In the "Methode Champenoise," grape juice is fermented in a wine bottle to produce sparkling wine. The reaction is given below.
C6H12O6(aq)--> 2 C2H5OH(aq) + 2 CO2(g)
Fermentation of 746 mL grape juice (density = 1.0 g/cm3) is allowed to take place in a bottle with a total volume of 825 mL until 12% by volume is ethanol (C2H5OH). Assuming that CO2 obeys Henry's law. Calculate the partial pressure of CO2 in the gas phase and the solubility of CO2 in the wine at 25°C. The Henry's law constant for CO2 is 32 L·atm/mol at 25°C with Henry's law in the form P = kC, where C is the concentration of the gas in mol/L. The density of ethanol is 0.79 g/cm3.
To solve this question, we need to determine the partial pressure of CO2 in the gas phase and the solubility of CO2 in the wine at 25°C.
Step 1: Calculate the amount of ethanol produced.
Given that the volume of grape juice is 746 mL and the density is 1.0 g/cm3, we can convert the volume to mass using the density:
Mass of grape juice = volume x density = 746 mL x 1.0 g/cm3 = 746 g
Since the molar mass of glucose (C6H12O6) is 180 g/mol, we can find the number of moles of glucose:
Number of moles of glucose = mass / molar mass = 746 g / 180 g/mol ≈ 4.14 mol
According to the balanced equation, 1 mole of glucose produces 2 moles of ethanol. Therefore, the number of moles of ethanol produced will be twice the number of moles of glucose produced:
Number of moles of ethanol = 2 x Number of moles of glucose ≈ 2 x 4.14 mol = 8.28 mol
Step 2: Calculate the volume of ethanol produced.
To find the volume of ethanol produced, we need to know the density of ethanol. Given that the density of ethanol is 0.79 g/cm3, we can convert the mass of ethanol to volume:
Volume of ethanol = mass / density = 8.28 mol x 46.07 g/mol / 0.79 g/cm3 ≈ 481 cm3 or 481 mL
Step 3: Calculate the total volume of the gas phase.
The total volume of the wine is given as 825 mL. Since ethanol makes up 12% of the volume, we can calculate the volume of the gas phase:
Volume of the gas phase = Total volume of wine - Volume of ethanol produced
= 825 mL - 481 mL ≈ 344 mL
Step 4: Calculate the concentration of CO2 in mol/L.
According to Henry's law, the partial pressure of CO2 is equal to the Henry's law constant (k) multiplied by the concentration of CO2. Rearranging the equation, we have:
C = P/k
Given that the Henry's law constant for CO2 is 32 L·atm/mol at 25°C, we can calculate the concentration of CO2 in mol/L:
Concentration of CO2 = Partial pressure of CO2 / Henry's law constant
= (Total pressure - Pressure of other gases) / Henry's law constant
= (1 atm - Partial pressure of ethanol) / 32 L·atm/mol
Step 5: Calculate the partial pressure of ethanol.
To calculate the partial pressure of ethanol, we need to know the mole fraction of ethanol in the gas phase. The mole fraction is defined as the moles of ethanol divided by the total moles of all gases present. The total moles of all gases present can be calculated using the ideal gas law:
PV = nRT
Where:
P = total pressure, which is 1 atm
V = volume of the gas phase, which is 344 mL
n = total moles of all gases present
R = ideal gas constant (0.0821 L·atm/mol·K)
T = temperature in Kelvin (25°C ≈ 298 K)
Rearranging the equation, we have:
n = PV / RT
We can find the partial pressure of ethanol using the mole fraction and the total pressure:
Partial pressure of ethanol = Mole fraction of ethanol x Total pressure
Step 6: Calculate the mole fraction of ethanol.
The mole fraction of ethanol can be calculated as the moles of ethanol divided by the total moles of all gases present.
Step 7: Calculate the concentration of CO2.
Using the values obtained in steps 5 and 6, we can substitute them into the equation from step 4 to calculate the concentration of CO2 in mol/L.
Step 8: Calculate the solubility of CO2 in the wine.
The solubility of CO2 in the wine can be determined by dividing the concentration of CO2 (in mol/L) by the density of wine (given as 1.0 g/cm3).
By following these steps, you can calculate the partial pressure of CO2 in the gas phase and the solubility of CO2 in the wine at 25°C.