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 problem, we need to follow a few steps:
Step 1: Find the moles of ethanol produced
Step 2: Calculate the concentration of ethanol in the wine (mol/L)
Step 3: Use Henry's law to find the partial pressure of CO2
Step 4: Calculate the solubility of CO2 in the wine
Let's start with step 1:
Step 1: Find the moles of ethanol produced
To find the moles of ethanol, we need the density of the grape juice and the volume of the grape juice.
Given:
Density of grape juice = 1.0 g/cm^3
Volume of grape juice = 746 mL
Density can be converted to mass using the formula:
Mass = Density x Volume
Mass of grape juice = 1.0 g/cm^3 x 746 mL
Next, we can convert the mass of grape juice to moles of ethanol using the molar mass of ethanol.
The equation shows that 1 mole of ethanol is produced from 1 mole of grape juice. So, the moles of ethanol produced will be the same as the moles of grape juice used.
Molar mass of ethanol (C2H5OH) = 2(12.01 g/mol) + 6(1.01 g/mol) + 1(16.00 g/mol) = 46.07 g/mol
Moles of ethanol produced = Mass of grape juice / Molar mass of ethanol
Step 2: Calculate the concentration of ethanol in the wine (mol/L)
The volume of the bottle is given as 825 mL. To calculate the concentration of ethanol in the wine, we need to convert this volume to liters.
Given:
Volume of bottle = 825 mL
Volume of bottle (in L) = 825 mL / 1000 mL/L
Concentration of ethanol = Moles of ethanol produced / Volume of bottle
Step 3: Use Henry's law to find the partial pressure of CO2
Henry's law states that the partial pressure of a gas above a liquid is directly proportional to the concentration of the gas in the liquid.
The equation for Henry's law in this case is: P = kC, where P is the partial pressure of CO2, k is the Henry's law constant for CO2, and C is the concentration of CO2 in mol/L.
The Henry's law constant for CO2 is given as 32 L·atm/mol at 25°C.
Partial pressure of CO2 = Henry's law constant x Concentration of CO2
Step 4: Calculate the solubility of CO2 in the wine
To find the solubility of CO2 in the wine, we need to use the ideal gas law equation: PV = nRT.
Given:
Temperature (T) = 25°C = 298 K (convert to Kelvin)
Partial pressure of CO2 (P) = calculated in Step 3
Assuming ideal gas behavior, we can rearrange the ideal gas law equation to solve for the concentration of CO2:
C = n/V = P/(RT)
Concentration of CO2 (solubility) = Partial pressure of CO2 / (Gas constant x Temperature)
The gas constant (R) is 0.0821 L·atm/(mol·K).
By following these steps and using the given values, you should be able to calculate the partial pressure of CO2 in the gas phase and the solubility of CO2 in the wine at 25°C.