A solution has `1:4` mole ratio of pentane to hexane . The vapour pressure of pure hydrocarbons at `20^@C`are `440` mm Hg for pentane and `120`mm Hg for hexane .The mole fraction of pentane in the vapour phase is

Options

A. `0.549`
B. `0.200`
C. `0.786`
D. `0.478`

Explanation

To find the mole fraction of pentane in the vapor phase of a mixture with a 1:4 mole ratio of pentane to hexane, we can follow these steps:

### Step 1: Determine the moles of each component
Given the mole ratio of pentane to hexane is 1:4, we can assume:
- Moles of pentane (C5H12) = 1
- Moles of hexane (C6H14) = 4

### Step 2: Calculate the total moles in the mixture
Total moles = Moles of pentane + Moles of hexane
Total moles = 1 + 4 = 5

### Step 3: Calculate the mole fractions of pentane and hexane
- Mole fraction of pentane (X_p) = Moles of pentane / Total moles
\[ X_p = \frac{1}{5} = 0.2 \]

- Mole fraction of hexane (X_h) = Moles of hexane / Total moles
\[ X_h = \frac{4}{5} = 0.8 \]

### Step 4: Apply Raoult's Law to find the total vapor pressure
According to Raoult's Law, the total vapor pressure (P_total) of the solution can be calculated as:
\[ P_{total} = P_a \cdot X_p + P_b \cdot X_h \]
Where:
- \(P_a\) = vapor pressure of pure pentane = 440 mm Hg
- \(P_b\) = vapor pressure of pure hexane = 120 mm Hg

Substituting the values:
\[ P_{total} = (440 \, \text{mm Hg} \cdot 0.2) + (120 \, \text{mm Hg} \cdot 0.8) \]
Calculating each term:
\[ P_{total} = 88 + 96 = 184 \, \text{mm Hg} \]

### Step 5: Calculate the mole fraction of pentane in the vapor phase
The mole fraction of pentane in the vapor phase (Y_p) can be calculated using:
\[ Y_p = \frac{P_a \cdot X_p}{P_{total}} \]
Substituting the known values:
\[ Y_p = \frac{440 \cdot 0.2}{184} \]
Calculating the numerator:
\[ Y_p = \frac{88}{184} \]
Now, simplifying:
\[ Y_p \approx 0.478 \]

### Final Answer
The mole fraction of pentane in the vapor phase is approximately **0.478**.

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