Question

In a container with a volumetric capacity of \(1 \mathrm{~L}\), we leave \(0.05 \mathrm{~mol}\) of \(\mathrm{PCl}_{5}\) and \(5 \mathrm{~mol}\) of \(\mathrm{PCl}_{3}\) reacting at \(760 \mathrm{~K}\). At equilibrium, we have \(0.043 \mathrm{~mol}\) of \(\mathrm{Cl}_{2}\). Find \(K_{c}\) for the reaction \(\mathrm{PCl}_{5} \rightleftarrows \mathrm{PCl}_{3}+\mathrm{Cl}_{2}\).

Short answer

Question: Calculate the equilibrium constant Kc for the following reaction at equilibrium: PCl5 ⇄ PCl3 + Cl2. The initial moles of PCl5, PCl3, and Cl2 are 0.05 mol, 5 mol, and 0 mol, respectively. The moles of Cl2 at equilibrium are 0.043 mol, and the reaction takes place in a 1 L container. Answer: The equilibrium constant Kc for the given reaction is approximately 31.35.

Step-by-step solution

Calculate the change in moles for PCl5, PCl3, and Cl2

At the beginning of the reaction: Initial moles of PCl5 = 0.05 mol, Initial moles of PCl3 = 5 mol, and Initial moles of Cl2 = 0 mol. At equilibrium: Moles of Cl2 = 0.043 mol. Since 1 mol of PCl5 dissociates to form 1 mol of PCl3 and 1 mol of Cl2, the change in moles of PCl5 and PCl3 will be equal to the change in moles of Cl2. Change in moles of PCl5 = -0.043 mol, Change in moles of PCl3 = +0.043 mol. #Step 2: Calculate moles of all substances at equilibrium#

Calculate the equilibrium moles of PCl5, PCl3, and Cl2

At equilibrium: Moles of PCl5 = (Initial moles of PCl5) + (Change in moles of PCl5) = 0.05 - 0.043 = 0.007 mol, Moles of PCl3 = (Initial moles of PCl3) + (Change in moles of PCl3) = 5 + 0.043 = 5.043 mol, Moles of Cl2 = 0.043 mol. #Step 3: Calculate concentrations of all substances at equilibrium#

Calculate the equilibrium concentrations of PCl5, PCl3, and Cl2

We have the reaction happening in a container with a volumetric capacity of 1L. Therefore, we can convert moles to concentrations by dividing moles by volume. Concentration of PCl5 = (Moles of PCl5) / 1L = 0.007 M, Concentration of PCl3 = (Moles of PCl3) / 1L = 5.043 M, Concentration of Cl2 = (Moles of Cl2) / 1L = 0.043 M. #Step 4: Calculate the equilibrium constant Kc#

Use the equilibrium concentrations to find Kc

The given reaction is: \(\mathrm{PCl}_{5} \rightleftarrows \mathrm{PCl}_{3}+\mathrm{Cl}_{2}\) The expression for the equilibrium constant Kc for the above reaction is: \(K_c = \frac{[\mathrm{PCl}_{3}][\mathrm{Cl}_{2}]}{[\mathrm{PCl}_{5}]}\) Substituting the equilibrium concentrations: \(K_c = \frac{(5.043)(0.043)}{(0.007)}\) #Step 5: Calculate Kc#

Compute the value of Kc

Finally, solve for Kc: \(K_c = \frac{(5.043)(0.043)}{(0.007)} = 31.350\) The equilibrium constant \(K_c\) for the given reaction is approximately 31.35.

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is central to predicting the behavior of chemical reactions in closed systems. It refers to a state where the rate of the forward reaction, which produces products from reactants, is equal to the rate of the backward reaction, in which products revert to reactants. At this point, though individual molecules continue to react, there is no net change in the concentrations of the reactants and products, hence the system is said to be in a 'dynamic' equilibrium.

In the exercise, the equilibrium is observed in the reaction between PCl₅, PCl₃, and Cl₂. Once the reaction reaches equilibrium, the amount of each substance remains constant over time. The equilibrium constant, K_c, is a numerical value that helps us understand the proportions of products and reactants present at equilibrium. It is defined based on the balanced chemical equation for the reaction and is only affected by changes in temperature.

The concept of equilibrium is pivotal in fields ranging from industrial chemical processes to environmental science, where it can be used to predict the behavior of pollutants and the fate of chemical compounds in natural systems.

Chemical Reaction

A chemical reaction involves the transformation of one or more substances, the reactants, into new substances, the products. This process occurs as a result of the breaking and forming of chemical bonds. It can be represented by a chemical equation, which provides information about the reactants, the products, and their stoichiometric relationships.

In our example, the decomposition of PCl₅ into PCl₃ and Cl₂ illustrates a chemical reaction. It's essential to balance the chemical equation to observe the conservation of matter — that is, the number of atoms of each element on the reactant side equals those on the product side. For the reaction PCl₅ ⇌ PCl₃ + Cl₂, we imply that one mole of PCl₅ will produce one mole each of PCl₃ and Cl₂ at equilibrium, a stoichiometry that informs the calculation of the equilibrium constant.

Le Chatelier's Principle

Le Chatelier's principle states that a dynamic equilibrium, when subjected to a change in conditions, will adjust to partially oppose the change and restore a new equilibrium condition. This principle can be applied to changes in concentration, temperature, and pressure for reactions involving gases.

For example, if more PCl₅ were added to the system described in the exercise, the principle predicts that the equilibrium would shift to produce more PCl₃ and Cl₂ in an effort to counteract the added PCl₅. Similarly, if the temperature is raised, the system will favor the reaction that absorbs heat, according to whether the reaction is exothermic or endothermic. Le Chatelier's principle is a valuable tool in the chemical industry, where it's used to increase the yield of desired products.