Methanol and ethanol froms nearly ideal solution at 300 K. A solution is made by mixing 32g methanol and 23 g ethanol. Calculate the partial pressure of its constituents and the total pressure of the solution. (at 300 K , `p_((CH_(3)OH))^(@) = 90` mm Hg, `p_((C_(2)H_(5)OH))^(@) = 51` mm Hg).

Explanation

To solve the problem of calculating the partial pressures of methanol and ethanol in a solution, as well as the total pressure, we can follow these steps:

### Step 1: Gather Given Data
- Mass of Methanol (CH₃OH) = 32 g
- Mass of Ethanol (C₂H₅OH) = 23 g
- Pure vapor pressure of Methanol (P₀(CH₃OH)) = 90 mm Hg
- Pure vapor pressure of Ethanol (P₀(C₂H₅OH)) = 51 mm Hg

### Step 2: Calculate Molar Masses
- Molar mass of Methanol (CH₃OH) = 12 (C) + 1*3 (H) + 16 (O) = 32 g/mol
- Molar mass of Ethanol (C₂H₅OH) = 12*2 (C) + 1*5 (H) + 16 (O) = 46 g/mol

### Step 3: Calculate Number of Moles
- Number of moles of Methanol (n₁) = Mass / Molar Mass = 32 g / 32 g/mol = 1 mole
- Number of moles of Ethanol (n₂) = Mass / Molar Mass = 23 g / 46 g/mol = 0.5 moles

### Step 4: Calculate Total Moles
- Total moles (n_total) = n₁ + n₂ = 1 mole + 0.5 moles = 1.5 moles

### Step 5: Calculate Mole Fractions
- Mole fraction of Methanol (χ₁) = n₁ / n_total = 1 / 1.5 = 0.67
- Mole fraction of Ethanol (χ₂) = n₂ / n_total = 0.5 / 1.5 = 0.33

### Step 6: Calculate Partial Pressures
- Partial pressure of Methanol (P₁) = P₀(CH₃OH) * χ₁ = 90 mm Hg * 0.67 = 60.3 mm Hg
- Partial pressure of Ethanol (P₂) = P₀(C₂H₅OH) * χ₂ = 51 mm Hg * 0.33 = 16.8 mm Hg

### Step 7: Calculate Total Pressure
- Total pressure (P_total) = P₁ + P₂ = 60.3 mm Hg + 16.8 mm Hg = 77.1 mm Hg

### Final Results
- Partial pressure of Methanol = 60.3 mm Hg
- Partial pressure of Ethanol = 16.8 mm Hg
- Total pressure of the solution = 77.1 mm Hg

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