Question

Consider a carbonated drink in a bottle at \(37^{\circ} \mathrm{C}\) and $130 \mathrm{kPa}$. Assuming the gas space above the liquid consists of a saturated mixture of \(\mathrm{CO}_{2}\) and water vapor and treating the drink as water, determine \((a)\) the mole fraction of the water vapor in the \(\mathrm{CO}_{2}\) gas and (b) the mass of dissolved \(\mathrm{CO}_{2}\) in a 200-ml drink. Answers: (a) \(4.9\) percent, (b) \(0.28 \mathrm{~g}\)

Short answer

= 130 kPa - 6.361 kPa = 123.639 kPa #tag_title#Step 2: Find the mole fraction#tag_content#Now we can find the mole fraction of water vapor in the CO₂ gas. We can use Dalton's law of partial pressures, which states that the mole fraction of a gas is equal to the ratio of its partial pressure to the total pressure. Mole fraction of water vapor (Y_H2O) = P_sat_water / P_total = 6.361 kPa / 130 kPa = 0.04893 (a) The mole fraction of water vapor in the CO₂ gas is approximately 0.04893. #tag_title#Step 3: Find the moles of dissolved CO₂ in the drink#tag_content#We can now find the moles of dissolved CO₂ in the drink using the given solubility and the partial pressure of CO₂ from Step 1. Solubility of CO₂ = 3.3 g CO₂ / (L H₂O * atm) Partial pressure of CO₂ (in atm) = 123.639 kPa * (1 atm / 101.325 kPa) = 1.220 atm Now, we can use the solubility of CO₂ to find the moles of dissolved CO₂ in 0.2 L of the drink: Dissolved CO₂ (g) = solubility * partial pressure * volume = 3.3 g/(L*atm) * 1.220 atm * 0.2 L = 0.806 g of CO₂ (b) The mass of dissolved CO₂ in a 200-ml drink is approximately 0.806 g.

Step-by-step solution

Find the partial pressures of CO₂ and water vapor

First, we need to find the partial pressures of CO₂ and water vapor in the gas space above the liquid. We can use the saturation pressure of water at the given temperature (37°C) and the given total pressure of the gas space (130 kPa). The saturation pressure of water at 37°C can be found in steam tables or using the Antoine equation: P_sat_water = 6.361 kPa Now, we can find the partial pressure of CO₂ by subtracting the saturation pressure of water from the total pressure: P_CO2 = P_total - P_sat_water