An ideal solution is formed by mixing of 460 g. of ethanol with xg. of methanol. The total vapour pressure of the solution is 72 mm of Hg. The vapour pressure of pure ethanol and pure methanol are 48 and 80 mm of Hg respectively. Find the value of x. [Given: Atomic mass H = 1, C = 12, O = 16]

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the moles of ethanol
The formula for ethanol is C2H5OH.
- The molar mass of ethanol (C2H5OH) can be calculated as:
- C: 2 × 12 = 24 g
- H: 6 × 1 = 6 g
- O: 1 × 16 = 16 g
- Total = 24 + 6 + 16 = 46 g/mol

Given mass of ethanol = 460 g.

Now, we can calculate the moles of ethanol using the formula:
\[ \text{Moles of ethanol} = \frac{\text{Given mass}}{\text{Molar mass}} = \frac{460 \text{ g}}{46 \text{ g/mol}} = 10 \text{ moles} \]

### Step 2: Set up the equation for moles of methanol
Let the mass of methanol be \( x \) grams. The formula for methanol is CH3OH.
- The molar mass of methanol (CH3OH) is:
- C: 12 g
- H: 4 g
- O: 16 g
- Total = 12 + 4 + 16 = 32 g/mol

Now, we can calculate the moles of methanol:
\[ \text{Moles of methanol} = \frac{x \text{ g}}{32 \text{ g/mol}} = \frac{x}{32} \text{ moles} \]

### Step 3: Calculate the mole fraction of ethanol
The total moles in the solution is:
\[ \text{Total moles} = \text{Moles of ethanol} + \text{Moles of methanol} = 10 + \frac{x}{32} \]

The mole fraction of ethanol (\( X_A \)) is given by:
\[ X_A = \frac{\text{Moles of ethanol}}{\text{Total moles}} = \frac{10}{10 + \frac{x}{32}} \]

### Step 4: Calculate the mole fraction of methanol
The mole fraction of methanol (\( X_B \)) is:
\[ X_B = \frac{\text{Moles of methanol}}{\text{Total moles}} = \frac{\frac{x}{32}}{10 + \frac{x}{32}} \]

### Step 5: Use Raoult's Law to find the total vapor pressure
According to Raoult's Law, the total vapor pressure (\( P \)) of the solution is given by:
\[ P = P^0_A \cdot X_A + P^0_B \cdot X_B \]
Where:
- \( P^0_A = 48 \) mm Hg (vapor pressure of pure ethanol)
- \( P^0_B = 80 \) mm Hg (vapor pressure of pure methanol)

Substituting the values:
\[ 72 = 48 \cdot \left(\frac{10}{10 + \frac{x}{32}}\right) + 80 \cdot \left(\frac{\frac{x}{32}}{10 + \frac{x}{32}}\right) \]

### Step 6: Solve the equation
To solve for \( x \), we can simplify the equation:
1. Multiply through by \( (10 + \frac{x}{32}) \) to eliminate the denominator:
\[ 72 \left(10 + \frac{x}{32}\right) = 48 \cdot 10 + 80 \cdot \frac{x}{32} \]
2. Expand and rearrange:
\[ 720 + \frac{72x}{32} = 480 + \frac{80x}{32} \]
3. Combine like terms:
\[ 720 - 480 = \frac{80x}{32} - \frac{72x}{32} \]
\[ 240 = \frac{8x}{32} \]
4. Solve for \( x \):
\[ 240 = \frac{x}{4} \implies x = 240 \cdot 4 = 960 \text{ g} \]

### Final Answer
The value of \( x \) is 960 g.

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