Henry’s law constant for CO2 in water is 10^{8}$$ Pa at 298 K. Calculate the quantity of CO2 in 500 mL of soda water when packed under 2.5 atm CO2 pressure at 298 K.
The quantity of CO2 in 500 mL of soda water is approximately 1.85 g.
Explanation
The question requires applying Henry's Law, which is provided in the context as p = K_H * x. To solve this numerical, one must first convert the given pressure from atm to Pa to match the units of the Henry's law constant. Then, the mole fraction is calculated. Since the mole fraction is the ratio of moles of solute to total moles, and the amount of dissolved gas is small, it can be approximated as the ratio of moles of CO2 to moles of water. Finally, the moles of CO2 are converted to mass using the molar mass of CO2 (a standard constant required for the calculation).
Solution Steps
Step 1: Convert pressure to Pascals (Pa). The given pressure is 2.5 atm. Using the conversion factor (1 atm = 101325 Pa), Pressure (p) = 2.5 atm × 101325 Pa/atm = 253312.5 Pa.
Step 2: Calculate the mole fraction of CO2. Using Henry's Law equation p = K_H * x: x(CO2) = p / K_H x(CO2) = 253312.5 Pa / (10^{8} Pa) x(CO2) $\approx$ $1.517 \times $10^{-3}
Step 3: Calculate moles of water (solvent). Assuming density of soda water is approximately 1 g/mL, 500 mL corresponds to 500 g of water. Molar mass of water (H2O) = 18 g mol^-1. Moles of water (nH2O) = 500 g / 18 g mol^-1 27.78 mol.
Step 4: Calculate moles of CO2. Mole fraction x(CO2) = n(CO2) / (n(H2O) + n(CO2)). Since n(CO2) is very small, x(CO2) n(CO2) / n(H2O). Rearranging for n(CO2): n(CO2) = x(CO2) × n(H2O) n(CO2) = (10^{-3}$$) × 27.78 mol n(CO2) 0.0421 mol
Step 5: Calculate the mass of CO2. Molar mass of CO2 = 44 g mol^-1. Mass of CO2 = n(CO2) × Molar mass Mass of CO2 = 0.0421 mol × 44 g mol^-1 Mass of CO2 1.85 g