The osmotic pressure of urea solution at `10^(@)C` is `200 mm`.becomes `105.3 mm` when it is diluted and temperature raised to `25^(@)C`. The extent of dilution is

Options

A. 8 times
B. 5 times
C. 4 times
D. 2 times

Explanation

To solve the problem, we will use the formula for osmotic pressure, which is given by:

\[ \pi = CRT \]

Where:
- \(\pi\) = osmotic pressure
- \(C\) = concentration of the solution (in moles per liter)
- \(R\) = universal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature in Kelvin

### Step 1: Convert temperatures to Kelvin
- For the first condition (10°C):
\[ T_1 = 10 + 273 = 283 \, K \]
- For the second condition (25°C):
\[ T_2 = 25 + 273 = 298 \, K \]

### Step 2: Write the equations for osmotic pressure
Using the osmotic pressure formula for both conditions:
1. For the first condition:
\[ \pi_1 = C_1RT_1 \]
2. For the second condition:
\[ \pi_2 = C_2RT_2 \]

### Step 3: Relate the two conditions
Since the number of moles of solute does not change during dilution, we can express the relationship as:
\[ C_1V_1 = C_2V_2 \]
Where \(V_1\) and \(V_2\) are the volumes before and after dilution, respectively.

### Step 4: Substitute for concentrations
From the osmotic pressure equations, we can express concentrations:
\[ C_1 = \frac{\pi_1}{RT_1} \quad \text{and} \quad C_2 = \frac{\pi_2}{RT_2} \]

### Step 5: Substitute into the relationship
Substituting the expressions for \(C_1\) and \(C_2\) into the relationship gives:
\[ \frac{\pi_1}{RT_1} V_1 = \frac{\pi_2}{RT_2} V_2 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ \frac{V_1}{V_2} = \frac{\pi_2 T_1}{\pi_1 T_2} \]

### Step 7: Substitute the known values
Now substituting the known values:
- \(\pi_1 = 200 \, mm\)
- \(\pi_2 = 105.3 \, mm\)
- \(T_1 = 283 \, K\)
- \(T_2 = 298 \, K\)

\[ \frac{V_1}{V_2} = \frac{105.3 \times 283}{200 \times 298} \]

### Step 8: Calculate the ratio
Calculating the right-hand side:
\[ \frac{V_1}{V_2} = \frac{29853.9}{59600} \approx 0.500 \]

### Step 9: Find the extent of dilution
This means:
\[ V_2 = 2V_1 \]
Thus, the extent of dilution is 2.

### Final Answer
The extent of dilution is **2**.

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