An ideal solution contains two volatile liquids `A(P^(@)=100` torr) and `B(P^(@)=200` torr). If mixture contain `1` mole of `A` and `4` mole of `B` then total vapour pressure of the distillate is :

Options

A. 150
B. 180
C. 188.88
D. 198.88

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the mole fractions of A and B in the liquid phase.

Given:
- Moles of A (NA) = 1
- Moles of B (NB) = 4

The total number of moles (N_total) = NA + NB = 1 + 4 = 5.

Now, we can calculate the mole fractions:
- Mole fraction of A (XA) = NA / N_total = 1 / 5 = 0.2
- Mole fraction of B (XB) = NB / N_total = 4 / 5 = 0.8

### Step 2: Calculate the partial pressures of A and B.

Using the formula for partial pressure:
- Partial pressure of A (PA) = XA * P°A
- Partial pressure of B (PB) = XB * P°B

Given:
- Vapor pressure of pure A (P°A) = 100 torr
- Vapor pressure of pure B (P°B) = 200 torr

Now, we can calculate the partial pressures:
- PA = 0.2 * 100 torr = 20 torr
- PB = 0.8 * 200 torr = 160 torr

### Step 3: Calculate the total vapor pressure of the mixture.

The total vapor pressure (P_total) is the sum of the partial pressures:
- P_total = PA + PB
- P_total = 20 torr + 160 torr = 180 torr

### Step 4: Calculate the mole fractions of A and B in the vapor phase.

Using Dalton's law of partial pressures:
- For A: PA = YA * P_total
- For B: PB = YB * P_total

Rearranging gives us:
- YA = PA / P_total
- YB = PB / P_total

Now, we can calculate:
- YA = 20 torr / 180 torr = 1 / 9
- YB = 160 torr / 180 torr = 8 / 9

### Step 5: Calculate the total vapor pressure of the distillate.

Using the mole fractions in the vapor phase:
- P_distillate = YA * P°A + YB * P°B

Substituting the values:
- P_distillate = (1/9 * 100 torr) + (8/9 * 200 torr)
- P_distillate = (100/9) + (1600/9) = (1700/9) torr

Calculating this gives:
- P_distillate ≈ 188.89 torr

### Final Answer:
The total vapor pressure of the distillate is approximately **188.89 torr**.

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