Two gases occupy two containers (A) and (B). The gas in (A) of volume `0.11 m^3` experts a pressure of `1.38 Mpa`. The gas in (B) of volume `0.16 m^3` experts a pressure of `0.69 Mpa. Two containers are united by a tube of negligible volume and the gases are allowed to intermingle. What is the final pressure in the container if the temperature remains constant ?

Explanation

To solve the problem, we can use the principle of conservation of moles and the ideal gas law. Here’s a step-by-step solution:

### Step 1: Understand the initial conditions
We have two containers with gases:
- Container A:
- Volume \( V_A = 0.11 \, m^3 \)
- Pressure \( P_A = 1.38 \, MPa \)

- Container B:
- Volume \( V_B = 0.16 \, m^3 \)
- Pressure \( P_B = 0.69 \, MPa \)

### Step 2: Calculate the total initial pressure-volume products
Using the ideal gas law \( PV = nRT \), we can express the number of moles of gas in each container as:
- For Container A:
\[ n_A = \frac{P_A V_A}{RT} \]

- For Container B:
\[ n_B = \frac{P_B V_B}{RT} \]

### Step 3: Calculate the total pressure-volume products
The pressure-volume products for both containers are:
- For Container A:
\[ P_A V_A = 1.38 \times 10^6 \, Pa \times 0.11 \, m^3 = 0.1518 \times 10^6 \, Pa \cdot m^3 \]

- For Container B:
\[ P_B V_B = 0.69 \times 10^6 \, Pa \times 0.16 \, m^3 = 0.1104 \times 10^6 \, Pa \cdot m^3 \]

### Step 4: Add the pressure-volume products
Now, we can sum the two pressure-volume products:
\[ P_A V_A + P_B V_B = 0.1518 \times 10^6 + 0.1104 \times 10^6 = 0.2622 \times 10^6 \, Pa \cdot m^3 \]

### Step 5: Calculate the total volume
The total volume after the gases intermingle is:
\[ V_{total} = V_A + V_B = 0.11 + 0.16 = 0.27 \, m^3 \]

### Step 6: Calculate the final pressure
Using the relationship \( P_{final} = \frac{P_A V_A + P_B V_B}{V_{total}} \):
\[ P_{final} = \frac{0.2622 \times 10^6}{0.27} = 0.9711 \times 10^6 \, Pa \]
Converting to MPa:
\[ P_{final} = 0.9711 \, MPa \]

### Final Answer
The final pressure in the container after the gases intermingle is approximately \( 0.97 \, MPa \).

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