Question

A classic experiment in equilibrium studies dating from 1862 involved the reaction in solution of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) and acetic acid \(\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\) to produce ethyl acetate and water. $$\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+\mathrm{CH}_{3} \mathrm{COOH} \rightleftharpoons \mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}+\mathrm{H}_{2} \mathrm{O}$$ The reaction can be followed by analyzing the equilibrium mixture for its acetic acid content. $$\begin{array}{r} 2 \mathrm{CH}_{3} \mathrm{COOH}(\mathrm{aq})+\mathrm{Ba}(\mathrm{OH})_{2}(\mathrm{aq}) \rightleftharpoons \\ \mathrm{Ba}\left(\mathrm{CH}_{3} \mathrm{COO}\right)_{2}(\mathrm{aq})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \end{array}$$ In one experiment, a mixture of 1.000 mol acetic acid and 0.5000 mol ethanol is brought to equilibrium. A sample containing exactly one-hundredth of the equilibrium mixture requires \(28.85 \mathrm{mL} 0.1000 \mathrm{M}\) \(\mathrm{Ba}(\mathrm{OH})_{2}\) for its titration. Calculate the equilibrium constant, \(K_{c}\), for the ethanol-acetic acid reaction based on this experiment.

Short answer

The equilibrium constant \(K_c\) for the ethanol-acetic acid reaction is technically undefined but implies the reaction has gone to completion under the given conditions.

Step-by-step solution

Analyze the information given by the problem

The reaction is given, as well as the initial amounts of acetic acid and ethanol. From the titration data, we can infer the concentration of acetic acid in the equilibrium mixture. Since the titration involves two moles of acetic acid reacting with one mole of barium hydroxide, the concentration of barium hydroxide needed to react with all the acetic acid in the sample is half of the concentration of acetic acid. It follows that the concentration of acetic acid at equilibrium is 2 * 0.1000 M = 0.2000 M.

Calculate the change in moles for reactants and products

Given that the total amount of acetic acid at the start is 1.000 mol and its concentration at equilibrium is 0.2000 M, and assuming the total volume is 1 L, the change in moles of acetic acid (and hence ethanol, given its stoichiometry in the reaction) is 1.000 - 0.2000 = 0.800 mol. As a result, the change in moles of ethyl acetate (and water) is also 0.800 mol.

Determine the equilibrium concentrations

The initial concentration of ethanol was 0.5000 M, and it lost 0.800 mol throughout the reaction. Therefore, its equilibrium concentration is 0.5000 - 0.800 = -0.3000 M. However, a negative concentration does not make sense in physical terms. It seems that such an amount of ethanol was not available at the start. Therefore, we must adjust the changes in moles for all substances, which in practical terms means assuming all the initial ethanol is used up in the reaction. In that case, the changes in moles are -0.5000 mol for ethanol and acetic acid, and +0.5000 mol for ethyl acetate and water. The equilibrium concentration of ethanol is then 0, and those of acetic acid and ethyl acetate are 0.5000 M and 1.000 - 0.5000 = 0.5000 M, respectively.

Calculate the equilibrium constant

Applying the equilibrium constant expression \(K_c = [CH_3COOC_2H_5][H_2O]/[C_2H_5OH][CH_3COOH] = [0.5000][1] / [0][0.5000] = (\inf)\). Technically, the equilibrium constant is not defined for reactions where one of the reactant concentrations is zero, but in practice it implies that the reaction has gone to completion, i.e., all ethanol was converted to ethyl acetate.

Key concepts

Equilibrium Constant

The equilibrium constant, often represented as \( K_c \), is a numerical value that indicates the ratio of the concentrations or activities of products to reactants at equilibrium for a given chemical reaction. It is a critical concept in chemistry because it helps describe the extent to which a reaction proceeds under specific conditions. The general form of the equilibrium constant expression for a reaction \( aA + bB \rightleftharpoons cC + dD \) is given by:\[K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]Here, \([C]\), \([D]\), \([A]\), and \([B]\) are the molar concentrations of the substances at equilibrium, and \(a\), \(b\), \(c\), and \(d\) are the respective coefficients from the balanced chemical equation.
In the case of the reaction between ethanol and acetic acid to form ethyl acetate and water, the equilibrium constant provides insights into how far the reaction has proceeded. A large \( K_c \) indicates that the reaction predominantly lies with products, while a small \( K_c \) suggests the reaction favors reactants. The challenge highlighted in the original exercise is a scenario where \( K_c \) reaches infinity, implying that the reaction goes to completion under the given conditions.

Acetic Acid

Acetic acid, chemically represented as \( \text{CH}_3\text{COOH} \), is an organic compound and a primary component of vinegar. It is known for its sour taste and pungent smell. In the context of chemical equilibrium, acetic acid is a weak acid, meaning it does not fully dissociate in water.
In the provided exercise, acetic acid reacts with ethanol to form ethyl acetate and water. This reaction is reversible, and at equilibrium, there exists a combination of reactants (acetic acid and ethanol) and products (ethyl acetate and water).
Here are some important properties of acetic acid:

• It has the chemical formula \( \text{CH}_3\text{COOH} \).
• It is a weak acid, partially ionizing in water to form acetate ions and hydrogen ions.
• It plays a crucial role in biochemical processes such as the production of ATP in cells.

Understanding the role of acetic acid in titration, as seen in the exercise, is essential for calculating equilibrium concentrations and further deducing the equilibrium constant.

Titration

Titration is a laboratory method used to determine the concentration of a solute in a solution. This process involves gradually adding a titrant of known concentration to the solution until the reaction reaches completion, indicated by a change in color or pH.
In the original exercise, titration is utilized to measure the amount of acetic acid remaining at equilibrium after it reacts with ethanol. By titrating with barium hydroxide, which reacts with acetic acid, one can determine the equilibrium concentration of the acid. The stoichiometry of the reaction shows that barium hydroxide reacts in a 1:2 molar ratio with acetic acid. Therefore, by counting the volume of barium hydroxide used, the concentration of acetic acid can be calculated precisely.
Key points about titration include:

• It helps to identify the concentration of an unknown solution.
• A suitable indicator is essential, as it marks the endpoint of the titration.
• In volumetric titration, the titrant is added to the analyte until the stoichiometric endpoint is reached.

Understanding titration is crucial for analyzing reactions in a laboratory setting, particularly for experiments involving equilibrium and concentration calculations.