A vessel of volume `3V` contains a gas at pressure `4 P_(0)` and another vessel of volume `2V` contains same gas at pressure `1.5 P_(0)`. Both vessels have same temperature. When both vessels are connected by a tube of negligible volume, the equilibrium is `IP_(0)`, where `I` is an integer. Find the value of `I`.

Options

A. `3.5`
B. `3.25`
C. `3.75`
D. `3`

Answer

To solve the problem, we will use the ideal gas law, which states that \( PV = nRT \). We will find the number of moles in each vessel, then combine them to find the equilibrium pressure when the vessels are connected.

### Step-by-step Solution:

1. **Identify Given Values**:
- Vessel 1: Volume \( V_1 = 3V \), Pressure \( P_1 = 4P_0 \)
- Vessel 2: Volume \( V_2 = 2V \), Pressure \( P_2 = 1.5P_0 \)
- Both vessels are at the same temperature \( T \).

2. **Calculate Number of Moles in Each Vessel**:
- For Vessel 1:
\[ n_1 = \frac{P_1 V_1}{RT} = \frac{(4P_0)(3V)}{RT} = \frac{12P_0V}{RT} \]
- For Vessel 2:
\[ n_2 = \frac{P_2 V_2}{RT} = \frac{(1.5P_0)(2V)}{RT} = \frac{3P_0V}{RT} \]

3. **Total Number of Moles**:
- The total number of moles \( n \) when both vessels are connected:
\[ n = n_1 + n_2 = \frac{12P_0V}{RT} + \frac{3P_0V}{RT} = \frac{(12 + 3)P_0V}{RT} = \frac{15P_0V}{RT} \]

4. **Total Volume**:
- The total volume \( V_{total} \) when both vessels are connected:
\[ V_{total} = V_1 + V_2 = 3V + 2V = 5V \]

5. **Apply Ideal Gas Law for the Combined System**:
- When the vessels are connected, the pressure \( P \) in the combined system can be expressed as:
\[ P V_{total} = n RT \]
- Substituting the values we have:
\[ P (5V) = \frac{15P_0V}{RT} RT \]
- Simplifying this gives:
\[ P (5V) = 15P_0V \]

6. **Solve for Pressure \( P \)**:
- Dividing both sides by \( 5V \):
\[ P = \frac{15P_0}{5} = 3P_0 \]

7. **Determine the Value of \( I \)**:
- The equilibrium pressure is given as \( I P_0 \), where \( I \) is an integer. From our calculation, we found that:
\[ P = 3P_0 \implies I = 3 \]

### Final Answer:
The value of \( I \) is \( 3 \).

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