Question

Consider the reaction \(\mathrm{A}(g)+2 \mathrm{B}(g) \rightleftharpoons \mathrm{C}(g)+\mathrm{D}(g)\) in a 1.0-L rigid flask. Answer the following questions for each situation (a-d): i. Estimate a range (as small as possible) for the requested substance. For example, [A] could be between \(95 M\) and \(100 M\),ii. Explain how you decided on the limits for the estimated range. iii. Indicate what other information would enable you to narrow your estimated range. iv. Compare the estimated concentrations for a through d, and explain any differences. a. If at equilibrium \([\mathrm{A}]=1 M,\) and then 1 mole of \(\mathrm{C}\) is added, estimate the value for [A] once equilibrium is reestablished. b. If at equilibrium \([\mathrm{B}]=1 M,\) and then 1 mole of \(\mathrm{C}\) is added, estimate the value for \([\mathbf{B}]\) once equilibrium is reestablished. c. If at equilibrium \([\mathrm{C}]=1 M,\) and then 1 mole of \(\mathrm{C}\) is added, estimate the value for \([\mathrm{C}]\) once equilibrium is reestablished. d. If at equilibrium \([\mathrm{D}]=1 M,\) and then 1 mole of \(\mathrm{C}\) is added, estimate the value for [D] once equilibrium is reestablished.

Short answer

Estimating the concentrations of A, B, C, and D when 1 mole of C is added to the system: a. [A]: Between 1M and 2M. Adding 1 mole of C pushes the reaction to the left, increasing [A]. b. [B]: Between 1M and 3M. Adding 1 mole of C pushes the reaction to the left, increasing [B]. c. [C]: Between 1M and 2M. After adding 1 mole of C, [C] decreases due to the reaction shifting towards reactants. d. [D]: Between 0M and 1M. Adding 1 mole of C pushes the reaction to the left, decreasing [D]. To narrow the estimated ranges, we need the initial concentrations of all substances, or the equilibrium constant (Kc).

Step-by-step solution

Problem (a)

Calculate the new equilibrium concentration of A after adding 1 mole of C. We are given that at equilibrium [\(\text{A}\)] = 1M. If we add 1 mole of C to the flask, the reaction will be pushed to the left, according to Le Chatelier's principle, resulting in increased [\(\text{A}\)] and decreased [\(\text{C}\)]. #tag_titlea# Step 1: Establish a range for the new [A] #tag_contenta# Since [\(\text{A}\)] will increase, a reasonable range for [\(\text{A}\)] could be between 1M and 2M, as it cannot exceed the initial concentration of 2M. #tag_titlea# Step 2: Explain the limits #tag_contenta# Here, the lower limit is the initial equilibrium concentration (1M). The upper limit of 2M is determined because [\(\text{A}\)] cannot go higher than the initial total concentration of the system. #tag_titlea# Step 3: Additional information for narrowing the range #tag_contenta# To narrow down our estimated range, we would need the concentrations of B, C, and D at the equilibrium or the equilibrium constant for the reaction (Kc).

Problem (b)

Calculate the new equilibrium concentration of B after adding 1 mole of C. We are given that at equilibrium [\(\text{B}\)] = 1M. Adding 1 mole of C will also cause the reaction to shift to the left, increasing [\(\text{B}\)] and decreasing [\(\text{C}\)]. #tag_titleb# Step 1: Establish a range for the new [B] #tag_contentb# Since [\(\text{B}\)] will increase, a reasonable range for [\(\text{B}\)] could be between 1M and 3M, as it cannot exceed double the initial concentration (since the reaction consumes two moles of B for each mole of C). #tag_titleb# Step 2: Explain the limits #tag_contentb# The lower limit is the initial equilibrium concentration (1M). The upper limit of 3M is determined because [\(\text{B}\)] cannot go higher than double the initial total concentration. #tag_titleb# Step 3: Additional information for narrowing the range #tag_contentb# To narrow down our estimated range, we would need the concentrations of A, C, and D at the equilibrium or the equilibrium constant for the reaction (Kc).

Problem (c)

Calculate the new equilibrium concentration of C after adding 1 mole of C. We are given that at equilibrium [\(\text{C}\)] = 1M. Adding 1 mole of C will cause the reaction to shift to the left, decreasing [\(\text{C}\)]. #tag_titlec# Step 1: Establish a range for the new [C] #tag_contentc# Since [\(\text{C}\)] will decrease after adding 1 mole, a reasonable range for [\(\text{C}\)] could be between 1M and 2M. #tag_titlec# Step 2: Explain the limits #tag_contentc# The lower limit is the initial equilibrium concentration (1M). The upper limit of 2M is determined because [\(\text{C}\)] cannot go higher than the initial total concentration + added moles. #tag_titlec# Step 3: Additional information for narrowing the range #tag_contentc# To narrow down our estimated range, we would need the concentrations of A, B, and D at the equilibrium or the equilibrium constant for the reaction (Kc).

