Three identical diatomic gases (`gamma`=1.5) are enclosed in three identical containers but at different pressures and same temperatures. These gases are expanded to double their volumes in first container the process is isothermal, in second container the process is adiabatic and in third container process is isobaric. Of The final pressures are equal in the three containers, find the ratio of the initial pressures in the three containers.

Answer

To solve the problem, we need to analyze the three processes (isothermal, adiabatic, and isobaric) for the three identical diatomic gases in their respective containers. We will derive the initial pressures in terms of the final pressure \( P_3 \) and then find the ratio of these initial pressures.

### Step-by-Step Solution:

1. **Isothermal Process (Container 1)**:
- For an isothermal process, the relationship between pressure and volume is given by:
\[ P_1 V = P_{1f} (2V) \]
- Since the final pressure \( P_{1f} \) is equal to \( P_3 \) (the final pressures are equal in all containers), we can write:
\[ P_1 V = P_3 (2V) \]
- Simplifying this gives:
\[ P_1 = 2 P_3 \]

2. **Adiabatic Process (Container 2)**:
- For an adiabatic process, the relationship is given by:
\[ P_2 V^\gamma = P_{2f} (2V)^\gamma \]
- Again, since \( P_{2f} = P_3 \), we can write:
\[ P_2 V^{1.5} = P_3 (2V)^{1.5} \]
- Simplifying this gives:
\[ P_2 V^{1.5} = P_3 (2^{1.5} V^{1.5}) \]
- Dividing both sides by \( V^{1.5} \):
\[ P_2 = 2^{1.5} P_3 = 2\sqrt{2} P_3 \]

3. **Isobaric Process (Container 3)**:
- For an isobaric process, the pressure remains constant:
\[ P_{3f} = P_3 \]
- Thus, the initial pressure \( P_3 \) remains the same.

4. **Finding the Ratio of Initial Pressures**:
- Now we have:
- \( P_1 = 2 P_3 \)
- \( P_2 = 2\sqrt{2} P_3 \)
- \( P_3 = P_3 \)
- The ratio of the initial pressures \( P_1 : P_2 : P_3 \) can be expressed as:
\[ P_1 : P_2 : P_3 = 2 P_3 : 2\sqrt{2} P_3 : P_3 \]
- Dividing each term by \( P_3 \):
\[ = 2 : 2\sqrt{2} : 1 \]
- To simplify, we can express this as:
\[ = 2 : 2\sqrt{2} : 1 = 2 : 2\sqrt{2} : 1 \]

### Final Answer:
The ratio of the initial pressures in the three containers is:
\[ 2 : 2\sqrt{2} : 1 \]