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Part 1: You run the chemical reaction \(\mathrm{C}(a q)+\mathrm{D}(a q) \rightleftharpoons\) \(2 \mathrm{E}(a q)\) at \(25^{\circ} \mathrm{C}\). The equilibrium constant \(K_{c}\) for the reaction at this temperature is \(2.0 .\) a. Write the equilibrium-constant expression for the reaction. b. Can you come up with some possible concentrations of \(\mathrm{C}\), \(\mathrm{D}\), and \(\mathrm{E}\) that you might observe when the reaction has reached equilibrium at \(25^{\circ} \mathrm{C} ?\) What are these values? c. A student says that only a very limited number of concentrations for \(\mathrm{C}, \mathrm{D}\), and \(\mathrm{E}\) are possible at equilibrium. Is this true? State why you think this is true or is not true. d. If you start with \(1.0 M\) concentrations of both \(\mathrm{C}\) and \(\mathrm{D}\) and allow the reaction to come to equilibrium, would you expect the concentration of \(\mathrm{C}\) to have decreased to zero? If not, what would you expect for the concentration of \(\mathrm{C} ?\) (An approximate value is fine.)

Step by step solution

01

Writing the Equilibrium Expression

For the reaction \( \mathrm{C}(aq) + \mathrm{D}(aq) \rightleftharpoons 2 \mathrm{E}(aq) \), the equilibrium constant \( K_c \) is expressed in terms of the concentrations of the products and reactants. The formula is: \[ K_c = \frac{[\mathrm{E}]^2}{[\mathrm{C}][\mathrm{D}]} \] where \([\mathrm{E}]\), \([\mathrm{C}]\), and \([\mathrm{D}]\) are the molar concentrations of \(\mathrm{E}\), \(\mathrm{C}\), and \(\mathrm{D}\) at equilibrium.

02

Determining Possible Equilibrium Concentrations

Since \( K_c = 2.0 \), we want values such that \( \frac{[\mathrm{E}]^2}{[\mathrm{C}][\mathrm{D}]} = 2.0 \). If we assume \([\mathrm{C}] = 0.5\, M\) and \([\mathrm{D}] = 0.5\, M\), then \([\mathrm{E}]\) can be found using: \[ 2.0 = \frac{[\mathrm{E}]^2}{0.5 \times 0.5} \] \[ 2.0 = \frac{[\mathrm{E}]^2}{0.25} \] \[ [\mathrm{E}]^2 = 0.5 \] \[ [\mathrm{E}] = \sqrt{0.5} \approx 0.71\, M \].

03

Assessing the Variety of Equilibrium Concentrations

Numerous combinations of concentrations for \(\mathrm{C}\), \(\mathrm{D}\), and \(\mathrm{E}\) can satisfy the equation \( \frac{[\mathrm{E}]^2}{[\mathrm{C}][\mathrm{D}]} = 2.0 \). Therefore, it is not true that only a very limited number of concentrations can exist. Variability in initial conditions or reaction pathways can lead to different concentrations at equilibrium.

04

Evaluating Concentration Changes from Initial Conditions

Starting with \([\mathrm{C}] = 1.0\, M\) and \([\mathrm{D}] = 1.0\, M\), we posit some \(x\) moles of \(\mathrm{C}\) react to reach equilibrium, making \([\mathrm{E}] = 2x\). At equilibrium, \([\mathrm{C}] = 1-x\) and \([\mathrm{D}] = 1-x\). Using the equilibrium constant expression: \[ 2.0 = \frac{(2x)^2}{(1-x)(1-x)} \] Solve for \(x\) to find that \([\mathrm{C}]\) will not decrease to zero, and calculating this evenly gives an approximate change. After solving, \([\mathrm{C}] \approx 0.5\, M \) at equilibrium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilibrium Constant Expression

In chemistry, the Equilibrium Constant Expression (Kc) is crucial for describing the state of a reversible reaction at equilibrium. For the given reaction, \( \mathrm{C}(aq) + \mathrm{D}(aq) \rightleftharpoons 2 \mathrm{E}(aq) \), the equilibrium constant expression is essential to determine the relationship between the concentrations of reactants and products. The formula for this specific reaction is:

• \( K_c = \frac{[\mathrm{E}]^2}{[\mathrm{C}][\mathrm{D}]} \)

Here, [E], [C], and [D] represent the molar concentrations of substances E, C, and D at equilibrium. The numerical value of \( K_c \) gives insight into the position of equilibrium. If \( K_c \) is greater than 1, the reaction favors product formation. On the other hand, if \( K_c \) is less than 1, the reactants are favored when equilibrium is established.
This expression allows us to predict the concentration of one participant if the others are known, helping chemists understand how equilibria shift under different conditions.

Reaction Quotient

The Reaction Quotient, represented as \( Q_c \), is calculated similarly to the equilibrium constant but applies to reactions not yet at equilibrium. It helps to predict which direction a reaction will shift to reach equilibrium. For the given reaction, the reaction quotient would be:

• \( Q_c = \frac{[\mathrm{E}]^2}{[\mathrm{C}][\mathrm{D}]} \)

By comparing \( Q_c \) with \( K_c \):

• If \( Q_c < K_c \), the forward reaction is favored, producing more products to achieve equilibrium.
• If \( Q_c > K_c \), the reverse reaction is favored, converting products back into reactants.
• If \( Q_c = K_c \), the system is already at equilibrium.

Understanding \( Q_c \) is fundamental for predicting how a reaction will behave immediately after it has started or if changes are made to the concentration of reactants or products.

Le Chatelier's Principle

Le Chatelier's Principle is a key concept in predicting how a change in conditions affects the position of equilibrium in a chemical reaction. According to this principle, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will shift to counteract the effect of the change.

• Concentration: Adding more of a reactant or product causes the system to shift towards the side that will consume the added substance. If a substance is removed, the system shifts to produce more of what was removed.
• Temperature: Increasing the temperature of an endothermic reaction shifts equilibrium to the right, favoring the formation of products, whereas for an exothermic reaction, it favors reactants. Decreasing the temperature has the opposite effect.
• Pressure: Alterations in pressure (considering gaseous reactions) affect equilibrium based on the number of moles of gas on each side. Increasing pressure shifts equilibrium to the side with fewer gas molecules, while decreasing pressure does the opposite.

Le Chatelier's Principle is a powerful tool in chemical engineering and laboratory settings for controlling reactions to optimize the yield and ensure safety by understanding the effects of manipulating external conditions.