In the "Méthode Champenoise," grape juice is fermented in a wine bottle to produce sparkling wine. The reaction is $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OH}(a q)+2 \mathrm{CO}_{2}(g) $$ Fermentation of \(750 . \mathrm{mL}\) 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 \(\left(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OH}\right)\). 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 \(0.79 \mathrm{~g} / \mathrm{cm}^{3}\).)

Step by step solution

01

Calculate the mass of grape juice

Given that the volume of grape juice is 750 mL and the density is 1.0 g/cm³, we can calculate the mass of the grape juice by multiplying the volume and density: Mass = Volume × Density = 750 mL × 1.0 g/cm³ = 750 g

02

Calculate the volume and mass of ethanol produced

According to the problem, 12% by volume is ethanol. Therefore, we can find the volume of ethanol produced: Volume of ethanol = 12% × 750 mL = 0.12 × 750 mL = 90 mL Now, we find the mass of ethanol produced using its density (0.79 g/cm³): Mass of ethanol = Volume of ethanol × Density = 90 mL × 0.79 g/cm³ = 71.1 g

03

Determine the moles of ethanol produced

To find the moles of ethanol, we need to divide the mass of ethanol by its molar mass: Moles of ethanol = Mass / Molar mass = 71.1 g / \(46.07\text{ g/mol}\) ≈ 1.542 mol

04

Determine the moles of CO₂ produced

Since the balanced chemical equation states that 1 mole of glucose produces 2 moles of ethanol and 2 moles of CO₂, we can find the moles of CO₂ produced: Moles of CO₂ = Moles of ethanol = 1.542 mol

05

Calculate the gas constant R

The gas constant R can be found using the following equation: R = \(0.0821\text{ L·atm/mol·K}\)

06

Calculate the volume of CO₂ produced

The volume of the bottle is 825 mL, and the volume of grape juice is 750 mL, so the volume available for CO₂ gas is: Volume of CO₂ = Total volume - Volume of grape juice = 825 mL - 750 mL = 75 mL To convert the volume to liters, we divide by 1000: Volume of CO₂ = 75 mL / 1000 = 0.075 L

07

Use the Ideal Gas Law to calculate the pressure of CO₂

Finally, we use the Ideal Gas Law (PV=nRT) to find the pressure of CO₂ inside the wine bottle. We have the number of moles (n), the gas constant R, the volume V, and the temperature in Kelvin (T = 273.15 + 25 = 298.15 K). P = nRT / V ≈ (1.542 mol × 0.0821 L·atm/mol·K × 298.15 K) / 0.075 L ≈ 53.14 atm The pressure of CO₂ inside the wine bottle at 25°C is approximately 53.14 atm.

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