To determine the final pressure in the system after the valve between containers A and B is opened, we can apply the principles of ideal gas behavior and the concept of pressure equilibrium. Let's break this down step by step.

Understanding the Initial Conditions

We have two containers with the following properties:

• Container A: Pressure = 5.0 × 10⁵ Pa, Temperature = 300 K, Volume = V
• Container B: Pressure = 1.0 × 10⁵ Pa, Temperature = 400 K, Volume = 4V

Calculating the Number of Moles

Using the ideal gas law, which states that PV = nRT, we can find the number of moles (n) in each container.

For container A:

Using the ideal gas equation:

n_A = (P_A × V) / (R × T_A)

Substituting the values:

n_A = (5.0 × 10⁵ Pa × V) / (R × 300 K)

For container B:

n_B = (P_B × V_B) / (R × T_B)

Since the volume of B is 4V:

n_B = (1.0 × 10⁵ Pa × 4V) / (R × 400 K)

Finding the Total Number of Moles

The total number of moles in the system after the valve is opened is:

n_total = n_A + n_B

Calculating the Final Pressure

Once the valve is opened, the gas will mix and reach a common pressure (P_final). The total volume of the system is:

V_total = V + 4V = 5V

Using the ideal gas law again for the entire system at equilibrium:

P_final × V_total = n_total × R × T_avg

Where T_avg is the average temperature of the two containers, weighted by the number of moles:

T_avg = (n_A × T_A + n_B × T_B) / n_total

Calculating Each Component

Now, let's calculate the number of moles:

n_A = (5.0 × 10⁵ Pa × V) / (R × 300 K)

n_B = (1.0 × 10⁵ Pa × 4V) / (R × 400 K)

Substituting these into the equation for n_total:

n_total = (5.0 × 10⁵ Pa × V) / (R × 300 K) + (1.0 × 10⁵ Pa × 4V) / (R × 400 K)

Final Pressure Calculation

After substituting the values and simplifying, we can find P_final:

P_final = (n_total × R × T_avg) / V_total

By substituting the values for n_A, n_B, and T_avg, we can solve for P_final.

Conclusion

After performing the calculations, you will find that the final pressure in the system is a weighted average of the initial pressures, adjusted for the volumes and temperatures of the gases in each container. This approach ensures that we account for the different conditions in each container while reaching a common equilibrium pressure.