The relationship between osmotic pressure at `273 K` when `10 g` glucose `(P_(1)), 10 g` urea `(P_(2))` and `10 g` sucrose `(P_(3))` are dissolved in `250 mL` of water is:

Options

A. `P_1 gt P_2 gt P_3`
B. `P_2 gt P_1 gt P_3`
C. `P_2 gt P_3 gt P_1`
D. `P_3 gt P_2 gt P_1`

Answer

To determine the relationship between the osmotic pressures of glucose, urea, and sucrose when dissolved in water, we can follow these steps:

### Step 1: Understand the Formula for Osmotic Pressure
The osmotic pressure (\( \pi \)) of a solution is given by the formula:
\[ \pi = iCRT \]
where:
- \( i \) = van 't Hoff factor (which is 1 for all three solutes since they do not dissociate in solution),
- \( C \) = concentration of the solution (in moles per liter),
- \( R \) = universal gas constant (constant for all),
- \( T \) = temperature in Kelvin (273 K in this case).

### Step 2: Identify the Given Data
We have:
- Mass of glucose = 10 g
- Mass of urea = 10 g
- Mass of sucrose = 10 g
- Volume of water = 250 mL = 0.250 L

### Step 3: Calculate the Number of Moles for Each Solute
1. **For Glucose**:
- Molar mass of glucose = 180 g/mol
- Moles of glucose (\( n_1 \)):
\[ n_1 = \frac{10 \text{ g}}{180 \text{ g/mol}} = \frac{1}{18} \text{ mol} \approx 0.0556 \text{ mol} \]

2. **For Urea**:
- Molar mass of urea = 60 g/mol
- Moles of urea (\( n_2 \)):
\[ n_2 = \frac{10 \text{ g}}{60 \text{ g/mol}} = \frac{1}{6} \text{ mol} \approx 0.1667 \text{ mol} \]

3. **For Sucrose**:
- Molar mass of sucrose = 342 g/mol
- Moles of sucrose (\( n_3 \)):
\[ n_3 = \frac{10 \text{ g}}{342 \text{ g/mol}} \approx 0.0292 \text{ mol} \]

### Step 4: Compare the Number of Moles
Now we can compare the number of moles:
- Moles of glucose (\( n_1 \)) = 0.0556 mol
- Moles of urea (\( n_2 \)) = 0.1667 mol
- Moles of sucrose (\( n_3 \)) = 0.0292 mol

### Step 5: Establish the Relationship Between Osmotic Pressures
Since osmotic pressure is directly proportional to the number of moles of solute, we can establish the following relationship:
\[ \pi_2 > \pi_1 > \pi_3 \]
This means:
\[ P_2 > P_1 > P_3 \]
where:
- \( P_1 \) = osmotic pressure of glucose,
- \( P_2 \) = osmotic pressure of urea,
- \( P_3 \) = osmotic pressure of sucrose.

### Conclusion
The relationship between the osmotic pressures at 273 K is:
\[ P_2 > P_1 > P_3 \]