Problem (d)

Calculate the new equilibrium concentration of D after adding 1 mole of C. We are given that at equilibrium [\(\text{D}\)] = 1M. Adding 1 mole of C will also cause the reaction to shift to the left, increasing [\(\text{D}\)]. #tag_titled# Step 1: Establish a range for the new [D] #tag_contentd# Since [\(\text{D}\)] will decrease, a reasonable range for [\(\text{D}\)] could be between 0M and 1M, as it cannot go lower than 0M nor higher than the initial equilibrium concentration. #tag_titled# Step 2: Explain the limits #tag_contentd# The lower limit is the lowest possible value (0M). The upper limit of 1M is determined because [\(\text{D}\)] cannot go higher than the initial equilibrium concentration. #tag_titled# Step 3: Additional information for narrowing the range #tag_contentd# To narrow down our estimated range, we would need the concentrations of A, B, and C at the equilibrium or the equilibrium constant for the reaction (Kc).

Comparing the estimated concentrations (a-d)

For problems a and b, the estimated concentrations increase with the addition of C, as expected per Le Chatelier's principle. In problem c, [\(\text{C}\)] increases since we directly added 1 mole of C. In problem d, [\(\text{D}\)] decreases due to the shift of the reaction towards reactants. The differences in the ranges are due to the stoichiometry of the reaction, affecting the response of each species to changes in concentration.

Key concepts

Understanding Le Chatelier's Principle

Le Chatelier's principle is a central concept in chemical equilibrium which provides insight into how a system at equilibrium responds to changes in concentration, temperature, or pressure. When a system in equilibrium is disturbed, for instance by adding or removing a reactant or product, the system will adjust in such a way as to counteract the disturbance and re-establish equilibrium.

For example, in the textbook exercise, when 1 mole of substance C is added to the equilibrium mixture, Le Chatelier's principle predicts that the system will favor the reverse reaction, where C is consumed along with product D, to produce more reactants A and B. This principle is crucial for predicting the direction in which the equilibrium will shift and for estimating the new concentrations of the components after the adjustment. Understanding this principle can help students to make educated guesses about the concentration ranges and the reasoning behind these ranges.

In practice, applying Le Chatelier's principle requires a detailed knowledge of the reaction conditions and the nature of the disturbance. It is one of the most powerful tools in chemistry for anticipating the outcome of changes in a reaction at equilibrium.

Equilibrium Constant (Kc) Explained

The equilibrium constant (Kc) quantifies the position of equilibrium for a chemical reaction. It’s a value that expresses the ratio of the concentrations of products to reactants, each raised to the power of their coefficients in the balanced equation, at equilibrium. The value of Kc is constant at a given temperature.

In the context of the textbook exercise, knowledge of the equilibrium constant for the reaction would allow a more accurate determination of concentration ranges after a perturbation. For instance, if 1 mole of C is added, the reaction will shift in response. With Kc, one can calculate the exact new equilibrium concentrations of A, B, C, and D by setting up an 'ICE' table (Initial, Change, Equilibrium) that includes the initial concentrations, the changes that occur as the system shifts to a new equilibrium, and the final equilibrium concentrations modelled by the expression for Kc. This helps refine the estimates and avoids the broader ranges that were initially proposed.

In essence, knowing the value of Kc would narrow the estimated ranges significantly, as it determines the extent to which a reaction can shift before reaching a new equilibrium state.

Applying Reaction Stoichiometry at Equilibrium

Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It is foundational to understanding chemical reactions and applies to systems in equilibrium as well. Stoichiometry informs the proportions in which substances react and are produced, making it an essential tool in estimating how much of one reactant is needed to produce a certain amount of product.

Within the context of our textbook problem, when 1 mole of C is added to the equilibrium system, the stoichiometry of the balanced reaction indicates that for every mole of C produced, 1 mole of A and 2 moles of B are consumed. This relationship is what drives the assumption that the concentration of B could potentially rise to twice its initial value upon the shift in equilibrium – it is based on the stoichiometric coefficients in the reaction (A(g) + 2B(g) ⇌ C(g) + D(g)).

Understanding the stoichiometric relationships allows us to explain why the estimated concentrations differ among the reactants and products when a change is made. It serves as a guide for predicting the quantitative changes in a system when it is subjected to a disturbance, complementing the qualitative predictions made with Le Chatelier's principle and providing a footing for calculations involving the equilibrium constant, Kc.