Problem 39 - Chemistry The Molecular Science

Question

Carbon tetrachloride can be produced by this reaction: $$ \mathrm{CS}_{2}(\mathrm{~g})+3 \mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{S}_{2} \mathrm{Cl}_{2}(\mathrm{~g})+\mathrm{CCl}_{4}(\mathrm{~g}) $$ Suppose \(1.2 \mathrm{~mol} \mathrm{CS}_{2}\) and \(3.6 \mathrm{~mol} \mathrm{Cl}_{2}\) are placed in a 1.00-L flask and the flask is sealed. After equilibrium has been achieved, the mixture contains \(0.90 \mathrm{~mol} \mathrm{CCl}_{4} \cdot\) Calculate \(K_{\mathrm{c}}\).

Step-by-Step Solution

Answer

The equilibrium constant \(K_c\) is approximately 3.70.

Step 1: Write the equilibrium expression

First, we need to write the equilibrium expression for the given reaction based on the concentrations of the products and reactants. The expression for the equilibrium constant \(K_c\) is given by: \[ K_c = \frac{[\text{S}_2\text{Cl}_2][\text{CCl}_4]}{[\text{CS}_2][\text{Cl}_2]^3} \] where the brackets denote the equilibrium concentrations of each component.

Step 2: Establish initial concentrations

Since the volume of the flask is 1.00 L, the initial concentrations of \(\text{CS}_2\) and \(\text{Cl}_2\) are the same as their initial moles. Therefore, the initial concentrations are: \([\text{CS}_2]_0 = 1.2 \, \text{M}\), \([\text{Cl}_2]_0 = 3.6 \, \text{M}\), and \([\text{CCl}_4]_0 = 0 \, \text{M}\).

Step 3: Use stoichiometry to find change in concentration

At equilibrium, \([\text{CCl}_4]\) is given as 0.90 M. Because the stoichiometric ratio between \(\text{CS}_2\) and \(\text{CCl}_4\) is 1:1, the change in the concentration of \(\text{CS}_2\) to form \(\text{CCl}_4\) is also 0.90 M. Thus, \([\text{CS}_2]_e = 1.2 - 0.90 = 0.30 \, \text{M}\).

Step 4: Calculate equilibrium concentration for Cl2

Using the stoichiometric ratio of \(3:1\) for \(\text{Cl}_2\) to \(\text{CCl}_4\), the change in \(\text{Cl}_2\) concentration is 3 times that of \(\text{CCl}_4\). Therefore, \([\text{Cl}_2]_e = 3.6 - 3(0.90) = 3.6 - 2.7 = 0.90 \, \text{M}\).

Step 5: Calculate equilibrium concentration for S2Cl2

Since \(\text{S}_2\text{Cl}_2\) and \(\text{CCl}_4\) are produced in a \(1:1\) ratio, \([\text{S}_2\text{Cl}_2]_e = 0.90 \, \text{M}\).

Step 6: Substitute equilibrium concentrations into Kc expression

Now, substitute the equilibrium concentrations into the \(K_c\) expression: \[ K_c = \frac{(0.90)(0.90)}{(0.30)(0.90)^3} \].

Step 7: Compute Kc

Calculate \(K_c\): \[ K_c = \frac{0.81}{0.30 \times 0.729} = \frac{0.81}{0.2187} \approx 3.70 \].

Key Concepts

Equilibrium Constant

In chemical reactions, the equilibrium constant, denoted as \(K_c\), is a crucial factor in understanding how a reaction behaves in a closed system. The equilibrium constant provides a ratio of the concentrations of products to reactants when the reaction has reached equilibrium.

This condition indicates that the rates of the forward and reverse reactions are equal, thus the concentrations remain constant over time. For the reaction involving carbon disulfide and chlorine to form carbon tetrachloride and sulfur dichloride, the \(K_c\) expression is formulated from the balanced chemical equation:

• The numerator comprises the concentrations of the products, \([\text{S}_2\text{Cl}_2]\) and \([\text{CCl}_4]\).
• The denominator includes the concentrations of the reactants, \([\text{CS}_2]\) and the cube of \([\text{Cl}_2]\).

The expression becomes:\[K_c = \frac{[\text{S}_2\text{Cl}_2][\text{CCl}_4]}{[\text{CS}_2][\text{Cl}_2]^3}\]By plugging in the equilibrium concentrations, one can solve for \(K_c\), providing insight into the reaction's dynamics and whether products or reactants are favored at equilibrium.

Stoichiometry

Stoichiometry is the science of measuring the quantitative relationships in chemical reactions. It is vital for calculating how much of a reactant is needed or how much product will be formed.

In the production of carbon tetrachloride, understanding stoichiometry helps determine the changes in concentration of reactants and products as the reaction approaches equilibrium. The stoichiometric coefficients, derived from the balanced chemical equation, dictate how reactant and product concentrations change.

• For this reaction, when 1.2 moles of \(\text{CS}_2\) react with chlorine, and 0.90 moles of \(\text{CCl}_4\) are produced, an equal amount of 0.90 moles of \(\text{S}_2\text{Cl}_2\) is formed, reflecting a \(1:1\) stoichiometry with \(\text{CCl}_4\).
• The change in chlorine concentration is three times more than \(\text{CCl}_4\) due to the \(3:1\) stoichiometric ratio, resulting in a reduction of 2.7 moles for chlorine (3 x 0.90).

These calculations highlight stoichiometry as a tool for predicting and balancing chemical equations, ensuring that the conservation of mass principle is observed in reactions.

Carbon Tetrachloride Production

Carbon tetrachloride (\(\text{CCl}_4\)) is a compound used in various industrial applications. It can be synthesized through the reaction of carbon disulfide (\(\text{CS}_2\)) with chlorine gas. This process involves a chemical reaction that reaches an equilibrium state where the amounts of reactants and products remain constant.

During this chemical reaction, the reactants \(\text{CS}_2\) and \(\text{Cl}_2\) transform into the products of \(\text{CCl}_4\) and sulfur dichloride \(\text{S}_2\text{Cl}_2\), facilitated by the aforementioned stoichiometric relationships. At equilibrium, it is essential to determine the concentration of the products, as it reflects the completeness of the carbon tetrachloride production.

• The initial moles of reactants, such as the 1.2 moles of \(\text{CS}_2\) and 3.6 moles of \(\text{Cl}_2\), set the stage for how much \(\text{CCl}_4\) will be produced based on the reaction's equilibrium dynamics.
• Upon achieving equilibrium, 0.90 moles of \(\text{CCl}_4\) signifies successful product formation, while the remaining reactant concentrations can offer insights into the reaction's efficiency.

Understanding the process of carbon tetrachloride production involves balancing the chemical equations, applying stoichiometry, and calculating equilibrium constants to obtain an optimal yield.