}_2\), it will increase \( Q_c \) temporarily beyond \( K_c \), pushing the reaction toward the reverse direction to decrease the \(\text{CO}_2\) concentration. Consequently, more \(\text{CaCO}_3\) will be formed.
• Conversely, if \(\text{CO}_2\) is removed, \( Q_c \) falls below \( K_c \), prompting the reaction to shift forward to produce \(\text{CO}_2\), thus decreasing \(\text{CaCO}_3\).

Le Chatelier provides a qualitative method for predicting the effect of changing conditions on equilibrium, offering an intuitive approach for evaluating how equilibrium can be altered by outside forces.