Question

In the "Méthode Champenoise," grape juice is fermented in a wine bottle to produce sparkling wine. The reaction is $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+2 \mathrm{CO}_{2}(g)$$ Fermentation of \(750 .\) mL grape juice (density \(=1.0 \mathrm{g} / \mathrm{cm}^{3}\) ) is allowed to take place in a bottle with a total volume of \(825 \mathrm{mL}\) until \(12 \%\) by volume is ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right) .\) Assuming that the \(\mathrm{CO}_{2}\) is insoluble in \(\mathrm{H}_{2} \mathrm{O}\) (actually, a wrong assumption), what would be the pressure of \(\mathrm{CO}_{2}\) inside the wine bottle at \(25^{\circ} \mathrm{C} ?\) (The density of ethanol is \(0.79 \mathrm{g} / \mathrm{cm}^{3} .\) )

Short answer

The pressure of CO2 inside the wine bottle would be approximately 61.99 atm.

Step-by-step solution

Calculate the mass of grape juice

First, we need to find the mass of grape juice. We know that the volume is 750mL and we can convert it to cubic centimeters using the equivalence \(1mL = 1cm^3\). Then, we can multiply the volume by density to get the mass: Mass = (Volume) x (Density) = (750cm^3)(1.0 g/cm^3) = 750 g

Calculate the amount of ethanol produced

We are given that 12% by volume of the final product is ethanol. We can use the total volume to determine the volume of ethanol produced. Volume_ethanol = (12% x Total Volume) = (0.12)(825cm^3) = 99 cm^3

Calculate the mass of ethanol produced

Next, convert the volume of ethanol to mass using the given density. Mass_ethanol = (Volume_ethanol) x (Density_ethanol) = (99cm^3)(0.79 g/cm^3) = 78.21 g

Calculate the moles of ethanol produced

Now we need to find the moles of ethanol produced using its molar mass (46.07 g/mol). Moles_ethanol = (Mass_ethanol) / (Molar_Mass_ethanol) = (78.21 g) / (46.07 g/mol) = 1.696 mol

Use stoichiometry to find the moles of CO2 produced

We know that for every 2 moles of ethanol produced, 2 moles of CO2 are produced (from the balanced chemical equation). So, the moles of CO2 produced are equal to the moles of ethanol produced. Moles_CO2 = Moles_ethanol = 1.696 mol

Calculate the volume of CO2 produced

We need to find the volume of CO2 in the headspace of the wine bottle. The headspace is the remaining volume not occupied by the liquid. Volume_headspace = Total_Volume - Volume_grape_juice = 825 cm^3 - 750 cm^3 = 75 cm^3 We can convert this volume to liters using the equivalence \(1 cm^3 = 0.001 L\) Volume_headspace (L) = 75 cm^3 x 0.001 L/cm^3 = 0.075 L

Calculate the pressure of CO2 using the Ideal Gas Law

Finally, we can use the Ideal Gas Law, PV = nRT, to compute the pressure of CO2 produced: Pressure_CO2 = (n_CO2 R T) / V_headspace where n_CO2 = 1.696 mol, R = 0.0821 L*atm/mol*K, T = 298K (25°C + 273), and V_headspace = 0.075 L: Pressure_CO2 = ((1.696 mol)(0.0821 atm L/mol K)(298 K)) / (0.075 L) = 61.99 atm The pressure of CO2 inside the wine bottle would be approximately 61.99 atm.

Key concepts

Understanding Méthode Champenoise

The Méthode Champenoise, also known as the traditional method, is a prestigious technique used to produce sparkling wines, most notably Champagne. The process begins with the primary fermentation of grape juice, which later undergoes a second fermentation inside the bottle. This second fermentation is crucial as it produces both the carbon dioxide responsible for the effervescence and additional flavor complexities.

During the secondary fermentation, a mixture of yeast and sugars, known as the 'liqueur de tirage', is added to the still wine. The bottle is then sealed with a cap, and the yeast begins converting the added sugars into alcohol and carbon dioxide. Since the bottle is sealed, the carbon dioxide dissolves in the wine, creating the bubbles that define sparkling wine. After the yeast has consumed all the sugars, it dies and forms a sediment, or lees, which is eventually removed through a process called 'disgorgement', leaving behind clear, bubbly wine. This artisanal technique is time-consuming and adds to the final product's quality and character.

Chemical Equation for Fermentation in Sparkling Wine Production

The chemical equation for fermentation is pivotal when understanding the production of alcoholic beverages, especially in the context of creating sparkling wines using the Méthode Champenoise. The simplified equation for the fermentation process is: \[\text{C}_6\text{H}_{12}\text{O}_6(aq) \longrightarrow 2\text{C}_2\text{H}_5\text{OH}(aq) + 2\text{CO}_2(g)\]

This equation represents the conversion of glucose (\(\text{C}_6\text{H}_{12}\text{O}_6\)), which is a type of sugar found in grape juice, into ethanol (\(\text{C}_2\text{H}_5\text{OH}\)) and carbon dioxide (\(\text{CO}_2\)) in the absence of oxygen. The process is catalyzed by yeast under anaerobic conditions. It's crucial to remember that each molecule of glucose yields two molecules of ethanol and two molecules of carbon dioxide, illustrating the one-to-one stoichiometry between the ethanol and carbon dioxide produced during fermentation. This relationship is key when calculating the volume and pressure of carbon dioxide in sparkling wine production.

Ideal Gas Law Application in Bottle Fermentation

The Ideal Gas Law is a fundamental equation in chemistry and plays a role in calculating the pressure of carbon dioxide in the wine bottle during Méthode Champenoise. The Ideal Gas Law is stated as \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin.

When applying this law to calculate the pressure of \(\text{CO}_2\) inside a sealed wine bottle, it's essential to first know the number of moles of \(\text{CO}_2\) produced during fermentation. Then, with the bottle's temperature given, one can solve for the pressure P inside the bottle. This calculation assumes that the \(\text{CO}_2\) behaves like an ideal gas and the wine bottle acts as a closed system. In the case of the exercise, the application of the Ideal Gas Law provides an estimation of the intense pressure (approximately 61.99 atm) created by the \(\text{CO}_2\) gas as a result of the fermentation process within the limited space of the bottle's headspace.

Stoichiometry in Fermentation

Stoichiometry is the branch of chemistry dealing with the quantitative relationships between reactants and products in a chemical reaction. During fermentation in winemaking, particularly in the Méthode Champenoise, stoichiometry helps predict the amount of ethanol and carbon dioxide produced from a given quantity of sugar.

The balanced chemical equation showing glucose turning into ethanol and carbon dioxide serves as the basis for these stoichiometric calculations. As mentioned earlier, the conversion ratio is 1:2:2, meaning one glucose molecule yields two ethanol molecules and two carbon dioxide molecules. By using the calculated mass of ethanol and its corresponding molar mass, we can determine the moles of ethanol produced. Then, since the ratio is one-to-one, the moles of ethanol directly give us the moles of \(\text{CO}_2\) produced, which can then be used in conjunction with the Ideal Gas Law to find the pressure exerted by the \(\text{CO}_2\) inside the wine bottle. Such quantitative analysis is crucial for maintaining the right levels of carbonation and ensuring the quality and consistency of the sparkling wine product.