Consider a setup of two urea solution of concentration `C_(1)` and `C_(2)(C_(2)gtC_(1))`, both at temperature `T`, separated by a semi permeable membrane. External pressure `P_(1)` and `P_(2)` respectively are applied on the two solutions. For what values of `P_(1)` and `P_(2)`, osmosis does not occur through the semi permeable membrane? `(R=` universal gas constant)

Options

A. `P_(1)=C_(1)RT` and `P_(2)=C_(2)RT`
B. `P_(1)=` zero and `P_(2)=(C_(2)-C_(1))RT`
C. `P_(1)=C_(2)RT` and `P_(2)=C_(1)RT`
D. `P_(1)=2C_(1)RT` and `P_(1)=(C_(2)+C_(1))RT`

Explanation

To determine the values of external pressures \( P_1 \) and \( P_2 \) for which osmosis does not occur through a semi-permeable membrane separating two urea solutions of concentrations \( C_1 \) and \( C_2 \) (where \( C_2 > C_1 \)), we can use the concept of osmotic pressure.

### Step-by-Step Solution:

1. **Understanding Osmotic Pressure**:
- The osmotic pressure (\( \Pi \)) of a solution can be calculated using the formula:
\[ \Pi = C \cdot R \cdot T \]
where \( C \) is the concentration of the solution, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

2. **Setting Up the Conditions**:
- We have two solutions:
- Solution 1 with concentration \( C_1 \) and external pressure \( P_1 \).
- Solution 2 with concentration \( C_2 \) and external pressure \( P_2 \).
- For osmosis to not occur, the external pressures must counterbalance the osmotic pressures of the respective solutions.

3. **Equilibrium Condition**:
- For no osmosis to occur, the following conditions must be satisfied:
\[ P_1 = \Pi_1 \quad \text{and} \quad P_2 = \Pi_2 \]
where \( \Pi_1 \) and \( \Pi_2 \) are the osmotic pressures of solutions 1 and 2, respectively.

4. **Applying the Osmotic Pressure Formula**:
- For Solution 1:
\[ P_1 = C_1 \cdot R \cdot T \]
- For Solution 2:
\[ P_2 = C_2 \cdot R \cdot T \]

5. **Final Values for Pressures**:
- Therefore, the values of \( P_1 \) and \( P_2 \) for which osmosis does not occur are:
\[ P_1 = C_1 \cdot R \cdot T \quad \text{and} \quad P_2 = C_2 \cdot R \cdot T \]

### Conclusion:
The correct answer is:
- \( P_1 = C_1 \cdot R \cdot T \)
- \( P_2 = C_2 \cdot R \cdot T \)