Question

In the "Methode 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 after the fermentation process, assuming CO₂ is insoluble in H₂O, would be approximately 24.8 atm at 25°C.

Step-by-step solution

Calculate the volume of ethanol produced

Given that the grape juice is converted into 12% by volume of ethanol after fermentation, we can calculate the volume of ethanol produced as follows: Volume of ethanol = (12 / 100) * Volume of grape juice = (12 / 100) * 750 mL = 90 mL

Calculate the amount of glucose consumed and ethanol produced

Now, let's find the amount (in moles) of glucose consumed and ethanol produced using their densities and molar masses: Density of grape juice = 1.0 g/cm³ Density of ethanol = 0.79 g/cm³ Molar mass of glucose (C₆H₁₂O₆) = 180.16 g/mol Molar mass of ethanol (C₂H₅OH) = 46.07 g/mol First, we calculate the mass of glucose consumed: Mass of grape juice = Volume of grape juice * Density of grape juice = 750 mL * 1.0 g/cm³ (since 1 mL = 1 cm³) = 750 g The mass of glucose consumed = mass of grape juice = 750 g. Now, we utilize the stoichiometry of the balanced equation to find the moles of glucose consumed: Moles of glucose consumed = mass of glucose consumed / Molar mass of glucose = 750 g / 180.16 g/mol = 4.164 moles From the balanced equation, we know that 2 moles of ethanol are produced for each mole of glucose consumed. So, moles of ethanol produced: Moles of ethanol produced = 2 * Moles of glucose consumed = 2 * 4.164 moles = 8.328 moles

Calculate the amount of CO₂ produced

From the balanced equation, we know that 2 moles of CO₂ are produced for each mole of glucose consumed. So, moles of CO₂ produced: Moles of CO₂ produced = 2 * Moles of glucose consumed = 2 * 4.164 moles = 8.328 moles

Find the pressure of CO₂ inside the wine bottle

Now, we have the moles of CO₂, the temperature, and the volume of the bottle, so we can calculate the pressure using the Ideal Gas Law: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (0.08206 L atm/mol K), and T is the temperature in Kelvin. First, convert the temperature from Celsius to Kelvin: T(K) = 25 + 273.15 = 298.15 K The volume of the bottle (V) = 825 mL = 0.825 L Next, plug the values into the equation and solve for P: P = (nRT) / V P = (8.328 moles * 0.08206 L atm/mol K * 298.15 K) / 0.825 L P ≈ 24.8 atm Thus, the pressure of CO₂ inside the wine bottle would be approximately 24.8 atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and amount of gas through the equation \( PV = nRT \). Here:

• \( P \) represents the pressure of the gas.
• \( V \) is the volume it occupies.
• \( n \) denotes the number of moles of gas.
• \( R \) is the universal gas constant (approximately 0.08206 L atm/mol K).
• \( T \) is the temperature in Kelvin.

The Ideal Gas Law assumes that gases behave ideally, meaning that the particles do not interact except through elastic collisions. It's used to solve many problems related to gases, such as calculating the pressure of a gas in a container when its temperature and volume are known. In the fermentation process of grape juice into sparkling wine, this law helps us find out the pressure exerted by \( CO_2 \) in the bottle, which is crucial for understanding the carbonation process in making sparkling wine.

Stoichiometry

Stoichiometry is the area of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It uses the balanced chemical equation to determine how much of each substance is involved.
In the given fermentation reaction:\[ \text{C}_6\text{H}_{12}\text{O}_6 (aq) \rightarrow 2\text{C}_2\text{H}_5\text{OH} (aq) + 2\text{CO}_2 (g) \]

• For every mole of glucose (\( \text{C}_6\text{H}_{12}\text{O}_6 \)) consumed, two moles of ethanol (\( \text{C}_2\text{H}_5\text{OH} \)) and two moles of carbon dioxide (\( \text{CO}_2 \)) are produced.

This stoichiometric relationship allows us to track how the amount of each reactant and product changes during the reaction. By converting masses into moles using molar masses, we can predict product formation and optimize reactions, such as ensuring all reactants convert completely during wine fermentation.

Glucose to Ethanol Conversion

Glucose to ethanol conversion is a key biochemical process during the fermentation of grape juice into wine. In this reaction, glucose serves as the primary sugar fuel for yeast.
Yeast catalyzes the conversion of glucose to ethanol and carbon dioxide using an anaerobic pathway. The balanced chemical equation for this conversion is:\[ \text{C}_6\text{H}_{12}\text{O}_6 \rightarrow 2\text{C}_2\text{H}_5\text{OH} + 2\text{CO}_2 \]

• For every 1 mole of glucose consumed, 2 moles of ethanol are formed.
• The ethanol accounts for the alcoholic content of the wine, whereas \( CO_2 \) provides the effervescence, or bubbles, in sparkling wine.
• This conversion process is essential in engaging yeast metabolism effectively for beverage production.

Understanding this conversion is crucial for winemaking because the proportion of ethanol produced affects the beverage's flavor and alcohol content.

Pressure Calculation in Gases

Pressure calculation in gases involves determining the force exerted by gas molecules against the walls of their container. The Ideal Gas Law is often applied, as it ties together the moles of gas, the temperature, the volume, and the resulting pressure.
In a practical scenario like the fermentation of grape juice, after converting masses into moles through stoichiometry, this law aids in calculating the pressure inside a sealed environment, such as a wine bottle.

• Start by converting any temperatures to Kelvin, as the Ideal Gas Law requires it.
• Substitute all known values into the equation \( PV = nRT \) to solve for \( P \) (pressure).
• Here, the pressure of \( CO_2 \) was found using the volume of the bottle (0.825L), the temperature (298.15 K), and the number of moles of \( CO_2 \).
• This type of calculation helps ensure the safety and quality of carbonated beverages, where excessive pressure could lead to bursting or undesirably high carbonation levels.