Two solutions of `KNO_(3)` and `CH_(3)COOH` are prepared separately. The molarity of both is `0.1 M` and osmotic pressure is `P_(1)` and `P_(2)`, respectively.
The correct relationship between the osmotic pressure is

Options

A. `P_(2)gtP_(1)`
B. `P_(1)=P_(2)`
C. `P_(1)gtP_(2)`
D. `P_(1)/(P_(1)+P_(2))=P_(2)/(P_(1)+P_(2))`

Explanation

To solve the problem, we need to analyze the osmotic pressure of two solutions: one of KNO₃ and the other of CH₃COOH, both at a molarity of 0.1 M. The osmotic pressure (π) is given by the formula:

\[ \pi = iCRT \]

Where:
- \( \pi \) = osmotic pressure
- \( i \) = van 't Hoff factor (the number of particles the solute dissociates into)
- \( C \) = molarity of the solution
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step 1: Identify the van 't Hoff factor for KNO₃
KNO₃ is a strong electrolyte and dissociates completely in solution into two ions:
- K⁺
- NO₃⁻

Thus, the van 't Hoff factor \( i \) for KNO₃ is:
\[ i_{KNO_3} = 2 \]

### Step 2: Identify the van 't Hoff factor for CH₃COOH
CH₃COOH (acetic acid) is a weak electrolyte and does not dissociate completely. In dilute solutions, it primarily exists as undissociated molecules with a small fraction dissociating into ions:
- CH₃COO⁻
- H⁺

For weak electrolytes like acetic acid, we approximate the van 't Hoff factor \( i \) to be:
\[ i_{CH_3COOH} \approx 1 \]

### Step 3: Calculate the osmotic pressure for both solutions
Given that both solutions have the same molarity (0.1 M) and assuming the temperature is constant, we can express the osmotic pressures as follows:

For KNO₃:
\[ \pi_{KNO_3} = i_{KNO_3} \cdot C \cdot R \cdot T = 2 \cdot 0.1 \cdot R \cdot T \]

For CH₃COOH:
\[ \pi_{CH_3COOH} = i_{CH_3COOH} \cdot C \cdot R \cdot T = 1 \cdot 0.1 \cdot R \cdot T \]

### Step 4: Compare the osmotic pressures
Now we can compare the two osmotic pressures:
\[ \pi_{KNO_3} = 0.2 \cdot R \cdot T \]
\[ \pi_{CH_3COOH} = 0.1 \cdot R \cdot T \]

Since \( 0.2 \cdot R \cdot T > 0.1 \cdot R \cdot T \), we conclude that:
\[ \pi_{KNO_3} > \pi_{CH_3COOH} \]

### Step 5: State the relationship between osmotic pressures
Thus, the correct relationship between the osmotic pressures is:
\[ P_1 > P_2 \]

Where \( P_1 \) is the osmotic pressure of the KNO₃ solution and \( P_2 \) is the osmotic pressure of the CH₃COOH solution.

### Final Answer
The correct relationship between the osmotic pressures is:
\[ P_1 > P_2 \]