The industrial solvents chloroform (CHCl₃) and dichloromethane (CH₂Cl₂) are prepared commercially by the reaction of methane with chlorine, followed by fractional distillation of the product mixture. At 25 °C, the vapor pressure of CHCl₃ is 205 mm Hg, and the vapor pressure of CH₂Cl₂ is 415 mm Hg. What is the vapor pressure in mm Hg at 25 °C of a mixture of 15.0 g of CHCl₃ and 37.5 g of CH₂Cl₂?

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1. Calculate the number of moles of CHCl₃ using its molar mass. The molar mass of CHCl₃ is approximately 119.38 g/mol. Use the formula: \( \text{moles of CHCl₃} = \frac{\text{mass of CHCl₃}}{\text{molar mass of CHCl₃}} \).
2. Calculate the number of moles of CH₂Cl₂ using its molar mass. The molar mass of CH₂Cl₂ is approximately 84.93 g/mol. Use the formula: \( \text{moles of CH₂Cl₂} = \frac{\text{mass of CH₂Cl₂}}{\text{molar mass of CH₂Cl₂}} \).
3. Determine the mole fraction of CHCl₃ in the mixture. The mole fraction is given by: \( X_{\text{CHCl₃}} = \frac{\text{moles of CHCl₃}}{\text{moles of CHCl₃} + \text{moles of CH₂Cl₂}} \).
4. Determine the mole fraction of CH₂Cl₂ in the mixture. The mole fraction is given by: \( X_{\text{CH₂Cl₂}} = \frac{\text{moles of CH₂Cl₂}}{\text{moles of CHCl₃} + \text{moles of CH₂Cl₂}} \).
5. Calculate the total vapor pressure of the mixture using Raoult's Law: \( P_{\text{total}} = X_{\text{CHCl₃}} \cdot P^0_{\text{CHCl₃}} + X_{\text{CH₂Cl₂}} \cdot P^0_{\text{CH₂Cl₂}} \), where \( P^0 \) represents the vapor pressure of the pure component.